University of Twente

Transcription

1 University of Twente Faculty of Electrical Engineering, Mathematics & Comuter Science Design of an audio ower amlifier with a notch in the outut imedance Remco Twelkemeijer MSc. Thesis May 008 Suervisors: rof. ir. A.J.M. van Tuijl dr. ir. R.A.R. van der Zee ir. F. van Houwelingen Reort number: Chair of Integrated Circuit Design Faculty of Electrical Engineering, Mathematics & Comuter Science University of Twente P. O. Box AE Enschede

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3 Abstract This reort is about the research and design of an audio ower amlifier with a notch in the outut imedance. A notch in the outut imedance could be beneficial if the amlifier is set in arallel with a class D amlifier. In this combination, the amlifier should remove the high frequency switching rile of the class D amlifier. Investigation of the outut imedance with a ole ero analysis determines the theoretical ossibilities about a notch in the outut imedance. A notch in the outut imedance can be created by a eak filter in the amlifier. An amlifier containing a eak filter is designed with ideal comonents, like transconductances and inductors. Different filters are investigated to imlement a eak filter in an IC rocess. These filters are created by ideal switches and caacitors. To evaluate their usability, simulations are done with amlifiers containing those filters in combination with a class D amlifier. The reort shows that a notch can be created at the cost of some increase in the distortion. This can be comensated by increasing the ower consumtion of the amlifier. Also a small eak in the outut imedance is introduced. The eak in the outut imedance results in a ringing when a ste resonse is alied. Simulations results in the conclusion that it seems not beneficial to use an amlifier with a notch in the outut imedance, if the notch should remove the switching rile of the class D amlifier.

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5 Table of contents Abstract Introduction...7 Amlifier requirements Gain in resect with stability Derivation of the outut imedance....3 Outut imedance in resect with stability Outut imedance secification Outut imedance using a Pole-Zero analysis Standard amlifier Adding two comlex eros to create a notch Limitations while neglecting the third ole Length of the two oles with resect to DC Shifting one real ole Limitations in a realistic situation System Design Amlifier design System Imlementation Realiing the notch in the outut imedance Amlifiers in arallel Amlifiers in arallel with an outut stage Filter imlementation Continuous time filters Switched caacitor filters Comonent simulation Digital second order Filter Peak filter using one caacitor Peak filter using eight equal caacitors Simulations Amlifier structure Different amlifiers Results Conclusion... 6 References Aendices A. Derivation of the outut imedance B. Coefficients of the Z filter C. Design of the five amlifiers... 67

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7 Chater - Introduction Introduction Audio ower amlifiers are widely used nowadays. The difficulty of audio amlifiers is to obtain high outut ower in combination with high efficiency and low distortion. Class AB amlifiers have low distortion but the efficiency is very low. Class D amlifiers have very high efficiency, but also high distortion. A arallel combination of a class D and class AB amlifier results in high efficiency and low distortion. The combination of the two amlifiers is shown in figure.. Figure. Parallel combination of a Class AB and Class D amlifier The outut ower should be delivered by the class D amlifier and the accuracy by the class AB amlifier. The class AB amlifier should be able to drain the switching rile. This toology is already investigated [], but there is no suitable design for the use in an IC rocess, because the outut imedance of a class AB amlifier is relatively high for the high switching frequencies. Consequently, it cannot comensate the rile. There are some otions to lower the outut imedance. In general, the gain of the amlifier should be imroved. It is not ossible to imrove the total amount of gain of the amlifier freely over the whole frequency by limitations of the stability of the amlifier in combination with the used IC rocess. More gain results in general in less distortion at the cost of stability or ower consumtion. A tyical amlifier design is a trade of between gain, stability and ower consumtion, which is set to the best needed erformance. A more realistic otion is to imrove the gain of the amlifier at a small bandwidth. This could be done by a eak filter in the gain stage of the amlifier. Another otion is to design an external notch filter at the outut of the amlifier. A filter at the outut of the amlifier should be able to drain and source large currents, which requires large external comonents. More suitable in IC design is a filter in the gain stage of the amlifier. By limitations of the accuracy in an IC rocess, some care should be taken by designing the eak filter. The research focuses on the ossibilities to create a notch in the outut imedance of the class AB amlifier to be able to remove the rile of the class D amlifier. The amlifier should be able to realie in an IC rocess without requiring many external comonents. Chater gives some background theory about the outut imedance of amlifiers. Also a derivation is done about the secifications of the amlifier in combination with the distortion, ower consumtion and stability. Remco Twelkemeijer Page 7 of 68

8 Chater - Introduction The outut imedance of the amlifier is derived in a ole ero analysis in chater 3. The chater deals with the stability in combination with a notch in the outut imedance. During the derivation, some limitations were derived. The results and limitations of the ole ero analysis is translated to an amlifier design. This design is done with ideal comonents and results in an amlifier toology and is described in chater 4. The amlifier toology contains some ideal comonents which are not suitable to imlement in an IC rocess. Esecially filters with inductors are not able to imlement. In chater 5, some filters are investigated to imlement the toology in an IC rocess. In Chater 6, simulations are done with the arallel combination of the AB amlifier and a class D amlifier. There are four ideal amlifiers designed which will be comared with an amlifier without a notch in the outut imedance. The last chater, chater 7 contains the conclusion of the roject. Remco Twelkemeijer Page 8 of 68

9 Chater - Amlifier requirements Amlifier requirements The arallel combination of a class AB and class D amlifier gives some more requirements to the class AB amlifier. The amlifier should be able to remove the switching residues of the class D amlifier. The switching frequency is relatively high comared to the audio bandwidth of 0 H to 0 KH. First the gain of amlifiers is described with resect to stability. The outut imedance should be low to remove the switching rile; the limitations of the outut imedance of amlifiers is described next. The outut imedance of the amlifier has a relation with stability which is described in aragrah.3. Finally, the requirements are given when the amlifier has a notch in the outut imedance.. Gain in resect with stability The design of audio amlifiers is widely described in literature and is not reeated in this section. This aragrah gives only a short summary about the gain of amlifiers in resect to stability. A two stage amlifier is shown in figure... First the Miller caacitance Cm is neglected. In that case, the amlifier contains two oles which are tyically close to each other. Each ole contributing 90 degrees hase shifts and consequently the amlifier asymtotically aroaches 80 degrees. The magnitude and hase characteristic of the amlifier is shown in figure..3a. The two oles without the miller caacitance are described by: = v = RC R L C L Figure.. Two stage Amlifier Alying negative feedback, a art of the outut is redirected to the inut. If the outut is fully redirected, a voltage buffer is created which is shown in figure... According to the stability criterion [8], the oen loo gain has to cross the unity gain bandwidth (UGB) or ero db before the hase shift reaches 80 degrees. The amlifier as sketched above without the Miller caacitance has a hase shift of 80 degrees at UGB. The Miller caacitance is needed which introduces a dominant ole. A bode lot containing Miller comensation is shown in figure..3b. The first ole is shifted more to the left, which results in ero db gain before the hase shift reaches the 80 degrees. Figure..: Voltage buffer Remco Twelkemeijer Page 9 of 68

10 Chater - Amlifier requirements Figure..3 a) Bode and hase lot b) Bode and hase lot with Miller caacitance The Miller comensation can be extended to more than two stages. Theoretically, it is ossible to extend the amlifier with an infinite number of stages. In literature different methods of Miller comensation is described, but it does not result in relatively high gain for high frequencies. In the most amlifier designs, the allowed hase is less than 80 degrees at UGB. The distance between the hase at UGB and 80 degrees is called hase margin (PM). The hase margin is indicated in figure..3b. Tyical hase margins are around 45 or 60 degrees [8]. The unity gain bandwidth of the amlifier of figure.., is described by []: gm UGB =. C m An otion to imrove the gain is to increase the transconductance of the outut stage. The outut stage of the two-stage amlifier in figure.. is gm. The transconductance can be increased by increasing the quiescent current in the outut transistors. This current leads to more ower dissiation and is not attractive in a low ower audio amlifier design. Remco Twelkemeijer Page 0 of 68

11 Chater - Amlifier requirements. Derivation of the outut imedance The interest in this reort lies on the outut resistance, because reducing the outut imedance will also reduce the distortion and switching rile []. The relation of the distortion and the outut imedance can be easily determined using figure... Figure.. Relation between distortion and outut imedance The amlifier is simlified by a voltage source with a resistance in series. The error or distortion signals are described by current source at the outut. The error signal could be introduced by the amlifier itself, but also by external factors for examle the class D amlifier in this case. If the outut imedance is ero, the error signal is shortcut by the voltage source. If the outut imedance is high, the error signal can not be dissiated by the voltage source and consequently the error signal is dissiated by the load. In this situation, the unwanted error current in the load results in distortion. The outut imedance of the voltage buffer of figure.. deending on the oen loo gain Ro is described by [4]: Z out = A oc The resistance R o, is the oen loo resistance at ero Hert. Increasing the gain, results in decreasing of the outut imedance and consequently in reducing the distortion. This is true in general, but it is also shown that it is ossible to decrease the outut imedance keeing the same oen loo gain by influence of the feed back Miller caacitances [3]. Next, the outut imedance of a simle amlifier toology is derived. The amlifier is shown in figure.., the outut is fed back to the inut and a current source is alied to be able to derive the outut imedance. In ractice, there are a lot of variations ossible, but the limitations are comarable. Figure.. Two stage amlifier Remco Twelkemeijer Page of 68

12 Chater - Amlifier requirements The outut imedance can be written by []: s ( ) C C R C C m m Z = out ( s) gm Cm gm s Cm Investigation of the outut imedance equations shows that it contains some constants and a ole and a ero. The ole and ero is written by: gm = v = R C C ( m ) C n For high frequencies, the outut imedance is written by: C Cm Z out ( ) = gm Cm gm Indeendent of the oles and eros, the outut imedance for high frequencies is always aroximately /gm, where gm is the transconductance of the outut stage. To have lower outut imedance at low frequencies, the ero should be set lower frequencies than the ole. The corresonding grah is shown in figure..3a. Also the situation is shown in figure..3b, when the ole is set to a lower frequency than the ero. In that case, the outut imedance for low frequencies is much higher than the final value of /gm for high frequencies. Figure..3 a) outut imedance with < b) outut imedance with > The low frequency outut imedance is written by: Z out ( 0) = gm gm R To have as low outut imedance as ossible, gm, gm and R should be as large as ossible. As described in [], this has some limitations. Increasing the resistance, results in decreasing the lace of the first ero and results in more hase shift. The hase shift in resect to stability is described in the next aragrah. The transconductance gm is limited by transient current and consequently by the ower dissiation. Remco Twelkemeijer Page of 68

13 Chater - Amlifier requirements.3 Outut imedance in resect with stability The amlifier in aragrah. contains one ero and one ole. The corresonding hase shift is shown in figure.3.. The hase shift is ero degrees for low frequencies, reaches the maximum value of 90 degrees and becomes ero degrees as well to high frequencies. A ositive amount of hase shift acts inductive. Loading the outut with a caacitor, there is a caacitor and inductor in arallel. This could results into oscillation and consequently in instability. In this aragrah, some limitations of designing an amlifier with resect to outut imedance and stability are described. Figure.3.: Phase shift of an amlifier with one ole and one ero With one ero and one ole the hase shift is always between ero and 90 degrees. In ractical designs, there are more oles and eros and consequently it is ossible to reach larger hase shifts. If the hase shift becomes negative, the amlifier acts like a caacitor. Zero degrees hase shift corresonds to a resistance. If the hase shift becomes more than 90 or less as -90 degrees, a negative resistance art is introduced. Figure.3. a) examle of Z out and a Z load b) imedances written as vectors A simlification of an amlifier with a load is shown in figure.3.a. The current source with outut imedance in arallel corresonds to the amlifier. In figure.3.b, the imedances are written as admittances. The imedance is equal to /admittance. The total admittance is the sum of the two vectors. If the hase of the amlifier is 90 degrees and the load is caacitive (-90 degrees) and Y out =Y load, Y total is ero. The outut imedance is /Y total, division by ero results in oscillation. Assuming the load can be every assive load, the hase of the load is always between /- 90 degrees. If the amlifier never reaches /-90 degrees, it is never ossible to get oscillation and stability can be guaranteed for every assive load. Remco Twelkemeijer Page 3 of 68

14 Chater - Amlifier requirements In ractical amlifier designs, the amlifier is only fully stable for limited loads [5]. With fully stable, it is meant that the amlifier does not resonate. The hase shift is normally ero or higher to get low outut imedance for low frequencies. Consequently, the amlifier is limited for caacitive loads. A grah is made containing lot of the outut imedance of an amlifier and two imedances of two caacitances, which are nf and 00nF. A caacitance of nf lies in the resistive art of the amlifier at 50 MH. Increasing the caacitance, results in decreasing the imedance of the caacitor to high frequencies. The 00nF caacitance lies in the inductive art of the amlifier and the amlifier could resonate. Figure.3.3 Outut imedance of amlifier in combination with a caacitive load The hase of the system could exceed the 90 degrees maintaining a stable system. The resistive load should be larger than the negative resistance which is created by the amlifier. If the load is Ω, the hase could exceed the 90 degrees between 0-00 KH in the examle of figure.3.3. Esecially for high frequencies when the outut imedance is high, the hase should be smaller than 90 degrees to guarantee a stable system. The amount of resonance is deendence on the hase of the outut imedance and the load. To be able to determine the stability of a system, a ste resonse can be alied. A ste contains all the frequency comonents. The ste resonse dislays overshoot and ringing. An examle of a ste resonse is shown in figure.3.4. The grah contains 3 lines with different ste resonses. The first line does not have ringing at all, the second line is damed out within 0 µs and the third line has a very long ringing time. The margin of the ringing time is not determined now, but a resonse like line 3 is not allowed. Figure.3.4 Examle of a ste resonse Remco Twelkemeijer Page 4 of 68

15 Chater - Amlifier requirements.4 Outut imedance secification An amlifier does not have low outut imedance for high frequencies by the limitations of the /gm of the outut stage. The switching frequency of the class D amlifier is high and consequently, it is not ossible to remove the switching rile. To be able to remove the switching rile, the outut imedance should be decreased. The idea is to do this by creating a notch in the outut imedance. A grah of the wanted curve is shown in figure.4.. Figure.4. Outut imedance with a notch The outut imedance of the amlifier at resonance frequency should be lower than the imedance of the loudseaker, because the current rile should be dissiated by the amlifier and not in the loudseaker. To be able to drive some caacitive load, the hase should not exceed 90 degrees above resonance frequency. Also the low frequency outut imedance should be low, to have as low distortion as ossible in the audio bandwidth. There is a relation between the oen loo gain of the amlifier and the outut imedance. Increasing the oen loo gain, results in decreasing the outut imedance. Creating a eak in the oen loo gain of the amlifier should results in a notch in the outut imedance. Some care should be taken, that the eak in the oen loo gain does not lead into instability with resect to the hase. The hase shift should always be smaller than 80 degrees if the gain is one. The switching frequency of the class D amlifier could be constant or variable, deending on the design. To be able to dissiate the switching rile, the resonance frequency should be known with certain accuracy. If there are design ossibilities, the resonance frequency should be variable. Filters which could be needed to create the notch should be able to imlement in an IC rocess and not requires much external comonents. Another imortant arameter is the ower consumtion. The extra comonents which are needed to create the notch in the outut imedance should consume less ower than simly increase the current in the outut transistors. Increasing the current in the outut transistors results in a higher transconductance and consequently in lower outut imedance. Next, the secifications as described in this aragrah are summaried. Remco Twelkemeijer Page 5 of 68

16 Chater - Amlifier requirements Secifications The amlifier should be stable Low outut imedance in the audio bandwidth Z out ( eak ) << Z seaker Phase should not exceed 90 degrees above resonance frequency The filter should be able to imlement in an IC rocess Power consumtion should be low in comarison with increasing the transconductance. Remco Twelkemeijer Page 6 of 68

17 Chater 3 - Outut imedance using a Pole-Zero analysis 3 Outut imedance using a Pole-Zero analysis This chater gives some theory and methods about reducing the outut imedance using a ole-ero analysis. First, a start is made with a standard amlifier described by a ole-ero ma. Next, two oles and two eros are added to create a notch in the outut imedance. The notch gives some limitations to the outut imedance. An aroach while the last ole is neglected is first described. Next, the outut imedance is investigated in a realistic situation, when the last ole is not neglected. Finally two examles of two system designs with resect to the tradeoff between the lace of the oles and eros is described. 3. Standard amlifier In this chater the outut imedance is derived in a very mathematical way. This means, there is a direct relation with an amlifier but the variables, oles and eros can be set to every value. With this aroach it is ossible to determine the limitations about reducing the outut imedance without using amlifier toologies. Designing an amlifier, it should deal with those limitations. The amlifier should have low outut imedance at low frequencies as well as a certain resonation frequency. Starting with the amlifier in aragrah. of figure.., the outut imedance is described with some constants, one ole and one ero. Simlifying the outut imedance to a general first order Lalace function, the outut imedance is written by: Z out () s = K ( s ) ( s ) K A certain roortionality factor for the outut imedance. First ero First ole From the function Z out, there are several methods to determine the magnitude and hase characteristics. One of them is using a ole-ero ma as shown in figure 3... A ole-ero ma is not the easiest way to determine the magnitude and hase characteristics of a first order system, but in the next aragrahs some oles and eros will be added to the function. With these extra oles and eros, the comlexity of the function increases. Using a ole-ero ma, it is ossible to simlify the function and it is easier to investigate the roblems when the comlexity increases. Figure 3.. Pole-Zero ma Remco Twelkemeijer Page 7 of 68

18 Chater 3 - Outut imedance using a Pole-Zero analysis The outut imedance at a frequency x is equal to the length of the vector of the ero divided by the length of the vector of the ole. The hase at this frequency is the angle of the vector of the ero Φ, minus the angle of the vector of the ole Φ. Z ( ) ( j x ) ( j ) r ( Z ( ) ) = Φ ( ) Φ ( ) out x = K = K, Φ out x r The length of the ole or ero is described as function of the frequency. It is the length of the vector from a ole or ero to a certain frequency. In figure 3.., the length of the ole at frequency x is r. Sometimes, the length of the vector is needed when the frequency is ero. This is described as the length at DC. Note that for real oles or eros, the length at DC is equal to the oles and eros. Starting at DC, is ero. The magnitude of the outut imedance becomes: Z out ( 0) = K. To get low imedance, it follows that the ero should be small and the ole should be large. This corresonds to the magnitude characteristic of the outut imedance as described in aragrah.. The grah is reeated in figure 3... Figure 3..: Curve of the outut imedance with one ero and one ole The contribution of a ero to the hase is 90 degrees and the contribution of a ole to the hase is -90 at maximum. Because the ero should be set to lower frequencies than the ole, the hase shift lies between ero and 90 degrees. Assuming the oles are real and the constant K is one, the outut imedance and hase shift deending on the frequency can be written by: Z out ( ) ( ( ) ) =, Φ Z = out tan tan Remco Twelkemeijer Page 8 of 68

19 Chater 3 - Outut imedance using a Pole-Zero analysis 3. Adding two comlex eros to create a notch In the revious aragrah, the outut imedance is described with a first order function. With a first order function it is not ossible to get low outut imedance to low frequencies as well as a notch in the outut imedance. To create a notch in the outut imedance, two comlex conjugated eros are needed. With the ero which is necessary to satisfy low outut imedance at low frequencies, at least a third order function is needed. With resect to stability, the hase of the amlifier should be between /- 90 degrees. To get ero degrees hase shift at high frequencies, the same amount of oles and eros are needed. To create a notch in the outut imedance as well low frequency outut imedance, at least three oles and three eros are needed. The ideal curve which is wanted is shown in figure 3... The curve can be written by: s s s 3 Z out () s = K s s s ( )( )( ) ( )( )( ) 3 K A roortionality factor for the outut imedance. First ero Second ero (comlex) :: = a b j 3 Third ero (comlex) :: = a b j First ole Second ole Third ole 3 The resonance frequency notch deending on the second and third ero is equal to the length of the eros at DC [9]: Z out ( j) = 0 notch a b = / Figure 3.. simlified lot of the outut imedance deendence on the frequency. With resect to stability, all the oles should be negative. The two comlex conjugated eros should be negative as well to comensate the oles. The first ero can be ositive or negative. A negative ero results in 90 degrees hase shift and a ositive ero results in -90 degrees hase shift. If the ero becomes ositive, the transfer function with s is ero becomes: ( ) ( )( )( 3 ) Z out 0 = K. ( )( )( ) 3 Remco Twelkemeijer Page 9 of 68

20 Chater 3 - Outut imedance using a Pole-Zero analysis The first ole has a minus sign, a minus sign in the outut imedance results in -80 degrees hase shift. A hase shift of -80 degrees results in a negative imedance, which could result in instability. With this aroach it can be concluded that all the oles and eros should be negative. Starting the third order system with the resonance frequency at MH and the first ero to 0 KH and the third ole to 00 MH, the oles can be directly determined. In aragrah 3., it was obtained that the oles should be as large as ossible to retain low outut imedance for low frequencies. Creating large oles, for examle if one ole is larger than the resonance frequency and one ole is smaller than the resonance frequency. It directly follows that the hase becomes larger than 90 degrees to frequencies larger than resonance frequency. The first ero is comensated by the first ole. At resonance frequency the hase shift is 80 degrees at maximum by the two eros and consequently more than 90 degrees. An examle with a ole at 00 KH and a ole at 0 MH is shown in figure 3... The hase shift has the maximum value after MH and becomes nearly 80 degrees. At this frequency the hase shift should be less than 90 degrees to be able to drive some caacitive load. Figure 3.. Combination of the notch and the first order system in a third order system The magnitude characteristic of figure 3.. corresonds to the wanted curve of figure 3... With resect to stability, it is not ossible to create this curve. To be able to create a stable system, the second ole should also be set to lower frequencies than the resonance frequency. This situation is shown in figure The two oles are able to comensate the two comlex eros if the system is designed roerly. This results automatically in 90 degrees hase shift at maximum. In aragrah 3., it was obtained that the oles should be as large as ossible to retain low outut imedance for low frequencies. With the need for stability, the oles are decreased and the outut imedance at low frequencies increases. Figure 3..3 simlified lot of the outut imedance deendence on the frequency. Remco Twelkemeijer Page 0 of 68

21 Chater 3 - Outut imedance using a Pole-Zero analysis With the first and second ole, large comlex conjugated oles can be formed. This is shown in figure Using the eaking behavior of the comlex oles; it is ossible to satisfy good outut imedance at DC and a hase shift between 90 and -90 degrees. For low frequencies and keeing the real value of the oles constant, the vector of the two oles becomes larger if the oles comlexity increases. Increasing the lengths of the oles, results in decreasing of the outut imedance. From this aroach it can be concluded that a eaking in the outut imedance is needed if the low frequency outut imedance has to be decreased. Figure 3..4 simlified lot of the outut imedance deendence of the frequency. Remco Twelkemeijer Page of 68

22 Chater 3 - Outut imedance using a Pole-Zero analysis 3.3 Limitations while neglecting the third ole It is seen that that the need for stability results in an increase of the outut imedance for low frequencies or introduce a eak in the outut imedance. One could ask how much the outut imedance increase and how large should the eak in the outut imedance be with a marginal lost in DC gain. In this aragrah, a derivation is done while the influence of the third ole is neglected. First the values of the oles are calculated to satisfy a stable system. Finally, the oles are suosed to be real. If the oles are real, they could be set to different frequencies. A calculation is done what haens to the outut imedance at low frequencies as well at resonance frequency Length of the two oles with resect to DC With resect to stability, the values of the oles are limited by the eros. Next these values will be calculated. The outut imedance for a third order system with the constant K is one can be written as: r ( ) ( ) r ( ) r 3( ) H = r r r ( ) ( ) ( ) 3 The length of the oles is written by: yx Im( ) ( ) = Re( ) ( Im( ) r yx = 0 ( ) = r yx yx The angle is determined by subtracting the angle of the oles and adding the angle of the eros which results in: Φ Z = Φ Φ Φ Φ Φ Φ ( ( )) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 The angle of the oles is written by: Im ( ) ( ) x Im x = 0 Φ = Φ = ( ) x tan x tan Re x x ( ) ( ) The magnitude of the function corresonds to certain outut imedance and should be as small as ossible. The length of the oles should be as large as ossible and the length of the eros should be as small as ossible. The lace, and consequently the length of the oles and eros are limited by the allowed hase shift. Starting again with the wanted curve shown in figure 3.., the formula for the hase and magnitude can be simlified. Using a resonance frequency of MH, the first ero, is far away (for examle 0Kh). The contribution of this ero results in 90 degrees hase shift. The third ole 3 is set to very high frequencies (for examle 00MH) which results in a contribution of ero degrees hase shift. The hase of the function can now be written as: Φ ( Z( ) ) 90 Φ ( ) Φ 3( ) Φ ( ) Φ ( ). yx yx Remco Twelkemeijer Page of 68

23 Chater 3 - Outut imedance using a Pole-Zero analysis In the calculations, the 90 degrees is left out. To maintain /- 90 degrees hase shift, the hase shift should be between -80 and 0 degrees. Also the contribution in magnitude of the first ero and last ole could be left out. This results in losing the low outut imedance in the grahs, but the transfer function is still useful and simler to read. If the transfer function becomes less than ero db, there is some imrovement and if the function becomes larger as ero db the outut imedance is decreased. The simlified outut imedance is written as: r ( ) ( ) r 3( ) H = r r ( ) ( ) As described in aragrah 3.3., the two oles should have a smaller length than the eros. This results in increasing the outut imedance at low frequencies. Next, it is ossible to calculate the influence of the lace of the oles and eros. Starting with comlex conjugated eros and comlex conjugated oles, the real art of the oles can be derived to maintain a stable system. Figure 3.3. Pole-Zero ma In figure 3.3., a ole-ero ma containing two comlex eros and the lace of the resonance frequency on the imaginary axis is given. On the imaginary axis there is a second oint, described as x. The variable x is only a multily factor and is the resonance frequency. The contribution of the first ole to the hase of the system deendence on the variable x can be described as: x Im( ) ( ) Φ = tan x Re( ) An examle of the hase characteristics deending on the variable x is shown in figure The hase should be limited to -80 and ero degrees. At x is 0., the hase is -80 degrees, around x is one (resonation frequency), the hase goes to ero degrees. It reaches ero degrees asymtotically at x infinity. In the calculations, the value x should be large. Remco Twelkemeijer Page 3 of 68

24 Chater 3 - Outut imedance using a Pole-Zero analysis Figure 3.3. Phase characteristics deending on x. If x becomes large (larger than 0), the contribution of the imaginary art to the hase becomes very small. The hase for two oles and two eros should be between -80 and ero degrees. The contribution of the two oles can be equal to the contribution of the two eros at high frequencies, to high values of x. The angle should be limited by: Im( ) Im( x x tan tan Re( ) Re( ) ) tan x Im( ) tan Re( ) x Im( Re( ) ) 0 If x is much larger than, the imaginary art can be neglected. The formula can be simlified to: x x x x tan tan Re( ) Re( ) Re( ) Re( ) Re( ) Re( ) The real art of the ero should always be larger or at least equal as the real art of the ole. This solution is useful if the contribution of the hase of the oles should always be larger or at least equal than the contribution of the eros. In a third order system, the hase can be larger by the influence of the first ero and third ole. The solution for this situation is described in aragrah 3.4. With the result that the real art of the ero should be larger or equal to the real art of the ole, it is ossible to make a grah of the imrovement in the outut imedance. The grah is shown in figure and contains 4 curves, two solid lines and two dotted lines. The two solid lines corresond to a system were the oles only contains a real art. The curve at DC and at resonance frequency is given. It shows clearly that if the eros are very comlex, the imrovement at resonance frequency is nearly 80 db. The imrovement at DC is -60dB, which corresonds to an increase in outut imedance with 60dB to low frequencies. This is unaccetable; also if the eros become less comlex the loss at DC is aroximately 40dB. The dotted curves corresonds to a system containing two oles with a real art equal to the ero, but an imaginary art is introduced to kee the loss 3dB for low frequencies. Comaring the two curves at resonance frequency, the imrovement at resonance frequency is decreased by aroximately 0dB, but the loss is at low frequencies is only 3 db. Remco Twelkemeijer Page 4 of 68

25 Chater 3 - Outut imedance using a Pole-Zero analysis Figure Imrovement of the outut imedance 3.3. Shifting one real ole If the oles are comlex, the real art of the oles is equal. Non comlex oles have the ossibility to have some distance between the oles. An examle of three magnitude bode lots are shown in figure The first ole is ket on the same lace and the second ole is shifted to the left. This results in increasing the outut imedance to low frequencies. The question is how much will the outut imedance increase and what haens around the resonation frequency. A ole-ero ma is shown in figure It contains two comlex eros and two real oles. Starting at a oint where the two oles are equal, the second ole can be decreased to ero at least. If the second ole decreases and the first ole is ket constant, the contribution of the oles to the hase is increases. From this, it follows that the comlexity of the eros can be increased and consequently the real art of the comlex conjugated eros can be decreased. Figure Magnitude lots with different distances between the oles The angle should be limited by: x Im( ) x Im( tan tan β Re( ) β Re( ) ) = tan x ) tan x γ Two new variables are introduced in this formula. The first one γ, is the decreasing factor of the second ole with resect to the first ole. The second ole is now described by = γ. The range for γ could be one to ero. Starting with a γ of one and decreasing this value, the real art of the comlex conjugated eros can be decreased keeing the hase between /- 90 degrees. This is done by the second new variable β. Remco Twelkemeijer Page 5 of 68

26 Chater 3 - Outut imedance using a Pole-Zero analysis Remco Twelkemeijer Page 6 of 68 Figure Pole-Zero ma with two real oles and two comlex eros Using the formula of the angle limitation, it is ossible to solve the value for β. This results in: ( ) ( ) γ β γ β >> ) Re(, ) Re( x x The outut imedance, deendence from can be written as the length of the eros divided by the length of the oles. In general, the outut imedance can be written for every second order system with two comlex eros and two real oles by: ( ) ( ) ( ) ) Im( ) Re( ) Im( ) Re( = = K r r r r K Z out Putting the two oles and eros in the formula and the constant factor K is one, the follow function is obtained at resonance frequency: ( ) ( ) ( )( ) ) Re( ) Re( ) Re( ) Re( ), ( out Z γ β β β β γ = The influence of the third ole is neglected in this aroach, consequently the interest lies on large values of x. For large values of x and using ( ) β γ Re( ), the formula can be simlified to: ( ) ) Re( ), ( γ ϖ γ out Z Next, the outut imedance can be comared with resect to γ is one (the first ole is equal to the second ole). This gives a relative imrovement or reduction in the outut imedance at resonance frequency, deending on the lace of the second ole. ( ) 0, ), (,) ( > γ γ γ out out Z Z

27 Chater 3 - Outut imedance using a Pole-Zero analysis A grah of this formula is made and is shown in figure Also a grah of the imrovement in outut imedance for low frequencies is given. For low frequencies, there is no imrovement at all because the imrovement factor is lower than one. It is clearly visible that if γ is aroximately 0. ( = 0. ), the low frequency erformance is decreased by 0dB and the benefit at resonance frequency is aroximately 6 db. Figure Imrovement at resonate frequency and reduction at low frequencies. Result In general the outut imedance around resonance frequency is imroved if the second ole becomes smaller keeing the first ole constant. The outut imedance at low frequencies increases too much, which is unaccetable. It can be concluded from figure in aragrah 3.3. in combination with the aroach in this aragrah, that the oles should be comlex to maintain low outut imedance for low frequencies. The real value of the oles should be equal or smaller than the real value of the eros under the condition that the last ole is neglected. Remco Twelkemeijer Page 7 of 68

28 Chater 3 - Outut imedance using a Pole-Zero analysis 3.4 Limitations in a realistic situation The aroach while the last ole is neglected shows clearly the limitations of a notch filter. The oles should be comlex to maintain low outut imedance for low frequencies. Alying a ste resonse to a system with two comlex oles, resonance occurs. It seems not ossible to design an amlifier without any resonation if it should have a notch and low outut imedance at low frequencies and be stable for any assive load. One could ask, how comlex should the oles be, using a good low frequency behavior and a dee notch. The hase limitation lies on high frequencies in aragrah A ractical value of the last ole could be around 0 MH. It is not allowed to neglect the influence of the last ole with this value. The hase influence by the last ole is described by α. Next, the minimum comlex conjugated oles will be simulated deending on the comlex conjugated eros and the variable α. The contribution of the first ero is 90 degrees again. This results in a hase of: Φ Z ( ) 90 Φ Φ 3 Φ Φ ( ) ( ) ( ) ( ) ( ) α The oles and eros have a real and imaginary art. The resonate frequency (notch) is set to MH. The resonate frequency of the oles (eak) should be smaller as MH to maintain a system between /- 90 degrees. This results in a loss of erformance in the low frequency region. The length of the oles is set to a loss of 3dB in the outut imedance at low frequencies. The length of the eros divided by the length of the oles at =0 results in the loss in outut imedance at low frequencies. A loss of 3dB and a resonate frequency of MH results in the length of the oles: ( Mh π ) Mh π H (0) = = 3dB = L = oles L ( ) 4 oles With the length of the oles and eros known, the oles and eros are described by: / / = L notch oles / / Re = Re ( ) ± j Re( ) / ( ) ± j L Re( ) / Comlex oles means resonation, if the oles become more comlex, the resonance increases. The real art of the oles should be as large as ossible to have as less resonation as ossible. The hase of the system should be between /-90 degrees, esecially to high frequencies. Using this limitation, the maximum real art of the oles can be calculated deendence on the real values of the eros. In this aroach with the extra variable the angle α, it is not ossible to solve the lace of the oles deending on the eros mathematically. But using the values of MH for the resonate frequency a loss of 3dB in the outut imedance at low frequencies; it is ossible to solve the equations numerically. In figure 3.4., four curves are shown, a real value of the ero of 0 3 π, 0 4 π, 0 5 π and the maximum real value of the ole which is ossible. The maximum real value of the ole is lotted deendence on the angle α. During the calculations, the length of the oles and eros is ket constant. Remco Twelkemeijer Page 8 of 68 notc oles / /

29 Chater 3 - Outut imedance using a Pole-Zero analysis Starting at α is ero, the corresonding real art of the oles are the same as the real art of the eros. This roves that the calculations in aragrah are correct, because the conclusion was Re( ) Re( ) under the condition α is ero which was not jet introduced at that moment. The grah shows a large increase in the real art of the ole to low values of α. When α becomes 5 to 0 degrees, the real art of the oles does not increase very vast anymore. The curve maximum value is equal to the length of the oles. The real art of the oles can never be larger as this value. At α is 45 degrees, none of the three curves will reach this maximum value of σ. From this, it can be concluded again that the oles are always comlex and there will always be some ringing when alying a ste resonse to the system. Figure 3.4. Maximum real value of oles deending on the extra hase variable α In figure 3.4. a tyical magnitude lot of the second order system is shown. There are three amlitudes given: A LF, A eak and A notch. This means the loss in outut imedance in the low frequency region, the relative amlitude of the eak and the relative amlitude of the notch. Because the third order system is aroached by a second order system with an extra hase variable α, the outut imedance is ero db to high frequencies and the loss value for low frequencies. Figure 3.4. Different calculated amlitudes Remco Twelkemeijer Page 9 of 68

30 Chater 3 - Outut imedance using a Pole-Zero analysis Next, the amlitude of the second order function deendence on an extra hase is calculated. The results are shown in figure It shows that an increase of the real value with a factor of 0, the notch decreases with a factor of 0. The difference of ero degrees and 45 degrees extra hase is aroximately a factor of two at maximum. Extra hase does not result in a big imrovement in the amlitude of the notch. Figure A notch deendence of extra hase variable α In figure the amlitude of the eak is shown. The eak should be as low as ossible to have as less as ossible resonation. The grah shows a large decrease in the eak value deendence by increasing α. For examle, if σ is 0 3 π, the eaking with α is ero degrees is more as 00. Increasing α to 0 degrees, the eaking left is only a factor of three. The extra hase results in less eaking. This is exactly what is wanted to be able to design an amlifier with a notch in the outut imedance. Esecially if the eros are large comlex, the difference is very large. Figure A eak deending of extra hase variable α The calculations with the extra hase variable α results in the conclusion, that it is ossible to design a system with a dee notch and a eak. The amlitude of the eak does not have to be as large as the notch if the system is designed in that way. With extra hase, it is not ossible to comensate the loss in the low frequency region. Remco Twelkemeijer Page 30 of 68

31 Chater 3 - Outut imedance using a Pole-Zero analysis 3.5 System Design With these results, it is ossible to do a system design. A ole-ero ma is made as shown in figure In the ma, there are also a ole and ero which results in low frequency gain. There is also a magnitude and hase lot in the figure. Those lots contain two curves, a dotted and solid line. The dotted line corresonds to a system which only has one ole and ero for the low frequency gain. This is utted in the grah for comarison uroses. The solid line has the same ole and ero as the system of the dotted line, but this system has also two comlex oles and eros to create to notch. The ole-ero ma describes the system with the solid line. The ero is set to 0 4 H and the ole to H. This results in 70 degrees at MH as can be read in the dotted line in the hase lot. This gives 0 degrees hase. From figure 3.4., it can be read that with a real value of 0 3 π of the ero results in aroximately a real value to the ole of π. From figure it can be read that the imrovement at MH is aroximately (-44dB). The amlitude of the eak is aroximately a factor of three (9,5dB) which can be read from figure The magnitude lot in figure shows that these values corresond. Figure 3.5. Pole-Zero ma with corresonding magnitude and hase characteristics Another otion is to decrease the hase of the system as shown in figure The eros are on the same lace but the last ole is set to π. To maintain a system between /- 90 degrees hase shift, the real art of the comlex oles should decrease to π. This decrement result in a larger eak, but with aroximately no change to the notch. Figure 3.5. Pole-Zero ma with corresonding magnitude and hase characteristics Remco Twelkemeijer Page 3 of 68

32 Chater 3 - Outut imedance using a Pole-Zero analysis For the systems of the ole ero mas as described in figure 3.5. and figure 3.3., a ste resonse is alied. The result is shown in figure 3.5.3, the solid line corresond to the oleero ma in figure 3.5. and the dashed line to the ole-ero ma of figure The lacement of the last ole of the two systems differs, and consequently the gain. This results in a different maximum value of overshoot. To comare those two systems, the maximum amlitude of overshoot of the resonse is normalied. Comaring the two curves, the daming of the system with the higher real value is much larger and the ste resonse becomes nearly ero within 0 μs. The other curve has low daming factor, which results in a very long ringing time. Figure Ste resonse corresonding to the two ole-ero mas. Remco Twelkemeijer Page 3 of 68

33 Chater 4 - Amlifier design 4 Amlifier design The aroach in the revious chater determines the theoretical limitations of the amlifier. In this chater ractical toologies are described to create a notch in the outut imedance. First the system imlementation is described briefly. Next the toology of an amlifier to create a notch in the outut imedance is described. With this amlifier a system design is done. Finally an imlementation is done with a outut stage. 4. System Imlementation In the revious chater it is determined that at least a third order outut imedance is needed to create a notch in the outut imedance. The outut imedance should contain two comlex oles and two comlex eros with one real ole and ero. Simlifying every amlifier stage as blocks, the amlifiers can be set in cascade or in arallel as shown in figure 4..a and figure 4..b resectively. In [] it is roven with simulations that adding an extra cascade stage normally results in an extra ole and ero. This is under the assumtion that the extra stage contains a caacitance and a resistance which is a first order function. From this, one of the two stages of the amlifier should contain two comlex oles and two comlex eros if the amlifiers are set in cascade. ( s ) ( s )( s 3 ) Z = v Z notch = s s s Z cascade ( ) = Z Z notch = ( s ) ( s ) ( )( 3 ) ( s )( s 3 ) ( s )( s ) 3 Creating two amlifiers with only one stage, the outut imedance of the two amlifiers are set in arallel. If the first amlifier has one ole and one ero and the second one has only two comlex eros and two real oles, the outut imedance is written by a third order function which contains comlex oles and two comlex eros. This can be obtained from the formula Z arallel, the denominator contains the eros and 3 which are assumed to be comlex. Solving the values of the three oles in the denominator, one real ole and two comlex oles is obtained. Z arallel = Z // Z notch = Z Z Z Z notch notch = ( s )( s )( s 3 ) ( s )( s )( s ) ( s )( s )( s ) 3 3 It seems easier to imlement two amlifiers in arallel, because only two comlex eros should be created. Another reason could be that the first amlifier contains always a linear ath, while the second amlifier can be designed with time discrete section in it without aliasing roblems. Figure 4.. a) Amlifiers in cascade b) Amlifiers in arallel Remco Twelkemeijer Page 33 of 68

34 Chater 4 - Amlifier design 4. Realiing the notch in the outut imedance The starting oint for the amlifier toology is determined in the revious aragrah. The roosed toology has two amlifiers in arallel. The design of an amlifier with one ole and one ero is already described in chater. The second amlifier with two comlex eros is described now. The two-stage amlifier of figure.. has a arallel combination of a resistor and a caacitor between the two stages. The arallel combination is shown in figure 4... The resistance for low frequencies is equal to R. Increasing the frequency, results in decreasing the imedance of the caacitance and consequently decreasing the imedance of the arallel combination. The gain and consequently the outut imedance is roortional to the inut voltage times the imedance of the arallel combination. The consequence of decreasing the imedance for high frequencies results in less gain. The imedance of the arallel combination can be written as: Z RC = C s R C ( ) Figure 4.. Parallel combination of a resistor and a caacitor The caacitance can be comensated at a certain frequency by an inductor. If the caacitance is comensated, the imedance becomes equal to the resistance. The frequency where the inductor comensates the caacitor is the resonance frequency. Adding an inductor in arallel, results in the network as shown in figure 4... For realistic modeling, there is also a resistance (R L ) in series with the inductance. The arallel combination including the inductance can be written as: R s L L Z RLC = C R L RL RLC s s RLCC L RLC LC Figure 4.. Parallel combination of a resistor, caacitor and an inductor Remco Twelkemeijer Page 34 of 68

35 Chater 4 - Amlifier design In figure 4..3 two bode lots are shown of the two arallel imedances which are indicated as Z RC and Z RLC. There also three resistances given, the imedance of Z RC is equal to R for low frequencies. The imedance of Z RLC is R L for low frequencies and Z LC (equal to R in this examle) at resonance frequency. The imedance at resonance frequency of the Z RLC network is much higher than the Z RC network, this result in more gain. Figure 4..3 Imedances of ZRC and ZRLC It is shown that adding an inductor in arallel results in a eak in the imedance at resonance frequency. Adding the inductor in a two stage amlifier, result in the toology as shown in figure In ractice it is not ossible to use an inductor in an integrated circuit. The design of an inductor or a second order filter in an integrated circuit is described in chater 5. The derivation of the outut imedance is described in aendix A. The contribution of the Miller caacitance at high frequencies is ignored first. Figure 4..4 Two stage amlifier with a RLC filter The outut imedance is written by: RL RL R s s C Cm = m m Z out gm Cm R L gm s s L Cm ( C C ) R L L ( C C ) Remco Twelkemeijer Page 35 of 68

36 Chater 4 - Amlifier design From the outut imedance, it can be obtained that the formula contains two comlex eros and two real oles. The first ole R L /L is normally set to low frequencies, the second ole gm /Cm is normally set to high frequencies. In the revious chater it is determined that the oles should be as large as ossible. It seems that the miller caacitance can be removed in this amlifier, because the outut imedance to high frequencies becomes very large in resect to the other amlifier. The miller caacitance is not needed for stability issues if the amlifiers are in arallel. The outut imedance to low frequencies and to high frequencies is written by: Z out ( 0 ) = v Z out ( ) = gm gm R L Figure 4..5 AC resonse of the two stage amlifier with a RLC filter and without Cm For low frequencies, the outut imedance is high. It is not the value R which have the major influence to the outut imedance, but the resistance R L. This value is much lower than the resistance R and consequently the outut imedance is much higher. A tyical AC resonse is shown in figure It shows clearly that the outut imedance around resonance frequency is low and the hase shift lies between /- 90 degrees. The interest of this amlifier lies around resonate frequency. The general second order equation is written by: s s Q Z out = K s ( ) Note that /Q is the bandwidth of the filter. Comaring the general equation with the equation of the amlifier it can be obtained that if R is very large, the contribution of /(C C m )R to the bandwidth is very small in resect to R L /L. The smallest bandwidth and consequently the highest quality of the notch is achieved if R is very large. The bandwidth left is equal to R L /L which is equal to the first ole. The outut imedance can now be written as: s s Z out = K s ( ) Remco Twelkemeijer Page 36 of 68

37 Chater 4 - Amlifier design Where, the ole and resonance frequency is written by: RL = v L L C C ( ) m With this aroach, it can be obtained that the outut imedance can only becomes ero if the first ole is ero. If the bandwidth becomes very small, it is very difficult to design the resonance frequency exactly at a certain frequency without any variations. The minimum bandwidth deends directly on how accurate it is ossible to create the eak filter. Remco Twelkemeijer Page 37 of 68

38 Chater 4 - Amlifier design 4.3 Amlifiers in arallel A toology when the amlifiers are set in arallel is shown in figure The outut imedance is equal to the arallel imedance of the two amlifiers. The low frequency outut imedance should be as low as ossible. In chater 3 it is described that a notch results in an increment in the outut imedance, but this increment should be minimal. Figure 4.3. Two stage amlifier The outut imedance is written by: // ( s )( s )( s 3 ) Z arallel = Z Z notch = s s s s s s ( )( )( ) ( )( )( ) 3 The outut imedance is now written in a more comlex way which is not easy to solve. To see what haens, ideal simulations are done when only one comonent is changed. First the transconductance gm L is simulated with the values of 0μS, 35μS and 70μS. Values gm = 70 μs gm L = variable gm = gm L = -00μS R =.5MΩ C = F C m = 5F C LC = 6 F L = 4.mH R L = kω R LC = The only frequency region where the outut imedance of Z notch is lower than Z is around resonance frequency. Starting with a resonance frequency of MH, the two amlifiers can be comared searately. In figure 4.3.a, three curves with a notch are shown and the last one is the outut imedance of Z alone with gm is 94 μs. From the figure, it can be determined that increasing the transconductance of gm L, results in lower outut imedance around resonance frequency in comarison with Z. The frequency when the outut imedances are equal also increases with the transconductance. With gm L =0 μs, the outut imedance becomes equal around. MH. Increasing the gml to 70 μs it is increased to aroximately MH. 3 Remco Twelkemeijer Page 38 of 68

39 Chater 4 - Amlifier design Figure 4.3. a) Outut imedance amlifiers searately b) Outut imedance amlifiers in arallel Combining the two amlifiers result in the grah as shown in figure 4.3.b. In this grah, also four curves are shown. Three of them are the amlifier of figure The last one is for comarison uroses and is only a two stage amlifier with the same values of Z, excet gm which is 94 μs. As can be obtained from the figure, increasing the notch in this way results in larger eaking and the outut imedance is lower to high frequencies. The larger eaking results in a larger ringing time when a ste resonse is alied. The last ole is shifted to the right when the outut imedance is lower to high frequencies, because the final imedance is the same (aroximately /gm ). Using the determined values in the simulations results in the conclusion that the third ole can be aroximated by: gm gml 3 Cm The high frequency art of amlifiers is limited to be able to drive some caacitive load. Increasing the gain of the amlifier with the notch and keeing the third ole constant results in increasing of the low frequency outut imedance. Another otion is to increase the quality of the notch by decreasing the series resistance R L of the inductance. Simulating it with the values of 00Ω, kω and kω result in the grah as shown in figure 4.3.3a. In this grah the outut imedance is determined searately again. Also the outut imedance with gm = 94 μs is shown for comarison uroses. Values gm = 70 μs gm L = 70 μs gm = gm L = -00 μs R =.5MΩ C = F C m = 5 F C LC = 6 F L = 4.mH R L = variable R LC = The figure shows clearly that increasing the quality of the notch only results in lower outut imedance around MH, the high frequency behavior is the same for every quality factor. This suggests that this is also the case when the two amlifiers are combined. The grah of the amlifier is shown in figure 4.3.3b which shows that the outut imedance to high frequencies is the same. The only disadvantage is that the eak becomes larger when the notch increases, but this is also exected from chater 3. Figure a) Outut imedance amlifiers searately b) Outut imedance amlifiers in arallel Remco Twelkemeijer Page 39 of 68

40 Chater 4 - Amlifier design Another imortant art of the stability of the amlifier is the frequency resonse of the oen and closed loo gain. Alying a voltage source to the inut and load the outut with kω, results in the grah as shown in figure 4.3.4a. The gain is increased as exected at MH, which could lead to instability if the hase reaches -80 degrees. The resonance frequency results in a ositive hase shift to aroximately 60 degrees. The hase becomes -90 degrees after resonance frequency. This hase shift does not lead to instability. Designing the amlifier, some care should be taken at the increasing of the gain with resect to the UGB. Figure a) Oen loo gain b) Closed loo gain In figure 4.3.4b, the closed loo gain of the amlifier is shown. The gain is 0dB to MH and decreases after this frequency. There is also a notch in the gain, which corresonds to the resonance frequency when a ste resonse is alied to the amlifier. In chater 3.4 it is seen that the increasing in the outut imedance for low frequencies directly corresonds to the eaking frequency. Now, an aroach is done to determine the eaking frequency. The eaking frequency and the notch is shown in figure Keeing all the variables constant excet gm L gm, it is ossible to determine a formula of the eaking frequency. Assuming the last ole is constant, gm L gm is constant. Increasing the notch, results in decreasing gm and consequently decreasing the eaking frequency because the low frequency outut imedance is increased. With this aroach and in combination with the results of chater 3.4, the eaking frequency is written by: gm eak = notch gm gm L Figure Bode lot of the outut imedance with a eak and a notch This is valid under the assumtion that gm =gm L and the outut imedance of Z out is lower than Z around resonance frequency. With the results in this aragrah, a toology design can be done. Setting the first ero to 0 KH, the resonance frequency to MH, the last ole to 3 MH and the increasing in the outut imedance to 3dB, results in the following values. Remco Twelkemeijer Page 40 of 68

41 Chater 4 - Amlifier design Values of outut imedance with notch gm = 66,5 μs gm L = 7,5 μs gm = gm L = -00μS R =.65 MΩ C = F C m = 5F C LC = 6 F L = 4. mh R L = 500 Ω R LC = Values of outut imedance without notch gm = 94 μs gm = -00μS R =.65 MΩ C = F C m = 5 F The magnitude and hase result of the simulation with these values is shown in figure 4.3.6a and 4.3.6b. The magnitude behavior of a two stage amlifier is also shown (gm =94uS). The grah clearly shows the increase of 3dB to low frequencies. The eak is much smaller than the notch and the high frequency outut imedance and hase are the same for both amlifiers. The eaking frequency is 835 KH, calculating this frequency with the formula above result in 84 KH which seems to be a good aroximation. Also a ole and ero simulation is done, which shows that the calculated values are corresond to the simulated one. Figure a) Magnitude lot b) Phase lot Simulated oles and eros = kh j kh = kh = kh - j kh = kh j.000 MH 3 = MH 3 = kh j.000 MH Remco Twelkemeijer Page 4 of 68

42 Chater 4 - Amlifier design 4.4 Amlifiers in arallel with an outut stage The toology of the revious aragrah can be extended by a outut stage as shown in figure An extra stage in cascade results in an extra ole and ero. Consequently, there is more freedom of lacing the oles and eros. The advantage of a outut stage lies in large signal behavior and it is ossible to set the last ole to higher frequencies than 3 MH which result in lower outut imedance. Figure 4.4. Two stage arallel three stage amlifier imlementation with RLC filter Keeing the arallel stage the same as in the simulation of the revious aragrah, the ero is set to 6.6 MH and the last ole to 3 MH. This should give an imrovement of 6 db in the outut imedance to low frequencies and around resonance frequency. The low frequency outut imedance in resect to a three stage without a arallel notch should have a difference of 3 db again. The caacitances of the outut stage are assumed to be large by the arasitic caacitances of the outut transistors. With resect to the toology of the revious aragrah, the gain is imroved by a factor of two. A simulation of this toology and a three stage toology without the arallel notch stage is done and the results are shown in figure 4.4.a and 4.4.b. It shows at MH an imrovement of 4 db in the outut imedance with only a eak in the outut imedance of 9 db. For low frequencies, the outut imedance is increased with 3 db as exected and the high frequency outut imedance is the same. Values gm = 66.5 μs gm = 0 ms gm 3 = -00μS C = F C m = 5 F C = 40 F C m = 00 F R =.65 MΩ R = 00 Ω gm L = 7.5 μs gm L = 0 ms C LC = 6 F L = 4. mh R L = 500 Ω R LC = Remco Twelkemeijer Page 4 of 68

43 Chater 4 - Amlifier design Values for comarison gm = 94 μs gm = 0 ms gm 3 = -00μS C = F C m = 5 F C = 40 F C m = 00 F R =.65 MΩ R = 00 Ω Figure 4.4.: a) Magnitude lot b) Phase lot Simulated oles and eros = kh j kh = kh = kh - j kh = kh j.000 MH 3 = MH 3 = kh - j.000 MH 4 = MH 4 = MH 5 = MH At high frequencies (higher than 00 MH), the outut imedance shows a caacitive behavior. This corresonds to the fifth ole which is equal to 85 MH. The ole is introduced by the high frequency influence of the miller caacitance. During the derivation of the outut imedances in this reort, the high frequency influence of the miller caacitance is always neglected. In ractice the influence of this caacitance introduce an extra ole and could lead to instability []. The extra ole can be aroximated by the intersection between the resistive art and the miller caacitance. The resistive art is the outut imedance at high frequencies. The ole can be aroximated by: C C g = gm C m m3 miller m sc m C Cm A bode lot of the outut imedance including the extra ole is shown in figure 4.4.3a. In this lot, the first ole and the ole of the miller caacitance are far away. Starting at low frequencies, this results in a resistive art, inductive art, a resistive art again and finally caacitive resectively. Figure a) Bode lot when m >> b) Bode lot when m and are comlex Remco Twelkemeijer Page 43 of 68

44 Chater 4 - Amlifier design The ole m is an aroximation which is valid if the oles are far away from each other. If the become closer to each other, the comlete derivation should be done. The derivation is also done in []. The interest lies on the high frequencies and consequently on the two oles. The two oles are written by: / m gm gm = C ± gm gm C gm gm C Cm From the two oles, it can be obtained that if the value of gm is close to gm, comlex oles are formed. A bode lot of the outut imedance is shown in figure 4.4.3b when the oles are comlex. The comlex oles results in a eak in the outut imedance, which results in resonation if a ste resonse is alied. If gm becomes larger than gm the comlex oles becomes ositive, which causes the amlifier to be unstable. The comlexity deends on how close gm and gm are. The outut imedance for the two simulated amlifiers (with and without a notch in the outut imedance) is the same for high frequencies. This results also in the same last ole and the same requirements to maintain a stable system. Remco Twelkemeijer Page 44 of 68

45 Chater 5 - Filter imlementation 5 Filter imlementation In the revious chater, inductors are used in the amlifier. It is imossible to create inductors in an IC rocess, if the inductor should work around the frequencies of interest. This roblem can be solved by filters which should have the same behavior as the function described in chater 3 and 4. First, a start is made with continuous time filters to imlement a filter in the amlifier. Finally switched caacitor filters are described in resect with some relevant amlifier toologies. 5. Continuous time filters The easiest way is to create a eak filter is an active comonent which acts like an inductor, This could be done by a gyrator loaded by a caacitor. A gyrator is an active comonent which gyrate a current into a voltage and a voltage into a current. Loading the gyrator results in an inverse caacitor which is actually a inductor. The inductor can just be relaced by the gyrator in the amlifier toology of figure A traditional gyrator loaded with a caacitor is shown in figure 5... The imedance of the active inductor is written by: Z = s L s inductor L eq Figure 5.. Gyrator loaded with a caacitor using two OTAs The inductance L eq is written by [7]: C0 L eq = g g m m The simulated inductance deends ideally on the caacitance and the two transconductance of the two Oerational Transconductance Amlifiers (OTAs). Combining the simulated inductance into the formula for the resonance frequency: g mg m = = L C C C eq 0 In an IC rocess there is a large sread of the transconductance of the amlifiers. Also the absolute values of caacitances contain a large sread. The resonance frequency deends on the absolute values of the caacitances and transconductance. If the total absolute sread is aroximately 50%, the resonance frequency lies between 700 KH and. MH. The imedance of a eak filter with this variation is shown in figure 5..a. It shows clearly that the notch should be much more accurate. Remco Twelkemeijer Page 45 of 68

46 Chater 5 - Filter imlementation Figure 5.. a) Peak filter with /- 50% accuracy b) Comensated filter with /- 0% accurcacy To reach certain accuracy in the outut imedance at resonance frequency, the eak should be reduced. An examle of a comensated filter with 0 ercent accuracy is shown in figure 5..b. The difference in outut imedance at resonance frequency is aroximately 6 db. It shows also the original curve with an outut imedance of MΩ around resonance frequency. The difference between the curves is very large and does not satisfy the requirements of the filter. In literature several other toologies of time continuous bandwidth filters are described, but these filters requires comensation methods like external bias currents to satisfy a high accuracy [0, ]. It is also ossible to design a filter without a direct relation shi of an inductor, but these filters deend on some way on the transconductance [7]. In chater 4 it is determined that the quality of the filter should be as high as ossible. It seems that continuous time filters does not satisfy this requirement. Remco Twelkemeijer Page 46 of 68

47 Chater 5 - Filter imlementation 5. Switched caacitor filters Switched caacitor filters are normally used in IC design and are widely described in literature. The accuracy of the filters in an IC rocess is much higher than continuous time filters. Switched caacitor filters can be designed in several ways. The first one which is described act like a inductor. Another ossibility is to design a Lalace or Z function which can be created in some way. A filter is described when only one caacitor is used to create the notch in the outut imedance. Finally, the filter is extended with more caacitors to reach a higher quality factor. 5.. Comonent simulation In comonent simulation, the inductors and resistances are simulated by active or assive SC networks. The design basically consists of relacing the inductors and resistors by these simulated networks. An examle of the simulation of an inductor is shown in figure 5... The equivalent inductance is written by [6]: T L eq = 6 C Figure 5.. Simulation of an inductor In comarison with the active gyrator in chater 5., the caacitance is now in the denominator. Placing a caacitance C 3 between V i and ground, a arallel combination of an inductor and caacitance is created. This results in a relative deendence of the caacitances in the resonance frequency which results in a much better accuracy in an IC-rocess. The resonance frequency can now be written as: 4 C = = LC T C 3 If the samling frequency is 0 MH and the caacitance C is F, the caacitance C 3 should be 40.5F to get a resonance frequency of MH. The caacitance of C is also F, but this caacitance does not influence the resonance frequency. Remco Twelkemeijer Page 47 of 68

48 Chater 5 - Filter imlementation The inductance is now created by an active comonent, an oerational amlifier. Ideally, the created inductance is very accurate, but it deends on the oen loo gain of the amlifier in ractice. A simulation is done when the oen loo gain of the amlifier is variable as shown in figure 5... It shows that an oen loo gain of 00 does not results in a resonance at all, but if the oen loo gain is 000, the resonance is aroximately 0dB which aroximately the minimum value which is needed. Figure 5.. Magnitude resonse deending on the oen loo gain of the amlifier 5.. Digital second order Filter A disadvantage of the revious design is that it is sensitive to the arasites of the caacitances. To solve this roblem, several stray insensitive toologies are roosed [6]. A toology of a stray insensitive circuit could be a biquad. With one biquad it is ossible to create every second order Z function. A Z function can be created by a Lalace function which is transosed to the Z domain. Starting with a band ass filter: s H ( s) = s s Q The function can be transosed to the Z-domain. To do this, several methods can be used, like imulse invariance method, matched- transformation, Backward and Forward Euler aroximation, Lossless Discrete Integrator (LDI) transformation or the bilinear transformation [6]. The Imulse Invariant transformation retains the exact imulse resonse of the analog system. It is desirable in cases where the imulse time resonse of the filter is imortant to be retained and is more comlex to calculate. The Matched Z transform emloys a basic simle method of translating analog oles and eros to digital oles and eros on the cascaded continuous transfer function. In the backward Euler Aroximation, the derivative is aroximated by the first backward difference. With forward Euler aroximation, the derivative is aroximated by the forward difference. The forward and backward Euler aroximations does not maed the imaginary axis in the s-lane into the unit circle. This means, a stable analog function with comlex oles or eros does not result in stable samled data transfer functions when Euler aroximation is used. Remco Twelkemeijer Page 48 of 68

49 Chater 5 - Filter imlementation Remco Twelkemeijer Page 49 of 68 The Bilinear uses traeoidal integration to imlement the digital Z transform. It is usually the most accurate digital imlementation for filtering a continuous variable. However, significant waring of the frequency sectrum occurs. The LDI transformation is, just as the bilinear transformation, derived from traeoidal integration method. But the LDI transformation uses midoint integration. However, significant waring of the frequency sectrum occurs. It is not ossible to do the backward and forward Euler aroximation, because comlex oles are needed. The bilinear and LDI transformation seems to be a good otion. To solve the roblem of waring of the frequency, re-waring techniques are available for the bilinear and LDI transformation. In this aragrah, the bilinear transformation is used. Calculation of the re-wared frequency and quality factor [6]: ( ) ( ) = = fs Q fs Q fs Q f fs s tan tan tan \ \ \ \ \ Setting the resonance frequency to MH and the quality of the filter to 0, a re-wared frequency of,034 MH and re-wared quality factor of 8 is calculated. Next, the filter can be transosed by [6]: ) ( = = u v q r f H H s In aendix B, the formulas for the variables are given. Solving the equation with a gain of 0 at resonance frequency, the formula results in: ) ( = H Figure 5..3 Frequency resonse of the Z-Filter

50 Chater 5 - Filter imlementation The frequency resonse of the Z-filter is shown in figure The grah shows clearly the resonance frequency of MH and the eak is aroximately 0. Solving the caacitances in a biquad, results in the conclusion that the total caacitance is higher than 00 times the unit caacitances. This is much too high in comarison with the sace which is needed for the caacitances Peak filter using one caacitor In literature, a simle but very good SC notch filter is described []. This filter is shown in figure 5..4a. The notch occurs exactly at samling frequency. The voltage at V could never contain this frequency because the inut voltage at V is every time the same. With this configuration it is very easy to control the notch frequency and the caacitance sie is not very imortant. The transient resonse when the inut frequency is equal to the samling frequency is shown in figure 5..5a. Figure 5..4 (a) SC notch filter; (b) SC eak filter. Changing the configuration into a eriodically inverse caacitor as shown in figure 5..4b, a eak filter is created. In this case, the inut is a current source at V, V is grounded and the eak occurs at samling frequency again. The idea is the same; the current over the ositive half of the sine wave is integrated by the caacitor in the even hase. In the odd hase, the caacitor is inverted, and the voltage at V becomes negative. Together with the negative half of the sine wave, the caacitor becomes more negative. An ideal simulation is done and the result is shown in figure 5..5b. The grah shows an increment in the outut voltage deending on the time. In comarison with the SC notch filter, this system has some disadvantages. The eak occurs exactly at Nyquist frequency and odd Harmonics of the samling frequency will be transosed to the samling frequency. Figure 5..5 a) Transient resonse SC notch filter b) Transient resonse SC eak filter Remco Twelkemeijer Page 50 of 68

51 Chater 5 - Filter imlementation A simle amlifier toology which contains the SC eak filter of figure 5..4b is shown in figure The resistor R s simulates the switching resistance and is assumed to be low. Now it is ossible to determine if this eak filter also generates a notch in the outut imedance. Alying some current I out to the amlifier, the voltage V is known because gm should drain this current. Switching the caacitance to the oosite direction results ideally in loading the caacitance with the required voltage directly. Consequently, the voltage over the caacitance is always known, and could not create a notch in the outut imedance. The behavior of the SC eak filter can be described as a low ass filter in this situation [6]. This is exactly what haens if the outut imedance is simulated in this situation as shown in figure 5..7a. The switching frequency is MH and the outut imedance increases by the caacitance behavior for high frequencies. Figure 5..6 Amlifier with SC eak filter. If the switching resistance is assumed to be much larger, for examle 00kΩ and the caacitance is 0F, a low ass configuration with a -3dB frequency of 60kH is created. The voltage over the caacitance cannot be change with the samling seed of MH and the voltage is ket constant when a signal of MH is alied. A simulation is done and the result is shown in figure 5..7b. It shows clearly a notch of several db s in the outut imedance around resonance frequency. Also some odd harmonics are filtered, but much the notches of the harmonics are much smaller. Figure 5..7 a) PAC Simulation with Rs=0Ω b) PAC Simulation with Rs=00kΩ The amlifier can be estimated as a bit adder or subtractor in this situation. The voltage deends on the inut amlitude. In the best case, the voltage over the caacitor becomes roortional to the average voltage of the sine wave over a half eriod. π Asin V caacitor Sin () t dt A sin π = π 0 Remco Twelkemeijer Page 5 of 68

52 Chater 5 - Filter imlementation The outut voltage of the amlifier is shown in figure There are two curves shown, the first one is the alied sine wave when there is only a resistor between the two stages of the amlifier. The second one is the amlifier with the SC eak filter; the sine wave is divided into two sections which results into a lower outut voltage and consequently lower outut imedance. Figure 5..8 Examle of the outut voltage of an amlifier with a SC eak filter. Integration of the absolute value of the sine wave which is left, results in the conclusion that the outut imedance can be decreased by aroximately 7.5 db at maximum in this situation. Integrating the sine wave with a hase difference of 90 degrees, results in ero and there is no imrovement at all. Consequently, if there is only one SC eak filter, the hase shift of the sine wave in relation with the clock hase should be ero to retain the best filter behavior. To avoid this roblem, two amlifiers are needed in arallel, when the second one has a SC eak filter with 90 degrees hase shift. With the same method as above it is ossible to integrate the sine wave deending on a certain hase difference: V V SC SC A π A π ϕ π sin ϕ Sin 3 ϕ π sin ϕ π Sin ( x) Cos dx = ( ϕ) Cos( ϕ π ) π A ( x) dx = Asin sin 3 Cosϕ π Cosϕ π π V SC V SC Φ Voltage over the first caacitor. Voltage over the second caacitor Phase difference in radials Now a bit adder or subtractor is created, with 4 different hases as shown in figure There are two curves shown, the first one is the sine wave for comarison with a hase shift of 45 degrees in resect to the two clocks. The second one is the outut voltage of the amlifier which has four non eriodic arts. Figure 5..9 Examle of the outut voltage using two amlifiers in arallel with a SC eak filter. Remco Twelkemeijer Page 5 of 68

53 Chater 5 - Filter imlementation With two SC eak filters and a variable hase, the minimum imrovement factor is not ero anymore. The imrovement factor deends on the hase difference. Integrating the outut function deending on the hase shift over the time, the benefit of this filter is calculated and shown is in figure The hase shift could only difference from 0 to 90 degrees, because the hase of the two eak filters has a hase difference of 90 degrees and more hase difference results in just a minus sign. The maximum benefit is reached at 45 degrees hase difference and is 9dB at maximum. The minimum benefit of the SC eak filter is aroximately 7.5 db, which is the same for only one amlifier with a SC eak filter with a hase shift of ero degrees. Figure 5..0 Imrovement when two SC eak filters are set in arallel with 90 degrees hase shift Peak filter using eight equal caacitors Extending the eak filter of the revious aragrah with more caacitances, a high Q filter can be created [3, 4]. Now the caacitances are not eriodically inverted anymore, but are set in arallel with a switch in series. Only one of the N switches is closed at a time. The roosed filter in [3, 4] is shown in figure 5.. The filter has eight caacitances; this is a trade of between comlexities in the control signals versus erformance. The quality factor is written in a simlified form as: Q = π N R C F O Figure 5.. High Q eak filter The formula of the quality shows that increasing the caacitance, the resistance and the number of caacitors results in increasing the quality factor. The control signals, x to x8 are shown in figure 5... Only one of the eight signals is one at a time. The signal x is only one between 0 and 0. us. A whole cycle of x to x8 (To) is equal to the resonance frequency of the filter. In this case, the resonance frequency is set to MH. The design of the control signals is not described here, but it could be done with a ring oscillator with exclusive or gates. Remco Twelkemeijer Page 53 of 68

54 Chater 5 - Filter imlementation Figure 5.. Control signals x to x8 The frequency resonse of the filter is shown in figure 5..3a. The magnitude resonse shows a eak at MH and the harmonics. Also a low as filter behavior is shown. The corresonding hase shift of the filter is also shown in figure 5..3a. The hase characteristic shows that the filter should be able to imlement in an amlifier. Figure 5..3 a) Frequency resonse filter b) Frequency resonse amlifier In figure 5..4, an amlifier toology is shown of the filter, also a caacitance C is introduced which simulate the gate source caacitances. Placing the amlifier in the same toology as in aragrah 4.4, the frequency resonse as shown in figure 5..3b can be obtained. The outut imedance has different notches of aroximately 0, 0 and 6 db to the frequencies of MH, MH and 3 MH resectively. Figure 5..4 Amlifier toology using the eak filter Remco Twelkemeijer Page 54 of 68

55 Chater 6 - Simulations 6 Simulations The revious chaters describe the theoretical limitations of the amlifier, some amlifier toologies and methods to imlement the filters in an IC rocess. With these results, simulations can be done with the amlifier in combination with a class D amlifier. First, the amlifier structure used in the simulations is described. There are several filters described in chater 5 which results in different amlifiers. The amlifiers used in the simulation are described next. With the structure and the amlifiers known, some simulations are done. The results of the simulations are described finally. 6. Amlifier structure A simle arallel combination of a class AB and a class D amlifier is shown in the introduction and reeated in figure 6... The class D amlifier in this structure is defined by the current delivered by the class AB amlifier. This is time indeendent, the class D switching frequency is variable which could be for examle 40 KH or MH. The filters which are designed in the revious chater are not variable. Consequently, the class D amlifier should have a fixed switching frequency to be able to remove the switching rile. The class AB amlifier should have a voltage source behavior and the class D amlifier should have a current source behavior, because it is difficult to have two voltage sources in arallel. Figure 6.. Parallel combination of a Class AB and Class D amlifier The switching frequency of the class D amlifier can be fixed if a ulse with modulator (PWM) is used. In a simle version, it is a comarator which comares a square wave with the outut current of the class AB amlifier. Also if the outut current is higher than the maximum value of the square wave, the class D amlifier should switch to satisfy a constant switching frequency. The class D amlifier is imlemented as ideal switches with an inductor at the outut to satisfy a current behavior. The linearity or the value of the inductance is not very imortant, because the accuracy is delivered by the class AB amlifier. The class AB amlifiers used in the simulations are described in the next aragrah. The amlifier structure which will be used in the simulations is shown in figure 6... The structure contains a PWM, a class AB and class D amlifier, an inductor and a loudseaker. The switching frequency of the PWM is MH, the inductance of the inductor (L ) is 0,8mH and the loudseaker imedance is assumed to be real and the resistance is 4Ω. The threshold of the PWM is set to 95mA. This means, if the amlifier delivers more than 95mA, the on time of the class D amlifier is maximal and satisfying a switching frequency of MH. Remco Twelkemeijer Page 55 of 68

56 Chater 6 - Simulations Figure 6.. Toology used in the simulations Remco Twelkemeijer Page 56 of 68

57 Chater 6 - Simulations 6. Different amlifiers Before the simulations can be done, the different amlifiers should be designed roerly to be able to comare the results. It is very easy to get wrong information from the simulations. There are three major arameters, the ower consumtion, the stability and the distortion. It is not ossible to kee all the three arameters equal. In chater three, the major ower consumtion (outut stage) and the stability (hase) are ket equal. This results in an increasing in the distortion deending on the amlifier design. In the simulations, the same criteria are used. This criteria is extended for one design, the hase shift should be 60 degrees or lower after a certain frequency (for examle 5 MH). The low frequency outut imedance of the amlifiers with a notch in the outut imedance is increased by 3dB. Consequently, the increasing in the distortion can also be aroximated by 3dB. The amlifiers can also be comared in another way. If the stability and the distortion should be equal, the ower consumtion is increased by 3dB. The toology of the amlifier is shown in figure 6... There are five different amlifiers containing different filters are designed for the simulations. The filter block in the figure contains a different filter for every amlifier. The first one is a standard three stage amlifier without a notch in the outut imedance for comarison uroses. The second one has a RLC filter with an ideal inductor. Continuous time filters does not have enough accuracy and are not simulated. The third circuit contains an inductor created by a switched caacitor filter. This filter could have some influence from arasites. The two last amlifiers have a eak filter using one and eight caacitors resectively. The amlifiers are numbered from one to five and the numbering will be used in the rest of this chater. Different filters used in the amlifier described in ) None (aragrah 4.4.4) ) Ideal inductor (aragrah 4.4.4) 3) Inductor created by a switched caacitor circuit (aragrah 5..) 4) Peak filter using one caacitor (aragrah 5..3) 5) Peak filter using eight equal caacitors (aragrah 5..4) Figure 6.. Two stage arallel three stage amlifier imlementation with a filter Remco Twelkemeijer Page 57 of 68

58 Chater 6 - Simulations The arameters of the comonents and filters are given in aendix C. Also the characteristics of the outut imedances are shown in the aendix. It shows clearly that the stability and ower criteria are satisfied at the cost of an increase of 3 db in the outut imedance for low frequencies. Summaried, the eak of amlifiers and 3 is aroximately 0dB, including the loss of 3dB for low frequencies. The notch as an amlitude comared with amlifier of: Amlifier Filter Notch None - Ideal inductor MH 3 SC inductor 9 MH 4 One caacitor MH 5 Eight caacitors 7 MH, MH, 3.5 3MH,.3 4 MH, MH. Remco Twelkemeijer Page 58 of 68

59 Chater 6 - Simulations 6.3 Results There are five different amlifiers which could be used in the simulation. The toology structure described in chater 6. is used. The main interest lies on the voltage rile over the loudseaker. The amlifiers one to five designed in the revious aragrah are used in structure called,,3,4 and 5 resectively. The four structures,3,4 and 5 should contain a lower voltage rile in comared with structure. The transient resonse over the loudseaker using the five structures is shown in figure The numbers of the structures are shown close to the transient resonse of the voltage over the loudseaker of the structure. There is no inut signal alied and the switching frequency is equal to MH for all the structures. The switching rile of the switched caacitor filters are clearly shown in curve 3, 4 and 5. The voltage of structure 4 seems to be relatively large in resect to structure. This corresonds to the outut imedance as shown in the revious aragrah. Figure 6.3. Transient resonse of the simulation with the 5 different amlifiers The voltage can be derived by calculating taking the root mean square (RMS) value of transient resonse. Comaring the RMS values of structures, 3, 4 and 5 with structure the imrovement factor can be calculated. The simulated RMS values and imrovement factors are: Structure Filter RMS value Imrovement factor None 3,35 mv Ideal inductor 0,9 mv,3db 3 SC inductor 0,98 mv 0,7dB 4 One caacitor 3,55 mv -0,5dB 5 Eight caacitors,78 mv 3,9dB The imrovement factors are much less than the factor of the notch. The reason for this lies in the square wave in the transient resonse of structure. A square wave with amlitude one contains a RMS value of aroximately 0,7 of the first harmonic. The other energy lies in the other harmonics. The logarithmic of -0,7 is aroximately 0,7dB which corresonds to the imrovement factor as shown in the table. A discrete Fourier transform (DFT) is done to determine the energy deending on the frequency. The results with a DFT of structure is shown in figure 6.3. a. It shows a large eak at MH and some smaller eaks in the odd harmonics. In figure 6.3.b, the DFT of the second structure is shown. The amlitude aroximately is 0 db smaller at MH and the harmonics seems to be the same. Remco Twelkemeijer Page 59 of 68

60 Chater 6 - Simulations Figure 6.3. a) DFT of the resonse of amlifier b) DFT of the resonse of amlifier The switched caacitor version of the inductor (structure 3) shows a small increase in the imrovement factor in comarison with structure. The contribution is mostly introduced by the influence of the switching eriods of the filter. Figure DFT of the resonse of amlifier 5 The result of structure 5 is not very good, the imrovement is only 4 db but the AC resonse of the corresonding amlifier is very good. It seems that this also come from the switching rile of the filter itself. To check this, also a DFT of the resonse of structure 5 is created and is shown in figure The results of the DFT deending on the frequency are shown in a table: Frequency Amlifier Amlifier Imrovement MH 4,6 mv mv 7, db 3 MH mv 0,6 mv 4,4 db 5 MH 0,43 mv 0,4 mv 0,6 db 7 MH 0,3 mv 0,6 mv -6 db 9 MH 0, mv 0,6 mv -9,5 db The table roves that there is some imrovement for the frequencies of, 3 and 5 MH. Those factors corresond to the AC results of the amlifier as shown in aragrah 6.. The imrovement factor for the frequencies of 7 and 9 MH is negative which means a reduction. The increment at those frequencies is the result from the switching residues of the filter. This increment results in the imrovement factor of 4 db. Remco Twelkemeijer Page 60 of 68

61 Chater 7 - Conclusion 7 Conclusion The outut imedance is investigated with a ole ero analysis. This aroach shows that a notch in the outut imedance results in a higher outut imedance for low frequencies. Increasing the outut imedance means increasing the distortion. The ower in the amlifier should be increased if the same distortion should be reached. Also a eak in the outut imedance is introduced, but the eak could be smaller than the notch. The quality factor of the notch should be as large as ossible and deends on the accuracy of the filter. With these limitations, an ideal amlifier design is made. This means, ideal comonents are used, like inductors and transconductances. The first one contains two, two stage amlifiers in arallel. The final toology contains an extra outut stage which could be beneficial for the ower consumtion or large signal behavior. Inductors are not able to imlement in an IC rocess and some filters have been investigated. It is not ossible to imlement continuous time filters, because they have a very bad accuracy. The accuracy of switched caacitor filters is much better and three amlifiers have been designed. The simulations of the amlifiers in combination with the class D amlifier in arallel shows a decreasing in the outut imedance of,3db at the cost of 3dB extra ower with the same stability and distortion. This result is obtained by an ideal amlifier with ideal comonents. The switched caacitor filter contributes some extra distortion by the switching of the caacitances itself. The maximum reached imrovement with a amlifier containing a switched caacitor filter is aroximately 0,5 db. This factor corresonds to the exected energy in the first harmonic of a squire wave. A real design of the amlifier will normally results in less imrovement. The advantages of removing aroximately db energy of the switching rile go at the cost of extra energy in the outut stage to maintain the same distortion, extra sace requirements on the chi by the caacitances and extra amlifier stages in arallel. It is ossible to create more notches in the outut imedance, but this requires extra ower consumtion to kee the distortion the same. The benefit of an extra notch is not very large, because most of the energy of a square wave lies in the first harmonic. It seems that the advantage of db not comensate the disadvantages. Under these conditions, it is not beneficial to design a real amlifier. Remco Twelkemeijer Page 6 of 68

62 Conclusion Remco Twelkemeijer Page 6 of 68

63 References References [] Zee, R.A.R. van der, High efficiency audio ower ams, Ph.D Thesis, University of Twente, ISBN , May 999. [] Schaink T., Outut imedance and stability of audio ower amlifiers, MSc. Reort, Universiteit of Twente, November 006. [3] Zee, R.A.R. van der, Heeswijk, R, Frequency Comensation of an SOI Biolar-CMOS- DMOS Car Audio PA", van. IEEE International Solid-State Circuits Conference (ISSCC), San Francisco, February 6-8, 006. ISSCC Digest, [4] Hamble A. R., Electronics, Prentica-Hall International, Inc., ISBN , 000 [5] Zee, R.A.R. van der, Frekwentiecomensatie van CMOS oams met klasse AB eindtra onder ware caacitieve en resistieve belasting, MSc. Reort, Universiteit of Twente, June 994 [6] Mohan P.V.A., Switched Caacitor Filters : theory, analysis, and design, Prentice Hall International (UK) Ltd, ISBN , 995. [7] Theodore D., Continuous-Time Active Filter Design, CRC Press LLC, ISBN , 999 [8] Raavi B., Design of Analog CMOS integrated circuits, Tata McGraw-Hill, Boston, ISBN , 00. [9] Schrage J., Regeltechniek voor HTO, HBuitgevers, Baarn, ISBN , 000. [0] Maruyama Y. et all, Simle CMOS /jl inductive transconductance amlifier, AP- ASIC, 00. [] Barthélemy H. and Fillaud M., A CMOS Current-Mode Band-Pass Filter with Small Chi Area, 00. [] Burt R. and Zhang J., A Microower Choer-Stabilied Oerational Amlifier Using a SC Notch Filter With Synchronous Integration Inside the Continuous-Time Signal Path, IEEE Journal of solid-state circuits, vol 4, no, December 006. [3] Paillot J.M. et all, Switched Caacitor Bandass Filter Tuned by Ring VCO in CMOS 0.35um, IEEE Radio Frequency Integrated Circuits Symosium, 003 [4] Paillot J.M. et all, Fully Integrated High-Q Switched Caacitor Bandass filter with Center Frequency and Bandwidth Tuning, IEEE Radio Frequency Integrated Circuits Symosium, 007 Remco Twelkemeijer Page 63 of 68

64 References Remco Twelkemeijer Page 64 of 68

65 Aendices A - Derivation of the outut imedance Remco Twelkemeijer Page 65 of 68 Aendices A. Derivation of the outut imedance In this aendix, the outut imedance of the amlifier as described in chater 4. is derived. During the derivations in this aendix, it is assumed that the miller caacitance will manifest itself at very high frequencies and therefore it is ignored. Figure A. Amlifier with RLC filter ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) = = = = = = = // // m L m L L m m m out m L L L m out out out out L L L L RLC m RLC m out RLC m out out C gm s L R s C C L R R L R R C C s s C C C gm Z sc gm R sl R R R R Cm C L s C C L R s R gm I V Z gm V I R R R R C L s L C R s R sl R C R L R Z sc Z sc gm V V Z sc V V V gm V

66 Aendices B - Coefficients of the Z filter Remco Twelkemeijer Page 66 of 68 B. Coefficients of the Z filter Starting with a second order Lalace band ass filter: ) ( Q s s s s H = With three variables: Resonance frequency Q Quality factor k Gain at resonance frequency (not shown in the Lalace function) Should be transosed by: ) ( = = u v q r f H H s First the frequency should be re-wared: ( ) ( ) = = fs Q fs Q fs Q f fs s tan tan tan \ \ \ \ \ Next, the variables u and v can be calculated by: = = \ \ \ \ \ \ \ \ \ fs Q fs Q fs v fs fs Q fs u The gain at resonance frequency is written by: v r k = The constant q is ero, and solving the variables and q results in: ( ) r q v k = = = 0 0.5

67 Aendices C - Design of the five amlifiers C. Design of the five amlifiers The amlifiers are designed under the assumtion that the stability and the major ower consumtion are equal. First the values of all the comonents of the amlifiers are given and finally the frequency resonses of the amlifiers are given. Different filters used in the amlifier ) None ) Ideal inductor 3) Inductor created by a switched caacitor circuit 4) Peak filter using one caacitor 5) Peak filter using eight equal caacitors Values amlifier gm = 94 μs gm = 0 ms gm 3 = -00μS C = F C m = 5 F C = 40 F C m = 00 F R =.65 MΩ R = 00 Ω Values amlifier gm = 66.5 μs gm = 0 ms gm 3 = -00μS C = F C m = 5 F C = 40 F C m = 00 F R =.65 MΩ R = 00 Ω Filter stage gm L = 7.5 μs gm L = 0 ms C LC = 6 F L = 4. mh R L = 500 Ω R LC = Values amlifier 3 gm = 66.5 μs gm = 0 ms gm 3 = -00μS C = F C m = 5 F C = 40 F C m = 00 F R =.65 MΩ R = 00 Ω Filter stage gm L = 00 μs gm L = 0 ms C LC = 40.5 F L eq = 65 μh R L = 5 kω R LC = Values amlifier 4 gm = 66.5 μs gm = 0 ms gm 3 = -00μS C = F C m = 5 F C = 40 F C m = 00 F R =.65 MΩ R = 00 Ω Filter stage gm L = 35 μs gm L = 0 ms C LC = F C filter = 50 F R switch = 0 kω R LC =.65 MΩ Values amlifier 5 gm = 66.5 μs gm = 0 ms gm 3 = -00μS C = F C m = 5 F C = 40 F C m = 00 F R =.65 MΩ R = 00 Ω Filter stage gm L = 7,5 μs gm L = 0 ms C LC = F C filter = 5 F R switch = 0 Ω R LC =.65 MΩ Remco Twelkemeijer Page 67 of 68

68 Aendices C - Design of the five amlifiers Figure C. Outut imedance characteristics of amlifier, and 3 Figure C. Outut imedance characteristics of amlifier and 4 Figure C.3 Outut imedance characteristics of amlifier and 5 Remco Twelkemeijer Page 68 of 68