Two integrator loop quadrature oscillators: A review

Transcription

1 Journal of dvanced Research (0) 4, Cairo University Journal of dvanced Research REVIEW Two integrator loop quadrature oscillators: review hmed M. Soliman * Electronics and Communication Engineering Department, Faculty of Engineering, Cairo University, iza, Egypt Received 8 January 0; revised 7 March 0; accepted 7 March 0 vailable online 9 pril 0 KEWRDS Two integrator loop oscillators; p mp; perational transresistance amplifier; Unity gain cells; Current conveyor; Current feedback operational amplifier bstract review of the two integrator loop oscillator circuits providing two quadrature sinusoidal output voltages is given. ll the circuits considered employ the minimum number of capacitors namely two except one circuit which uses three capacitors. The circuits considered are classified to four different classes. The first class includes floating capacitors and floating resistors and the active building blocks realizing these circuits are the p mp or the TR. The second class employs grounded capacitors and includes floating resistors and the active building blocks realizing these circuits are the DCVC or the unity gain cells or the CF. The third class employs grounded capacitors and grounded resistors and the active building blocks realizing these circuits are the. The fourth class employs grounded capacitors and no resistors and the active building blocks realizing these circuits are the T. Transformation methods showing the generation of different classes from each other is given in details and this is one of the main objectives of this paper. ª 0 Cairo University. Production and hosting by Elsevier B.V. ll rights reserved. Introduction Sinusoidal oscillators are basic building blocks in active circuits and have several applications in electronics and communication circuits [ ]. The oscillator circuits are available in different forms which includes Colpitts oscillator, Hartely oscillator, two integrator loop oscillators, Wien bridge oscillator and * Tel.: / address: asoliman@ieee.org ª 0 Cairo University. Production and hosting by Elsevier B.V. ll rights reserved. Peer review under responsibility of Cairo University. Production and hosting by Elsevier phase shift oscillators [ ]. This paper is limited to the two integrator loop oscillators realized using different active building blocks. The first active building block used in this paper is the perational mplifier (p mp) and four circuits are given [ 5]. Next the perational Transresistance mplifier (TR) [ 8] is used as the active element and two circuits are reviewed both employ two floating capacitors and four floating resistors [9]. Both of the p mp and the TR oscillators represent the first class of oscillators considered in this paper. The second class of oscillator circuit considered employ grounded capacitors [0] and includes floating resistors. The active element employed is the Differential Current Voltage Conveyor (DCVC) [] also known as the Current Differencing Buffered mplifier (CDB) [], the unity gain cells and the Current Feedback perational mplifier (CF) []. The third class of oscillator circuit considered employ grounded capacitors and grounded resistors. The active element employed is the single output current conveyor () [4] or the two output

2 .M. Soliman. The fourth class considered employs grounded capacitors and no resistor, the active element employed is the transconductance amplifier (T) [5]. Transformation methods using Nodal dmittance Matrix (NM) expansion to show how the CF oscillator circuits leads to the generation of oscillators are given. p mp based oscillators The analysis in this section is based on assuming ideal p mp having infinite gain which forces the two input voltages to be equal. The two input currents are zero due to the very high input impedance at both input nodes. Four alternative circuits are given in Fig. a d and will be summarized next. (i) Three p mp oscillator circuits 4 The first circuit given in this paper is the three p mp two integrator loop circuit shown in Fig. a. It employs three single input p mps, two floating capacitors and four floating resistors [], its NM equation is given by: ¼ sc ðþ s The radian frequency of oscillation is given by: sffiffiffiffiffiffiffiffiffiffiffiffi x o ¼ ðþ This circuit has the disadvantage of having no independent control on the oscillation condition. modification to the circuit of Fig. a to provide control on the oscillation condition was given in Refs. [4,5]. Fig. b represents the recently reported three p mp oscillator circuit with independent control on both the condition of oscillation and on the frequency of oscillation, its NM equation is given by: ¼ sc þ 4 ðþ s Fig. b Modified three p mp oscillator circuit [5]. 5 C 4 Fig. c Three p mp oscillator circuit []. / Fig. a Three p mp oscillator circuit [ 4]. Fig. d Two p mp oscillator circuit []. It should be noted that the signs of and are opposite to their signs in Eq. ().The characteristic equation in this case is given by:

3 Quadrature scillators Table Summary of the class I oscillator circuits. Circuit figure number ctive element Number of C Number of Condition of oscillation x q ffiffiffiffiffiffiffiffiffi Ref. (a) p mp Floating 4 Floating None [ 4] (b) p mp Floating Floating 4 = [5] 45 (c) p mp Floating 5 Floating C C C [] (d) p mp rounded Floating rounded 4 Floating = [] (a),(b) TR Floating 4 Floating 4 = [9] s þ s ð 4 Þþ ¼ 0 ð4þ The condition of oscillation is given by: ¼ 4 ð5þ The condition of oscillation is controlled by or 4 without affecting the frequency of oscillation which is given by Eq. () and is independently controlled by or without affecting the condition of oscillation. The third circuit is shown in Fig. c which was recently reported by Horng []. This circuit is non-canonic as it employs three capacitors. The condition of oscillation and the radian frequency of oscillation are summarized in Table. The condition of oscillation is controlled by or 4 or 5 without affecting the frequency of oscillation. There is no independent control on the frequency of oscillation. nother non-canonic quadrature oscillator circuit which uses five capacitors was also introduced by Horng []. (ii) Two p mp oscillator circuit The first circuit shown in Fig. a is the two TR two integrator loop originally introduced in [9]. It uses two floating capacitors and four floating resistors; its NM equation is given by: ¼ sc þ 4 ð0þ s The characteristic equation in this case is given by: s þ s ð 4 Þþ ¼ 0 ðþ The condition of oscillation is given by: ¼ 4 ðþ The condition of oscillation is controlled by or 4 without affecting the frequency of oscillation and the radian frequency of oscillation is given by Eq. () and is controlled by or without affecting the condition of oscillation. The fourth circuit shown in Fig. d is the two p mp two integrator loop circuit based on using Deboo integrator []. It uses one grounded capacitor, one floating capacitor, one grounded resistor and four floating resistors []; its NM equation is given by: ¼ sc þ ðþ s The characteristic equation in this case is given by: s þ s ð Þþ ¼ 0 ð7þ The condition of oscillation is given by: ¼ ð8þ sffiffiffiffiffiffiffiffiffiffiffiffi x o ¼ ð9þ The condition of oscillation is controlled by without affecting the frequency of oscillation and the frequency of oscillation is controlled by without affecting the condition of oscillation. TR based oscillators The second active building block considered in this paper is the TR which in the ideal case has the two inputs virtually grounded leading to circuits that are insensitive to stray capacitances. lso ideally the transresistance gain, R m approaches infinity which forces the input currents to be equal. R m 4 R m Fig. a Two TR oscillator circuit I [9]. R m 4 Fig. b Two TR oscillator circuit II [9]. R m

4 4.M. Soliman The second circuit which employs the same number of elements is shown in Fig. b [9]; its NM equation is given by: ¼ sc ðþ s þ 4 The condition of oscillation and the frequency of oscillation are the same as in the previous circuit and are summarized in Table. The oscillators reported in the next sections of the paper employ grounded capacitors which provide advantages in integrated circuits [0]. DCVC based oscillators DCVC DCVC The DCVC was introduced in [] as a four terminal building block and defined by: Vx Ix Vx 7 4 Iz 5 ¼ Ix 7 7 ð4þ Vz 5 Vo Io The currents I and I are pointing inwards at nodes and whereas the current I is out words from node as shown in Fig. a. The DCVC was independently introduced and defined as a CDB by car and zoguz []. The first oscillator circuit considered in this section is shown in Fig. a which uses two DCVC, two grounded capacitors and four virtually grounded resistors and was introduced by Horng [7]; its NM equation is given by: ¼ sc þ 4 ð5þ s The condition of oscillation is controlled by varying or and the frequency of oscillation is controlled by varying or 4 without affecting the condition of oscillation. The second oscillator circuit considered in this section is shown in Fig. b which uses two DCVC, two grounded capacitors and three virtually grounded resistors was introduced [8,9]; its NM equation is given by: ¼ sc þ ðþ s The condition of oscillation is controlled by varying without affecting the frequency of oscillation and the frequency Fig. b Two DCVC oscillator circuit II [8,9]. of oscillation is controlled by varying without affecting the condition of oscillation. Unity gain cells based oscillators The four types of the unity gain cells are defined as follows [0,]. The Voltage Follower (VF) is defined by: I ¼ 0 0 V ð7þ 0 I The Voltage Inverter (VI) is defined by: I ¼ 0 0 V ð8þ 0 I The Current Follower (CF) is defined by: ¼ 0 0 I ð9þ I 0 The Current Inverter (CI) is defined by: ¼ 0 0 I ð0þ I 0 scillators using unity gain cells were introduced in the literature []. The first oscillator circuit considered in this section is shown in Fig. 4a. This is equivalent to the circuit shown in Fig. a and Eq. (5) applies to this circuit. 4 4 I DCVC DCVC CF VF CI VF I I Fig. a Two DCVC oscillator circuit I [7,9]. Fig. 4a [5,9]. Unity gain cells oscillator circuit equivalent to Fig. a

5 Quadrature scillators 5 CF VF CF VI V Fig. 4b [9]. Unity gain cells oscillator circuit equivalent to Fig. b Fig. 5b Two CF grounded capacitor oscillator circuit II [5]. The second oscillator circuit considered in this section is shown in Fig. 4b. This is equivalent to the circuit shown in Fig. b and Eq. () applies to this circuit. CF based oscillators The CF is a four-terminal active building block and is described by the following matrix equation []: Vx Ix Iy 7 4 Iz 5 ¼ Vy 7 7 ðþ Vz 5 Vo Io The first oscillator circuit considered in this section is shown in Fig. 5a [ 5]; its NM equation is given by: ¼ sc þ ðþ s The condition of oscillation is given by Eq. (8) and the radian frequency of oscillation is given by Eq. (9). The second oscillator circuit considered in this section is shown in Fig. 5b [5]. This is equivalent to the circuit shown in Fig. d and Eq. () applies to this circuit. The results of this class of oscillators are summarized in Table. Single output based oscillators The single output is defined by: Vx 4 Iy Iz 7 5 ¼ 0 0 Ix Vy ðþ 0 0 Vz The positive sign in the third row applies to whereas the negative sign is for. In this section, the conventional systematic synthesis framework using NM expansion [ 0] to synthesize oscillator circuits is used to transform the CF oscillators to single output oscillators using grounded capacitors and grounded resistors. Starting from Eq. () and add a third blank row and column and then connect a nullator between columns and and a current mirror (CM) between rows and in order to move from the, position to the diagonal position, to become as follows: ð4þ dding a fourth blank row and column to the above equation and then connect a nullator between columns and 4 and a CM between rows and 4 in order to move from the, position to the diagonal position 4, 4 to become, the following NM is obtained: ð5þ dding a fifth blank row and column to the above equation and then connect a nullator between columns and 5 and a norator between rows and 5 in order to move from the, position to the diagonal position 5, 5 the following NM is obtained: ðþ Fig. 5a Two CF grounded capacitor oscillator circuit I [ 5].

6 .M. Soliman Table Summary of the class II oscillator circuits. Circuit Figure Number ctive Element Number of C Number of Condition of oscillation x q ffiffiffiffiffiffiffiffiffi Ref. 4 (a) DCV rounded 4 Floating = [7,9] (b) DCV rounded Floating = [8,9] 4 4(a) VF,CF,CI rounded rounded Floating = [5,9] 4(b) VF, CF, VI rounded Floating = [9] 5(a) CF rounded rounded Floating = [ 5] 5(b) CF rounded rounded Floating = [5] V Fig. a Three oscillator circuit I [,]., the circuit shown in Fig. a using two and one is obtained. It should be noted that the number of resistors is increased to four instead of three in the circuit of Fig. 5a. Note also that the resistors at nodes and 5 are of the same value. This circuit has been introduced originally [] with four grounded resistors of different values. Changing the magnitude of at node 5 to become 4 [] adds more degrees of freedom in the tuning of the oscillator circuit and the NM equation will be modified to: " ¼ sc # þ 4 ð7þ s This is similar to Eq. (5) except for the signs of and 4. In this case the condition of oscillation is controlled by varying or and the frequency of oscillation is controlled by varying or 4 without affecting the condition of oscillation. Table summarizes these results. The second oscillator circuit considered in this section is shown in Fig. b [,] and is generated from Fig. d or its equivalent CF oscillator circuit shown in Fig. 5b both are represented by Eq. () and using NM expansion as follows. Starting from Eq. () and add a third blank row and column and then connect a nullator between columns and and a current mirror (CM) between rows and in order to move from the, position to the diagonal position, to become therefore: ð8þ dding a fourth blank row and column to the above equation and then connect a nullator between columns and 4 and a norator between rows and 4 in order to move from the, position to the diagonal position 4, 4 the following NM is obtained: ð9þ Fig. b Three oscillator circuit II []. The above equation is realized as a five node circuit using three nullators, one norator and two CM. Noting that the nullator and norator with a common terminal realize a, the nullator and CM with a common terminal realize a dding a fifth blank row and column to the above equation and then connect a nullator between columns and 5 and a CM between rows and 5 in order to move from the, position to the diagonal position 5, 5 the following NM is obtained:

7 Quadrature scillators 7 Table Summary of the class III oscillator circuits. Circuit figure number ctive element Number of C Number of Condition of oscillation x q ffiffiffiffiffiffiffiffiffi Ref. 4 (a), rounded 4 rounded = [,] 4 (b), rounded 4 rounded = [] 7(a) D, rounded rounded = [4] 7(b) B, rounded rounded = [4] 4 8(a), rounded 4 rounded = [] 8(b), rounded 4 rounded = 4 [] ð0þ The above equation is realized as shown in Fig. b using two and one. Changing the magnitude of at node 5 to become 4 adds more degrees of freedom in the tuning the oscillator circuit and the NM equation in this case will be modified to: ¼ sc þ 4 ðþ s This is similar to Eq. (5) except for the signs of and. Table summarizes these results. It is worth noting that the two circuits of Fig. a and b can also be generated from Fig. b as given in Soliman []. They can also be generated from the circuits given in Soliman [4] as explained in the following section. Two-output based oscillators The first type of the two-output is the double output which is defined by: Vx Ix Iy Vy ¼ ðþ 4 Izþ Vz 7 5 Izþ Io The oscillator circuit shown in Fig. 7a using a D and a was first introduced by Soliman [4]; its NM equation is given by: ¼ sc þ ðþ s The second type of the two-output is the balanced output which is defined by: Vx Ix Iy Vy ¼ ð4þ 4 Izþ Vz 7 5 Iz Io D The oscillator circuit shown in Fig. 7b using a B and a was first introduced by Soliman [4]; its NM equation is given by: ¼ sc þ ð5þ s Replacing the D by its equivalent two single output as demonstrated by Soliman [] results in the circuit of Fig. 8a. Similarly the circuit of Fig. 8b can be generated from Fig. 7b by replacing the B by its equivalent single output and single output. Note that at node 5 has been taken as 4 to increase degrees of freedom in the tuning of the oscillator circuit. The results of this circuit are summarized in Table. 4 - Fig. 7a rounded passive elements oscillator circuit I [4]. B - 4 Fig. 7b rounded passive elements oscillator circuit II [4].

8 8.M. Soliman Fig. 9a Fig. a. Four T grounded capacitors oscillator generated from Fig. 8a Single output equivalent circuit to Fig. 7a []. V Fig. 9b Fig. b. 4 Four T grounded capacitors oscillator generated from Fig. 8b Single output equivalent circuit to Fig. 7b. []. Transconductance amplifier based oscillators The T is a very powerful active building block [5 40] in realizing oscillator circuits. Three different types of T are considered in this section. (i) Single input single output T based oscillators There are two types of the single input single output T (SIS T) defined as follows. The T has current pointing outwards and the T has current pointing inwards. It is well known that the with a conductance connected to its terminal realizes a T with the same magnitude as the conductance. Similarly a with a conductance connected to its terminal realizes a T. The first circuit considered in this section is shown in Fig. 9a and is generated from Fig. a by replacing the three by three T. The conductance is realized by the T shown to the left of Fig. 9a. The four SIS T circuit is shown in Fig. 9a and has the same equation as that of Fig. a and Eq. (7) applies to it as well. It is worth noting that the T of transconductance realizes a negative grounded conductance at node of magnitude. This was realized by the number in Fig. a acting as a Negative Impedance Converter (NIC), with acting as its load. The second SIS T circuit considered in this section is shown in Fig. 9b and is generated from Fig. b; and Eq. () applies to this circuit. T circuit with the same topology has been reported in [40,4]. (ii) Single input two-output T based oscillators The circuit shown in Fig. 0a is generated from Fig. 7a and it employs a Double utput T with two positive outputs (DT) and two-sis T. Its NM equation is the same as given by Eq. (). The condition of oscillation is controlled by the T ( ) without affecting the frequency of oscillation and the frequency of oscillation is controlled by the T ( ) without affecting the condition of oscillation. The circuit shown in Fig. 0b is generated from Fig. 7b and it employs a Balanced utput T (BT) and two-sis T. Its NM equation is the same as given by Eq. (5).

9 Quadrature scillators 9 Fig. 0a Three T oscillator circuit I. Fig. Three T oscillator with one two input T [4 44]. has been demonstrated on a current conveyor oscillator published by Soliman and Elwakil [47]. Discussion of non-idealities The circuit has the same condition and the radian frequency of oscillation as the previous circuit and is given in Table 4.Tcircuit with the same topology has been reported by Swamy et al. [4]. (iii) Two-input single-output T based oscillator The circuit shown in Fig. employs a two input T and two SIS T and its NM equation is given by: ¼ sc þ ðþ s This circuit has been reported [4 44] and is the adjoint [45,4] of the circuit shown in Fig. 0b. Nonlinearity Fig. 0b Three T oscillator circuit II. It is well known that oscillators are nonlinear circuits, and the reported oscillators are based on a two integrator loop. Regarding amplitude control it should be noted that most oscillators rely on output voltage saturation of the active devices as a nonlinear mechanism for amplitude control. Regarding the amplitude control and stability of limit cycle this topic lthough the main objective of the paper is to review quadrature oscillators using various active building blocks and to show how they are related to each other, it may be useful to discuss the non-idealities of the active building blocks in the following sections. (i) p mp circuits The NM equations of the p mp oscillators shown in Fig. a d are based on assuming ideal p mps having infinite gain are used. In the actual case the single pole model of the p mp gain should be used. This model is represented by the following equation []: ¼ ox o ffi x t ð7þ s þ x o s where x t is the unity gain bandwidth of the p mp; the use of this model in the oscillator circuit analysis affects the NM equation [5] and hence both the condition of oscillation and the frequency of oscillation are changed from their ideal values. (ii) TR circuits The main parameter which is affecting the performance of the oscillator circuits of Fig. a and b is the finite and Table 4 Summary of the class IV oscillator circuits. Circuit figure number ctive element Number of C Condition of oscillation x q ffiffiffiffiffiffiffiffiffi Ref. 4 9(a) 4 SIS T rounded = 4 9(b) 4 SIS T rounded = 0(a) SIS T DT rounded = 0(b) SIS T BT rounded = SIS T TIS T rounded = [4 44]

10 0.M. Soliman frequency dependent nature of the transresistance Rm which is represented by: m ¼ ox o s þ xo ð9þ R m ¼ R ox o ð8þ s þ x o where R o is the DC value of the transresistance and x o is the pole radian frequency of the TR. (iii) DCVC circuits The non-ideality of the DCVC is mainly due to the parasitic elements that are represented by the two terminal resistances R,R and the terminal capacitance C. For the circuit of Fig. a all the parasitic elements can be absorbed in the circuit parameters except R of the first DCVC which affects the circuit operation. n the other hand for the circuit of Fig. b all the parasitic elements can be absorbed in the circuit parameters. (iv) CF circuits The CF parasitic elements are mainly represented by the terminal resistance R and the terminal capacitance C and the terminal resistance R. For the circuit of Fig. 5a the resistor R value should be adjusted to accommodate the added parasitic resistance R, the resistor R value should be adjusted to accommodate the added resistance (R R ). Similarly and should be adjusted to accommodate the parasitic capacitances C and C respectively. For the circuit of Fig. 5b the resistor R value should be adjusted to accommodate the added parasitic resistance R, the resistor R value should be adjusted to accommodate the added parasitic resistance R and the resistor R value should be adjusted to accommodate the added output terminal resistance R. Similarly and should be adjusted to accommodate the parasitic capacitances C and C respectively. Thus it is seen that the two circuits of Fig. 5a and b have the advantage of absorbing all parasitic elements within the design values of the circuit components. (v) circuits The parasitic elements are mainly represented by the terminal resistance R and the terminal capacitance C. For the circuit of Fig. a the resistor R should be adjusted to accommodate the added parasitic resistance R, the resistor R value should be adjusted to accommodate the parasitic resistance R and the resistor R 4 should be adjusted to accommodate the parasitic resistance R. Similarly and should be adjusted to accommodate the parasitic capacitances (C C ) and C respectively. Similarly for all other oscillator circuits presented in Figs. b 8b. Thus it is seen that the oscillator circuits presented in this paper have the advantage of absorbing all parasitic elements within the design values of the circuit components. (vi) T circuits The main parameter which is affecting the performance of the T oscillator circuits is the finite and frequency dependent nature of the transconductance which is represented by: where o is the DC value of the transconductance m and x o is the pole radian frequency of the T. Conclusions Quadrature oscillators are reviewed in this paper using several active building blocks namely; operational amplifier, operational transresistance amplifier, differential current voltage conveyor, unity gain cells, current feedback operational amplifier, current conveyors and transconductance amplifiers. eneration methods including nodal admittance matrix expansion shows that many oscillators that are reviewed in this paper can be obtained from each other. Simulation results or experimental for most of the oscillator circuits reviewed are available in the proper references indicated below and are not included to limit the paper length. It is worth noting that the chip size and the power dissipation for each of the reported oscillators will depend on the circuit design and the CMS analog circuit realizing the active building block used. References [] Sedra S, Smith KC. Microelectronic circuits. 4th ed. xford University Press; 998, p [] Van Valkenburg M. nalog filter design. 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