Class Notes, 31415 RF-Communication Circuits Chapter VII MIXERS and DETECTORS Jens Vidkjær NB235
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Contents VII Mixers and Detectors... 1 VII-1 Mixer Basics... 2 A Prototype FET Mixer... 2 Example VII-1-1 Square-Law FET mixer... 7 Example VII-1-2 Square-Law FET mixer - continued... 9 VII-2 Differential Stage Mixers... 15 VII-3 Gilbert-Cell Mixers... 18 Problems... 22 References and Further Reading... 25 Index... 27 iii
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1 VII Mixers and Detectors The fundamental operation of a mixer is to multiply two signals. One of them is often a modulated signal and the other a stable local oscillator output, and the purpose of the process is to shift the carrier frequency in the modulated signal while keeping the information carrying envelope - real or complex - intact. If one of the carriers is at zero frequency, DC, this means that we are either shifting from or to baseband signals. In that case the circuits may also be called modulators and demodulators respectively. Demodulator circuits are sometimes included in a group of circuits known as detectors, although this term also comprises cases where baseband information is extracted from a modulated signal without guidance from a local oscillator. Finally, circuits that shifts from one carriers frequency to another are sometimes also called frequency converters. Mixing processes were considered from a system point of view in sec.i-5. It was seen that even an ideal multiplication process had undesirable effects that must be dealt with around a multiplying circuit. There was removal of either unwanted sum or unwanted difference frequency signal components by post filtering. Protection against image responses by prefiltering was another problem. In this chapter we shall present more details about circuit implementations and consider limitations on ideal performances due to noise and non ideal components. This chapter is preliminary. You may find additional information on the subject in ref s [1],[2], or [3].
2 VII-1 Mixer Basics Multiplying voltage or current waveforms is a nonlinear process. A simple example is the resultant response if two signals are applied to a nonlinear device. We have already seen some consequences in the discussion of limitations in almost linear circuits in section VI-4. There, considerations were made from the point of view, that the mixing process deteriorates linear performance by introducing distortion and intermodulation signal components at frequencies, which may differ from the input signal frequencies. Dealing with mixers, however, we shall now organize the circuits to provide the best possible translation from one frequency range to another. A Prototype FET Mixer Fig.1 FET transistor that is biased and driven for mixer applications. Figure (b) shows a simplified transistor transfer characteristic. In the preceding section IV-4 we saw how signal distortion was introduced when device characteristics departed from linearity. In mixers it is the same nonlinear device characteristics that are used to give the desired frequency translations. To fix ideas, Fig.1 demonstrates in principle how a FET may be employed as a mixer when two input signals are applied to the gate. We assume that the transistor is biased, driven, and loaded in a way that prevents saturation effects, so its operation may be described solely by its nonlinear transfer
VII-1 Mixer Basics 3 characteristic in Fig.1(b). With both input signals set to zero, the transistor DC bias conditions are fixed by where the bias current to the drain flows through the RF-choke L chk.v GG is the gate bias voltage from the battery. When signals are applied, we suppose that the corresponding current components are related to the input signal through a Taylor series expansion of the transfer characteristic around the gate bias voltage V GG. Voltage v g denotes the deviation voltage according to Following the notation that was introduced in section IV-4, the drain current, when the two time varying signals V rf and V lo are applied, is expressed (1) (2) Inserting the two signal components, where (3) (4) (5) it is seen, that the second order term in the Taylor expansion provides a drain current component, which is proportional to the product of the two input signals,v rf and V lo.ifwe let the RF signal be represented by a single tone so both applied signals are sinusoidal, the generation of frequency components in the drain current follows basically the scheme that formerly gave the distortion components in Eq.IV-136 for Taylor expansion components up to third order. In present terms this equations reads, (6)
4 Mixers, Modulators, Demodulators, and Detectors (7) This result required repeated use of the following basic relations (8) Had we proceeded to even higher order of terms, recursive use of the last identities would reveal that each order of expansion terms introduces components where, in double sided representation based on the identity, the frequencies become all possible combinations that may be written by (9) (10)
VII-1 Mixer Basics 5 The resultant frequency from this process may be simpler than the expression above suggests at a first glance. For instance, the left term of the third order components in Eq.(7)(c1) develops as Here the resultant frequency keeps no indication of the Taylor expansion order in the nonlinear mixing characteristic, N=3, that caused this particular frequency component. Note, however, that the amplitudes hold information about the order. In the example above, the two contributions are both of third order in voltages, V 3 RF3 and V RF V 2 LO respectively. A common way of expressing the resultant frequency components from a mixing process in literature is 1 (11) (12) According to the discussion above, there is no simple one to one relationship between the integer set n,m and Taylor expansion order(s) for the mixing nonlinearity. The only point that can be states that a mixing frequency given by Eq.(12) cannot originate from an expansion term of order less than n + m. Returning to basic mixing properties, the product term from Eq.(4) provides both the sum and the difference frequency drain current components in Eq.(7)(b2). Which one to use depends on the application at hand. In a receiver the typical mixer function is to shift the frequency region of an incoming RF-signal down to a lower intermediate frequency, IF, range. Had we used the diagram in Fig.1 directly we would get output voltage components at all combination frequencies in the Taylor expansion of the current. A simple way to sort out the desired component is indicated by Fig.2, where the load of the FET is made by a parallel circuit that is tuned to IF at the difference frequency If the Q-factor for the tuned circuit is high enough, the IF current component provides the dominant contribution to the output voltage while all other current components are practically short circuited. In this case, the output voltage is given by the IF component I d,if in the drain current, (13) (14) Consult ref s [1], [2], or [3], for more elaboration on this subject.
6 Mixers, Modulators, Demodulators, and Detectors Fig.2 Simple FET transistor mixer, where the parallel circuit L, C is tuned to the IF frequency. Like the case of distortion in almost linear circuits, residual signal components at all but the desired IF frequencies are called spurious signals. Sometimes, the ratio of the current amplitude at the desired output frequency, here the IF, over the input RF voltage amplitude is called the conversion transconductance 2. The ratio of the desired frequency output voltage amplitude over the input RF voltage amplitude is correspondingly called the conversion voltage gain or just the conversion gain if it is clear from the context that we are dealing with voltages, not powers. In the present case we have (15) (16) Introducing conversion quantities, we may now express (17) 2 ) The Danish term is "blandingsstejlhed".
VII-1 Mixer Basics 7 so G cnv and A cnv are used the same way in amplitude calculations as the traditional transconductance and voltage gain factors are used in conventional circuit characterization. It must be kept in mind, however, that conversion transconductances and conversion gains relate current and voltage components of different frequencies and, furthermore, these quantities depends not only on simple device and DC biasing properties, but also on the local oscillator signal, V LO. In order not to distort the information in a modulated signal envelope when the carrier frequency is changed by a mixer, the conversion transconductances and gains are supposed to be independent of the RF signal level, V RF. Like transconductance and gain, concepts like compression, intermodulation, or intercept points are inherited from the almost linear amplifier terminology to characterize distortions introduced in mixers, now with the complication, that the input and output frequency ranges of significance are no longer coincident. Example VII-1-1 Square-Law FET mixer In ideal form the transfer characteristic of a n-channel FET that is biased for normal operation is a part of a parabola,[4]. It ranges from zero current at the so-called pinch-off or threshold voltage V P - a negative voltage - and the saturation current I DSS at zero gatesource voltage as expressed by (18) First and second order derivatives of the drain currents are (19) and all higher order derivatives are zero, so the Taylor expansion in Eq.(3) for a given gate biasing, V GG, is given through (20) As seen, the second order b-term is independent of the bias voltage V GG, i.e. constant across the whole gate voltage range where the transistor conducts current. In this range the b term determines the conversion transconductance, which becomes (21) The maximum conversion transconductance limit in the expression is approached if the RF signal is small compared to the local oscillator signal, which is set to the maximum amplitude,
8 Mixers, Modulators, Demodulators, and Detectors where the transistor is conducting all times. That is V LO = ½ V P, gate bias V GG =½V P, and assuming V RF << V LO as sketched in Fig.3. Fig.3 Local oscillator driving of a square-law FET mixer for max. continuous mode conversion transconductance. The RF signal, V RF, is much smaller than V LO. Example VII-1-1 end The assumption of a small RF signal compared to the local oscillator signal - which gave the maximum conversion gain in the example above, applies commonly to the initial mixers in radio receivers. The assumption provide the background for a method of calculating conversion gains in mixer circuits by the so-called time-varying transconductance approach 3, which is an alternative to the Taylor expansions we have considered so far. Here the transconductance, which is calculated like the "a" term in the Taylor series expansion, is taken as a function of both the DC bias and the large local oscillator time dependent signal. When the small RF signal is applied, the resultant mixing components in the drain current are calculated like conventional small signal current components by multiplying the RF gate voltage by a transconductance. However, the latter is now time dependent, so we may express the mixing components in the drain current by (22) To find a particular mixing component with a given time dependent transconductance waveshape, it must be expanded in a Fourier series, 3 ) The approach is also called "Large Signal - Small Signal" analysis.
VII-1 Mixer Basics 9 (23) Here, it is assumed that the transconductance waveshape G d,mix (t) is symmetrical with respect to t=0. If this is not the case we must elaborate the Fourier expansion correspondingly. To find a particular mixing components with a given RF component of frequency ω RF, each Fourier coefficient gives rise to two mixing products, one sum and one difference frequency between the RF and one of the local oscillator components, usually the fundamental local oscillator frequency component at ω LO corresponding to m=1 in (24) Conversion transconductance is still defined like Eq.(15) by the ratio of the desired frequency component in the drain current over the RF signal amplitude, so we have to select one of above component by filtering. In time-varying transconductance terminology, the conversion transconductance becomes (25) where g mix,m is the Fourier coefficient of appropriate order in the expansion of the transconductance waveshape. Example VII-1-2 Square-Law FET mixer - continued There are several practical drawbacks in the square-law mixer circuit that was considered in the preceding example. One of them is that gate bias and local oscillator voltage depends on the pinch voltage of the FET. This parameter is commonly subject to large spreadings, say 100% or more. Precise tracking would therefore require adjustment capabilities in the circuit and tuning in production. We may reduce the direct dependencies of the pinch voltage with a higher local oscillator amplitude than we had before. In consequence, the FET must be non conducting - i.e. cut off - in part of the oscillator period. Assume that the RF signal is small compared to the local oscillator amplitude, so it totally dominates the drain current waveshape and the switching instants of the FET. The drain current pulse train is illustrated in Fig.4, and the corresponding time varying transconductance is shown in Fig.5. It is given though
10 Mixers, Modulators, Demodulators, and Detectors (26) Since the current characteristic I d (V gs ) is square-law when the transistor conducts current, the corresponding transconductance is a linear function of V gs. Therefore - using a sinusoidal local oscillator - the transconductance waveshape becomes a train of sine-tips. We have already considered the Fourier expansion of this waveshape in conjunction with the power amplifier discussion in chapter 5. The transistor opening angle θ, i.e. the portion of a period where the transistor conducts, may still be used as the controlling parameter. Thereby the conversion transconductance is expressed (27) Fig.4 Current pulses in square-law FET mixer with large sinusoidal LO drive. The RF signal is small, V RF V LO.
VII-1 Mixer Basics 11 Fig.5 Time varying transconductance in square-law FET mixer with large, sinusoidal local oscillator drive. The ratio y m /y p represents the Fourier coefficient and it is taken from Table 5-1 using the appropriate local oscillator mixing harmonic component number m. The peak value in the transconductance pulse train is g m (V gs,max ). If V gs is driven up to 0 V, the peak value is (28) A few extracts from Table 5-1 in the case of fundamental frequency local oscillator mixing, m=1, are summarized in Table I. It is seen here that in the limit case of full conduction, θ=360, we get a conversion conductance in agreement with the maximum limit in the previ- Table I Fourier coefficient of sinetips. Extracts from table 5-1. θ y 1 /y p y 2 /y p y 3 /y p 0 0.0000 0.0000 0.0000 90 0.3102 0.2562 0.1811 180 0.5000 0.2122 0.0000 270 0.5326 0.0439-0.0311 360 0.5000 0.0000 0.0000
12 Mixers, Modulators, Demodulators, and Detectors Fig.6 Time-varying transconductance in square-wave driven FET mixer. ous result from Eq.(21). Between full conduction and half-time conduction at θ=180, the conversion transconductance exceeds this value by a small amount. If the conduction angle goes below θ=180, conversion transconductance reduces correspondingly. However, the smaller the conduction angle, the smaller is the significance of the actual pinch voltage parameter size, V P, since the local oscillator amplitude V LO must rise correspondingly. With a maximum gate to source voltage of zero volts, which implies that the gate bias is V GG =- V LO, the relationships between the two are expressed through Instead of driving by a sinusoidal local oscillator, the transconductance could be square-waved using a square-waved local oscillator signal. In the simple case sketched in Fig.6, where the conduction angle is 180, the conversion transconductance becomes (30) Here the 4/π factor is the Fourier coefficient of a square-wave that is normalized to the interval [+1,-1], so the amplitude to be used is ½g m (V gs,max ). Square-wave local oscillator signals are feasible in many cases, since they resemble outputs from limiter circuits that may be used to minimize the effects of local oscillator amplitude noise and fluctuations in sensitive equipment like radio receivers.
VII-1 Mixer Basics 13 Driving the mixer with a local oscillator signal that is so large that the transistor becomes non-conducting, like it was done in this example, has the consequence, that mixing takes place around harmonic components of the local oscillator signal. We have assumed that any undesired or spurious frequency component is removed by subsequent filtering, but clearly the more spurious components generated, the more efforts must be given to the filtering problem. When the FET mixer was used in continuous mode in the foregoing example, no higher order mixing took place, so the price paid by making the mixer less sensitive to parameter variations by enlarging the local oscillator drive could be that the filtering requirements are tightened. Example VII-1-1 end The FET mixer prototype we have considered in this section was introduced as a vehicle to exemplify how the required multiplying function may be realized through a nonlinear characteristic. Besides this very basic property there are several more concerns for employing mixers in RF-circuits which we shall deal with below. If we should use the prototype FET mixer as an outset for a practical application, we should probably rearrange the circuit as sketched in Fig.7, where the RF and LO signals are separated so each of them gets a ground terminal. Z RF and Z LO represent the generator impedance of the RF and LO sources respectively and they are applied through coupling capacitors C cpg and C cps. Biasing of the transistor is made through R G and R S1,R S2. We ascribe no DC voltage across R G as the ideal FET requires no gate bias current. The gate source biasing is establish by the DC current through the transistor and R S1 +R S2. It is assumed that the FET has a negative threshold voltage V P. Fig.7 Practical realization of a FET mixer including biasing. A negative threshold voltage is assumed.
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15 VII-2 Differential Stage Mixers Fig.8 Differential stage mixer coupling. C cpli and C cple are coupling and decoupling capacitors. R 1 to R 3 are bias resistors. Filtering is made outside this circuit. The differential amplifier structure that was considered in Chapter 5 may also serve as a mixing circuit. One way of doing this is sketched in Fig.8. The local oscillator signal is applied to the differential input terminals while the common tail current, which formerly was a constant bias current, now is overlaid by the RF signal through a conventional common emitter stage around transistor Q 3. Assuming small signal conditions for the RF signal, the tail current is expressed through (31) Here, appropriate bias resistor settings establish the DC current level, I 0. Inserting into the large signal expression for the differential amplifier, Eq.5-138, the differential output current may now be written (32)
16 Mixers, Modulators, Demodulators, and Detectors To see the mixing properties, we assume initially that both the RF and the local oscillator signals are sinusoidal, and, furthermore, that the local oscillator amplitude is small compared to V t, so if suffice to use the first, linear term in the Taylor series expansion (33) (34) (35) Now the differential current may be written (36) No mixing is associated with the first term since it contains only the local oscillator frequency component. The second term holds the mixing product (37) By subsequent filtering, the desired intermediate frequency component, either must be selected. It is represented by the differential current (38) (39) The corresponding conversion parameters now take the forms (40) (41)
VII-1 Differential Stage Mixers 17 It is often desirable to use a local oscillator signal that is larger than the size assumed by the approximation in Eq.(35). With growing local oscillator signal we shall, like the limiter development in Section 5-3, reach a limit where the two differential transistors Q 1 and Q 2 are operated as antagonistic switches. In that case we may approximate the result of the hyperbolic tangent function by a square-wave and approximate, (42) Now the differential current becomes (43) Again, the first term contributes nothing to the mixing function. It holds only fundamental and harmonic frequency components of the local oscillator signal. The last term, however, hold mixing products around the local oscillator frequency and higher harmonic components. Isolating one component from mixing with local oscillator fundamental frequency gives (44) The corresponding conversion parameters, which are independent of the local oscillator signal level, now become (45) (46) If the output of a mixing circuit includes unmixed fundamental and higher harmonic frequency components of both the RF and the LO signals before the IF filter is applied, the mixer is called unbalanced. If one of these family of components are suppressed before filtering, the mixer is called single balanced, if both families are absent, the mixer is called double balanced. In this terminology, the differential stage mixer above is single balanced. Besides selecting the proper IF components, the subsequent IF filter has to suppress the local oscillator components down to levels where they do no harm. An alternative is to employ two cross-coupled differential stages to achieve double balancing by the so called Gilbert Cell mixer, which is discussed next.
18 Mixers, Modulators, Demodulators, and Detectors VII-3 Gilbert-Cell Mixers Fig.9 Gilbert-Cell. The input terminals must be properly biased in addition to differential inputs V LO and V RF. All current expressions assume transistor current gains α f equal to one. Cross coupling two differential stage mixers that are current biased through a common DC tail current gives the structure in Fig.9. It is called a Gilbert Cell after its inventor. Compared with the single differential stage mixer, also the RF signal is now applied to a differential input port. As seen in the figure, there are three differentially operated transistor pairs in this configuration, and to investigate the mixing function, we shall make repeated use of the differential stage results from Section 5-3. It is convenient, therefore, to introduce the input signal normalization that was used formerly, i.e. (47) The common tail current is here kept at a constant DC value, I 0. The differential current in the bottom transistor pair, which is driven by the RF signal, becomes (48)
VII-1 Differential Stage Mixers 19 where α f is the common base current gain of the transistors. Commonly it has a value slightly below one, say 0.98. Including the effect of the bottom differential current, ΔI 0, the differential and tail currents for the two LO signal operated transistor pairs are expressed (49) (50) Finally, the two differential terms above subtracts to the final output differential current ΔI out, (51) We may use the Taylor series expansions and assumptions from Eqs.(34),(35) to approximate (52) Compared with the similar expressions from the single differential stage mixer like Eq.(36), it is seen that only a product term remains. There is no separate LO signal terms, so the Gilbert Cell has clearly doubly balanced mixer function. For the same reason, a Gilbert Cell driven by small input signals is sometimes called a pure four-quadrant multiplier. With sinusoidal input signals (53) the frequency component of desired intermediate frequency IF, where has the amplitude (54) (55)
20 Mixers, Modulators, Demodulators, and Detectors Thereby, the conversion parameters for a Gilbert Cell Mixer operated by small signals at both the RF and the LO ports become, (56) (57) With a large signal input to the local oscillator port, i.e. x lo >6 or V LO >150 mv, the upper differential pairs are operated like switches as sketched in Fig.10. The direction of the RF-signal controlled differential current, ΔI 0, is changed according to the sign and in turns the frequency of the LO signal. In the limit, where we assume ideal instant switching, the LO controlled hyperbolic tangent factor in Eq.(51) should be replaced by the Fourier expansion of a square-wave like Eq.(42). With this replacement we get (58) Maintaining the assumption of a small RF signal, and mixing around the fundamental LO frequency component, the output differential current IF terms have amplitudes of Fig.10 Gilbert-Cell circuit used as a switching mixer. The RF differential current is reverted when the local oscillator voltage V LO shifts between high positive (a) and negative (b) levels.
VII-1 Differential Stage Mixers 21 (59) The mixing parameters in switched operation now become (60) (61)
22 Mixers, Modulators, Demodulators, and Detectors Problems P.VII-1 Fig.11 Fig.11 shows the principle for a mixer with a square-law FET. Both the RF and the LO signals are sinusoidal at frequency f RF = 450MHz and f LO = 520 MHz respectively. The load circuit has quality factor Q IF = 100, and it is tuned to f IF = 70 MHz. The load resistor is R L =1kΩ. Capacitor C dcp is a decoupling capacitor. It is assumed that the RF signal is much smaller then the local oscillator amplitude. Find the bias resistor R E and the local oscillator amplitude V LO that give mean drain current equal to 10 ma and forces the transistor peak current to I DSS. Sketch the time varying transconductance - minimum and maximum values - and find the conversion transconductance. What is the IF output voltage if an RF signal of 2mV is applied to the RF input port? Fig.12
VII-1 Differential Stage Mixers 23 In practice, the RF signal is applied through a transformer as shown in Fig.12. The input circuit is tuned to f RF and has quality factor Q RF = 20 including the effect of the generator resistance R g. What is the image frequency of the mixer. Estimate the IF output voltage if an image frequency input signal of 2mV amplitude is applied to the RF input port? P.VII-2 Fig.13 Fig.13 shows a differential stage mixer. The bias current without any RF signal is set by resistor R 0, which is chosen to give I 0 =5mA. The local oscillator frequency is 120 MHz and the amplitude V LO is set to provide a differential amplitude of ΔI c,lo = 2.5 ma at the local oscillator frequency if no RF signal is applied. The RF signal is of frequency 100 MHz is applied through coupling capacitor C 0. The load circuit L, C is tuned to a intermediate frequency of 20 MHz. Including the load resistance R L =1kΩ, the load circuit has Q factor equal to 100. What is the conversion transconductance of the stage and the corresponding local oscillator voltage amplitude, V LO? Find the IF output voltage V IF if the RF signal is an unmodulated carrier of 5 mv. Note, it is not assumed in this problem that V LO is << V t. An amplitude modulated RF signal, where m is the modulation index and the baseband frequency f BB << f RF, is applied. The resultant intermediate frequency output is written (62)
24 Mixers, Modulators, Demodulators, and Detectors where the frequency dependency of the resultant modulation index, m IF, is caused by the frequency characteristic of the tuning circuit. At which baseband frequency is the resultant modulation index m IF reduced 3dB compared to the input index m? The envelope of the IF output is distorted due to the nonlinear transfer characteristic of the mixer. Assume a low baseband frequency and show that the 2nd and 3rd harmonic contribution to the total harmonic distortion of the IF envelope for small RF signals, V RF << V t = 25mV are (63) (64) The exponential function series expansion may be useful to answer the last question, (65)
25 References and Further Reading [1] R.S.Carson, Radio Communication Concepts: Analog, Wiley 1990. [2] S.A.Maas, Microwave Mixers, 2nd ed., Artech House 1993. [3] S.A.Maas, Nonlinear Microwave Circuits, Artech House 1988. [4] A.S.Sedra, K.C.Smith, Microelectronic Circuits, 3rd ed., Saunders, 1991.
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27 Index Conversion Gain... 6 Conversion Transconductance... 6 Conversion Voltage Gain... 6 Demodulators... 1 Detectors... 1 Double Balanced Mixer...17 Frequency Converters... 1 Gilbert-Cell...18 Large Signal - Small Signal Analysis... 8 Mixer balanced...17 Conversion Gain... 6 double balanced...17 unbalanced...17 Mixers... 1 FET basics... 2 Gilbert-Cell...18 Modulators... 1 Single Balanced Mixer...17 Spurious Signals... 5 Switching Conversion Gain differential stage mixer...17 Gilbert cell mixer...21 Time-Varying Transconductance... 8