ChapVolterra2

ChapVolterra2 - Wireless Communications Circuits Lawrence...

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Unformatted text preview: Wireless Communications Circuits Lawrence E. Larson January 7, 1999 c L. Larson January 7, 1999 2 Chapter 1 Volterra Series 1.1 Introduction to Volterra Series So far, we have assumed that the nonlinearities in the circuit we are modeling are purely resistive," in the sense that there is no memory in the circuit. All of the instantaneous currents and voltages in the circuit depend only on the present values | so there are no capacitors or inductors in the circuit. This assumption works well at low frequencies, and also works well in certain frequency dependent circuits in limited cases. However, in the case where frequency dependent elements are introduced | where the circuit exhibits memory" | the use of a memoryless powerseries approximation to the nonlinearities in the circuit loses accuracy. For example, at the high frequencies typically encountered in a radio frequency circuit, the reactive components of the circuit impedances are at least as large as the resistive components of the impedance. This creates an enormous complication in the analysis of nonlinear circuits, which the use of Volterra analysis neatly resolves. In fact, Volterra techniques have been applied to a wide variety of problems in nonlinear circuits, all the way from radio frequency receiver design 1 to the improvement of high- delity speaker performance 2 . However, the complexities introduced by the use of Volterra techniques are themselves quite involved, and have historically been limited to a handful of specialists in the eld. I hope that this chapter will introduce enough of the topic to allow for some sophisticated analysis of nonlinear circuits, without the tedium of learning multidimensional Fourier Transform techniques, upon which the technique is based. 3 1.1. INTRODUCTION TO VOLTERRA SERIES1. VOLTERRA SERIES CHAPTER The following example illustrates some of the problems encountered when frequency dependent elements are added to the analysis of even a simple nonlinear circuit. Example 1.1 Series Inductive Feedback in an n-channel MOSFET Figure 1.1 shows an example of the use of inductive feedback in the source of a long-channel n-MOSFET. The feedback can be used to linearize the device, as well as improve its impedance match for noise gure optimization. We are particularly interested in calculating the input intercept point of the device in this con guration, in particular with comparison to the case of resistove series feedback analyzed in the previous Chapter. In this case, the instantaneous drain current through the device in the saturation region can be approximated by ignoring channel-length modulation and short-channel e ects. 1.1 Now, if we assume that the instantaneous gate voltage vGS consists of a small-signal component vgs and a dc component VGS , then Equation  1.1 can be expressed as 2 iD = kvGS , VT  iD = kVGS + vgs , VT  iD = kVGS , VT  + 2vgsVGS , VT  + vgs 2 2 2 1.2 and the small-signal drain current id is id = k2vgsVGS , VT  + vgs = a vgs + a vgs 2 1 2 2 1.3 so the linear gain coe cient |a1 | for the long-channel MOSFET is simply 2kVGS ,VT  | the transconductance of the device | and the quadratic gain coe cient | a2 | for the device is k. Now, in this case, the voltage across the inductor reduces the e ective gate-to-source voltage, providing the negative feedback. Therefore, idt = a vint , L did + a vint , L did  dt dt 1 2 2 1.4 c L. Larson January 7, 1999 4 CHAPTER 1. VOLTERRA SERIES 1.1. INTRODUCTION TO VOLTERRA SERIES In a sinusoidal steady-state condition, like a two-tone test, the input voltage will consist of two sinusoidal signals, each of a di erent frequency and phase. The output current will consist of multiple sinusoidal signals, whose frequencies are linear combinations of the input frequencies. When the output current is viewed as a sum of sinusoidal signals, Equation  1.4 must be satis ed at each of the output frequencies of the circuit. The magnitude and phase of this feedback voltage L did will be frequency dependent with a value dt j!aLid at each frequency !a. This can not be accomodated in a straightforward way with the frequency indpendent power-series coe cients derived in the previous chapter. So, this very simple demonstration illustrates the necessity for accounting for the frequency dependence of the various voltages and currents in the circuit. The use of Volterra Series provides us with a general method for accounting for the frequency dependent behavior of nonlinearities in active circuits as long as the nonlinearities are weak. This is most often the case in receiver and up-converter circuits. The assumption of a weak nonlinearity is less valid in the case of power ampli ers, which are strongly nonlinear circuits, and will be covered in a separate Chapter. For example, the circuit of Figure 1.2 shows a linear lter which has storage elements like inductors and capacitors preceding a memoryless nonlinearity. In this case, we can use the power-series approximation to calculate the output of the nonlinear circuit, but the input to the nonlinear circuit has been ltered by the linear circuit preceding it. Now, the input voltage to the whole circuit sin can be expressed as a sum of sinusoidal signals, each of amplitude Sn as sin = Q X q=1 Q 1 X S ej!q t Sq cos!q t = 2 q q ,Q = 1.5 where q 6= 0. When these input signals are passed through a linear time-invariant lter, the output of the lter and the input to the nonlinear ampli er becomes si = Q X q=1 Q 1 X S H ! ej!q t Sq jH !qj cos!q t + H !q  = 2 q q q ,Q   = 1.6 c L. Larson 5 1.1. INTRODUCTION TO VOLTERRA SERIES1. VOLTERRA SERIES CHAPTER where H !q  is the linear transfer function through the lter. Since H !q  is a complex quantity, dependent on j!, H !q  = H ,!q , and H !q = argH !q. The output of the nonlinear ampli er can be represented by a power series of the form   sout = a si + a si + a si +    = N X 1 2 2 3 3 n=1 ansn i 1.7 and the output of the circuit becomes sout t = = 3n 2 Q 1 X S H ! ej!q t5 an 4 2 q q n=1 q=,Q Q Q Q Q N X an X X X X ::: Sq1Sq2 : : : SqnH !q1H !q2 : : : n n=1 2 q1 =,Q q2 =,Q q3 =,Q qn =,Q : : : H !qnej!q1 +j!q2+:::j!qnt 1.8 N X where N is the maximum order of the nonlinearity, and Q is the number of input tones. As before, t`he summation in the above equation does not not include q = 0. This general model works well only when there is no ltering of the output intermodulation or harmonic terms, i.e. in the case of a linear circuit preceding a memoryless nonlinearity. If there is ltering of these terms, then their output amplitude will depend on their frequencies as well as the frequencies that created them. The result of Equation  1.8 does not accommodate this | the output amplitude at each frequency depends on the amplitude of the input frequencies only. The e ect of any subsequent ltering, which will change the amplitude at a given frequency at the output, can not be predicted by this Equation. As an alternative, consider the case of a memoryless nonlinearity followed by a linear frequency dependent circuit. This is shown in Figure 1.3. Now, with this circuit, the output of the memoryless nonlinearity can be characterized by the power series expansion as before, i.e. c L. Larson January 7, 1999 6 CHAPTER 1. VOLTERRA SERIES 1.1. INTRODUCTION TO VOLTERRA SERIES si = a sin 2+ a sin + 3 a sin +    n Q N X 1 X S ej!q t 5 = an 4 2 q n q ,Q Q Q Q Q N X an X X X X = ::: Sq Sq : : : n n 2 q1 ,Q q2 ,Q q3 ,Q qn ,Q : : : Sqne j!q1 j!q2 :::j!qn t 1 2 2 3 3 =1 = 1 2 =1 = = = =  + +  2 1 1 2 3 2 1 1 1.9 2 When a two-tone input is applied to the circuit, it produces third-order intermodulation products at 2! , ! , 2! , ! , 2! + ! , and 2! + ! , all of the same amplitude IM 3 = 3=4a =a Si , as well as the desired output frequencies of ! and ! , both of the same amplitude. But now, these intermodulation products are ltered by the action of H j!, and they will have have di ering amplitudes and phases, depending on their exact frequencies, as Figure 1.4 illustrates. In this case, each of the output tones is applied to the linear lter, and the resulting output is 3 1 1 2 soutt = Q Q Q N X X an X X ::: Sq1Sq2 : : : SqnH !q1 + !q2 : : : n n=1 2 q1 =,Q q2 =,Q qn =,Q +!qnej!q1 +j!q2+:::j!qnt 1.10 In contrast with Equation  1.8, Equation  1.10 can not accommodate any linear ltering at the input of the circuit, which would have to be proportional to H !q . So, the goal of a perfectly general power series expansion that can accommodate any frequency dependent nonlinearity still eludes us. Notice that the forms of Equation  1.10 and Equation  1.8 are identical, except for the transfer functions, which are either of the form H !q +!q : : :+ !qn or H !q H !q  : : : H !qn. In the rst case, this composite transfer function can not model frequency dependence at the input of the circuit, and in the second case, it can not model frequency dependence at the output of the circuit. A general nonlinear circuit will exhibit a combination of these two transfer function forms; so we typically specify the nonlinear transfer function as H !q ; !q : : : ; !qn, which is su ciently general to model any potential form of the circuit behavior. 1 2 1 2 1 2 c L. Larson 7 1.1. INTRODUCTION TO VOLTERRA SERIES1. VOLTERRA SERIES CHAPTER Based on this general discussion, a more general form of the nal output equation is given by sot = Q Q Q N X X an X X ::: Sq1 Sq2 : : : SqnH !q1; !q2 : : : n n=1 2 q1 =,Q q2 =,Q qn =,Q !qnej!q1+j!q2+:::j!qnt 1.11 This approach is quite general, in the sense that it can predict the behavior of any weak nonlinearity, with or without memory. This model" of a weakly nonlinear frequency dependent system was elaborately justi ed by Wiener and Volterra about fty years ago, and seems to work well for a variety of front-end communications circuit problems ? . It is essentially equivalent to taking the N-dimensional Fourier Transform of the nonlinear signal 4 . Equation 1.11 is a little hard to follow, as it involves multiple summations, so it can be rewritten as Q Q Q 1 X X ::: X S S :::S H ! ; ! :::!  ej !q1 !q2 :::!qn t sot = q1 q2 qn n q1 q2 qn n n 2 q1 ,Q q2 ,Q qn ,Q Q X1 = Sq H !q1  ej!q1t q1 ,Q 2 Q Q X X1 S S H !q ; !q  ej !q1 !q2 t + 4 q1 q q1 ,Q q2 ,Q Q Q Q X X X1 j !q1 !q2 !q3 t 1.12 + 8 Sq1 Sq2 Sq3 H !q1 ; !q2 ; !q3  e q1 ,Q q2 ,Q q3 ,Q N X  + +  =1 = = = 1 1 = 2 2 1 2  +  = = 3  + +  = = = where Hn: : : = anH : : :. This appears to be somewhat less complicated than the original form of the expression, but is still fairly intimidating unless you happen to be a computer. As a result, researchers in the eld developed a short-hand notation to simplify circuit calculations. Equation 1.12 can be rewritten as So = H !a  Si!a 1 c L. Larson January 7, 1999 8 CHAPTER 1. VOLTERRA SERIES 1.1. INTRODUCTION TO VOLTERRA SERIES +H !a; !b Si !a + !b + H !a; !b; !c Si !a + !b + !c + : : : 2 2 3 3 1.13 n frequencies | where Hn !a; !b; : : : !k  is the nth order sinusoidal steady-state transfer function, and where each !a ; !b; !c is at one of ! ; ! ; : : : !Q. Notice that the !a ; !b; : : : !k are meant to function as placeholders" for the input frequencies ! ; ! ; : : : !Q. Each one of them can take on any of their values We have also introduced a new operator | the Volterra operator " | which has the following meaning: z 1 2 1 2 1. Multiply the magnitude of each frequency component in Sin by jHn!a; !b; !cj and 2. Shift the resulting phase by the phase of Hn!a; !b; !c 3. Sum all the resulting 2Qn terms The total output is then said to be composed of the sum of N individual responses. The linear portion of the circuit | characterized by the rstorder transfer function H !a| generates the rst-order component of the response. The quadratic portion of the circuit | characterized by the secondorder transfer function H !a; !b| generates the second-order component of the response. In this model, components above the N th order are neglected, since their contribution is assumed to be small. This approach is illustrated schematically in Figure 1.5. In this case, the total output is expressed as the sum of the contributions from each order of the nonlinear transfer function. It is important to note that nal output is independent of the order of the frequencies appearing in the Hn operators. As an example, H !a ; !b = H !b; !a by de nition. Although this is apparently somewhat mysterious, it is really only a restatement of the obvious fact that the output of the circuit is una ected by the ordering of the frequencies that we choose for the nonlinear transfer function. Now, a real sinusoidal input signal is composed of both positive and negative exponential frequency terms, whose magnitudes are equal but which spin in opposite directions in the complex plane. This implies that a single sinusoidal input at ! will generate frequencies at ! ; 2! ; 3! ; : : : due to the various distortion products. In a Volterra Series analysis, the second-order 1 2 2 2 1 1 1 1 c L. Larson 9 1.2. VOLTERRA OPERATIONS FOR CIRCUIT1. VOLTERRA SERIES CHAPTER ANALYSIS transfer function H !a; !b will have ! as its two input frequencies, and the third-order transfer function H !a; !b; !c will also have ! as its three input frequencies. But these will generate only integral multiples of the single input frequency | resulting in harmonic distortion at 2! and 3! . In addition, a sinusoidal input containing two sinusoidal inputs at ! and ! will generate frequencies at dc, 2! , and 2! the dc due to the second order nonlinearity generating ! , !  and ! , ! , as well as ! , !  and ! + ! . The second-order transfer function H !a; !b will have ! and ! as its input frequencies, and the third-order transfer function H !a; !b; !c will also have ! and ! as its input frequencies, resulting in intermodulation distortion at frequencies 2! , !  and 2! , ! . 2 1 3 1 1 1 1 2 1 2 1 1 2 2 1 2 1 2 2 1 2 3 1 2 1 2 2 1 Example 1.2 Nonlinear transfer function of a linear RC lowpass lter As an example, consider the simple linear single pole low-pass lter of Figure 1.6. In this case, the circuit is perfectly linear, and !x is the only frequency that a ects the amplitude of the output signal at frequency !x. As a result, the nonlinear transfer functions for this circuit are given by As an example, the second order intermodulation product at !1 , !2 is evaluated at !a = !1 and !b = ,!2 . The resulting transfer function is H2!a ; !b = 1 + j ! 1 ! RC 1.15 1, 2 1 H !a = 1 + j! RC a H !a; !b = 1 + j ! 1+ ! RC a b 1 H !a; !b; !c = 1 + j ! + ! + ! RC a b c 1 2 3 1.2 Volterra Operations for Circuit Analysis The manipulation of the Volterra operators is absolutely crucial for the calculation of circuit responses, but the vast array of possible responses complicates matters considerably. In order to simplify these calculations, various rules" have been developed to ease hand calculation. We will now derive c L. Larson January 7, 1999 10 CHAPTER 1. VOLTERRA SERIES 1.2. VOLTERRA OPERATIONS FOR CIRCUIT ANALYSIS some of the rules, and state others, with the aim of providing the reader with a complete tool box" of techniques for subsequent calculations. As a simple example of the kind of calculations that are required, let's examine the case of the series interconnection of two nonlinear circuits, as shown in Figure 1.7. In this case, the transfer functions are given by So = H !a  Si!a + H !a ; !b Si !a + !b + H !a ; !b; !c Si !a + !b + !c + : : : 1 1 2 3 2 3 1.16 and So = G !a So !a  + G !a; !b So !a + !b + G !a; !b; !c So !a + !b + !c + : : : 1 2 3 1 2 1 3 1 1.17 We are interested in nding the overall transfer function | let's call it K | in terms of G and H. Inserting Equation1.16 into Equation1.17 yields So = G !a  H !a Si !a + H !a ; !b Si !a + !b + H !a ; !b; !c Si !a + !b + !c + : : :!a 1 1 2 3 2 3 + G !a ; !b H !a Si!a  + H !a ; !b Si !a + !b + H !a ; !b; !c Si !a + !b + !c + : : :!a 2 1 2 3 2 3 2 + G !a ; !b; !c H !a Si !a + H !a ; !b Si !a + !b + H !a ; !b; !c Si !a + !b + !c + : : :!a 3 1 2 3 2 3 3 1.18a 1.18b 1.18c 1.18d 1.18e 1.18f 1.18g 1.18h 1.18i 1.18j 1.18k Now, several terms in Equation ??eq:SeriesVolterra3 will require some illumination. The rst is the square of the rst-order term Equation 1.18 c L. Larson 11 1.2. VOLTERRA OPERATIONS FOR CIRCUIT1. VOLTERRA SERIES CHAPTER ANALYSIS H !a Si!a  In terms of Volterra operators, this can be expressed as 1 2 1.19 H !a  Si !a = H !a H !b Si !a + !b 1 2 1 1 2 1.20 since H !a  Si !a = 1 2 = = H !a H !b Si !a + !b 1 1 2 2 32 3 Q Q X1 X1 4 Sq1 H1 !q1 ej!q1 t 5 4 Sq1 H1 !q2 ej!q2 t 5 q1 =,Q 2 q2 =,Q 2 Q Q X X1 Sq1 Sq2 H1 !q1 H1 !q2 ej!q1 +!q2 t q1 =,Q q2 =,Q 4 1.21 Similarly, the cube of the rst-order term Equation 1.18 can be expressed as H !a Si !a 1 3 = H !a H !b H !c Si !a + !b + !c1.22  1 1 1 3 since 2 3 Q X1 j!q1 t 5 H1 !a Si !a 3 = 4 2 Sq1 H1 !q1 e q =,Q 2 3 21 3 Q Q X1 X1 j!q2 t 5 4 j!q3 t 5 4 2 Sq2 H !q2 e 2 Sq3 H !q3 e q2 =,Q q3 =,Q = Q X Q X Q X q1 =,Q q2 =,Q q3 = 1 H ! H ! H ! S q1 q2 q3 q1 ,Q 8 Sq Sq ej!q1 !q2 !q3  1 1 1 + + 1.23 1.24 2 3 c L. Larson January 7, 1999 12 CHAPTER 1. VOLTERRA SERIES 1.2. VOLTERRA OPERATIONS FOR CIRCUIT ANALYSIS 1.25 1.26 1.27 = H !a  H !b H !c Si !a + !b + !c 1 1 1 3 Another third-order term occurs with the cross-product of the rst-order and second-order terms, i.e. H !a Si !a  H !a ; !b Si !a + !b = H !aH !b; !c Si !a + !b + !c 1 2 2 1 2 3 h i 1.28 Now, the magnitude of the transfer function can not depend on the order of the frequencies as they appear in the operator. Note that the symmetry of the nal Volterra operator | H !a H !b ; !c | is maintained here, though not explicitly. This is because !a, !b, and !c each represent all of the frequencies in the summation. If we were to express this result as an equivalent third-order transfer function G !a; !b; !c, we would have to maintain the symmetry in the overall expression by the following manipulation 1 2 3 G !a; !b; !c = 1=3 H !a H !b; !c + H !b H !a ; !c + H !c H !a; !b = H !a H !b; !c Si !a + !b + !c 1.29 3 1 2 1 2 1 2 1 2 3 This last expression | G !a; !b; !c | is often expressed as H !a H !b; !c to emphasize the symmetry required in the nal argument. With these results, we can express the nal transfer function of the series combination of two nonlinear circuits as 3 1 2 So = K !a So !a  + K !a; !b So !a + !b + K !a; !b; !c So !a + !b + !c + : : : 1 2 3 1 2 1 3 1 1.30 where K !a = H !a  G !a  1 1 1 c L. Larson 13 1.3. HARMONIC DISTORTION AND INTERMODULATION USING VOLTERRA SERIES CHAPTER 1. VOLTERRA SERIES K !a ; !b = G !a + !bH !a ; !b + G !a; !b H !a H !b K !a; !b; !c = G !a + !b + !cH !a; !b; !c + G !a; !b; !c H !aH !bH !c 2 + 3 G !a; !b + !c H !aH !b; !c + G !b; !a + !c H !bH !a; !c + 2 1 1 3 2 2 1 1 3 3 1 1 1 2 1 2 2 1 2 A similar situation can occur with the shunt combination of the nonlinear frequency dependent elements, as shown in Figure 1.8. In this case, if So = H !a Si!a  + H !a; !b Si !a + !b + H !a; !b; !c Si !a + !b + !c + : : : 1 1 2 3 2 3 1.32 and So = G !a Si!a  + G !a; !b Si !a + !b + G !a; !b; !c Si !a + !b + !c + : : : 2 1 2 3 2 3 1.33 then So + So = G !a  + H !a Si!a + G !a ; !b + H !a; !b Si !a + !b + G !a ; !b; !c + H !a; !b; !c Si !a + !b + !c + 1.34 ::: 1 2 1 2 3 1 2 2 3 3 1.3 Harmonic Distortion and Intermodulation Using Volterra Series Once the nonlinear transfer functions have been derived for the circuit in question, it is relatively straightforward to calculate the harmonic distortion and intermodulation distortion. The advantage of the technique now is that these quantities can now be calculated as a function of frequency, in contrast to a memoryless power series approximation of the circuit. c L. Larson January 7, 1999 14 1.3. HARMONIC DISTORTION AND INTERMODULATION USING CHAPTER 1. VOLTERRA SERIES VOLTERRA SERIES For example, the third-order harmonic distortion can be calculated by examining the third-order response of the Volterra series with a single sinusoidal input. The full expansion of the third-order term, derived from Equation  1.12 is sot rd = 3 Q X Q X Q X 1.35 If we apply a single sinusoidal input of amplitude S to the circuit | say at frequency ! | then Q = 1 in this case, and the third-order harmonic output will be at 3! . There will be only one term at this frequency, and so the output at frequency 3! will be 1 1 1 1 q1 =,Q q2 =,Q q3 = 1 S S S H ! ; ! ; !  ej !q1 q1 q2 q3 q1 q2 q3 ,Q 8 3  + !q2 +!q3 t h sot !1 = 1 S H ! ; ! ; ! ej 8 h = 1 S H ! ; ! ; ! ej 8 3 3 1 3 1 3 1 1 1 3 1 1 1 3 1  ! t+H ! t+ 3 ,! ; ,! ; ,! e,j 1 1 1 3 1 1 1 3 1  3 1  !t i 3 1  H ! ; ! ; ! ej 1 i !t 1.36 1.37 Therefore, the magnitude of the output at frequency 3! is 1.38 Now, this gives us the magnitude of the third-order harmonic distortion as a function of frequency. The magnitude of the desired signal as a function of frequency is given by 3 1 1 1 jsoHD j = jH ! ; ! ; ! j S 4 3 3 1 So the third-order fractional harmonic distortion is given by: the Third Harmonic Output HD = Amplitude ofof the Fundamental Output Amplitude = jH ! ; ! ; ! jS 4jH ! jS !; ; = jH 4jH!! ! jjS  3 3 1 1 1 3 1 1 1 1 3 1 1 1 2 1 1 1 jsodes j = jH ! jS 1 1 1 1.39 1.40 c L. Larson 15 1.3. HARMONIC DISTORTION AND INTERMODULATION USING VOLTERRA SERIES CHAPTER 1. VOLTERRA SERIES It is illuminating and reassuring to compare this result to that obtained for the power-series expansion of the third-order harmonic distortion in the previous Chapter the Third Harmonic Output HD = Amplitude ofof the Fundamental Output Amplitude = a S =4 aS = aS 1.41a 4a So, the two relationships are very similar, with the power series coe cients being replaced by Volterra Series coe cients in this case. In a similar manner, we can derive the third-order intermodulation distortion in the Volterra case. If we apply two sinusoidal inputs of amplitude S and S to the circuit | say at frequencies ! and ! | then Q = 2 in this case, and the third-order intermodulation output will be at 2! , ! and 2! , ! . There will be three terms at this frequency, and so the output at 2! , ! will be 3 3 3 1 2 1 1 1 3 1 1 2 1 2 1 2 2 2 1 1 Now, this gives us the magnitude of the third-order intermodulation distortion as a function of frequency. The magnitude of the desired signal as a function of frequency is given by ; jsotIM j = 3jH ! ; ! 4 ,! jS S 3 3 2 2 1 2 2 1 1.42 jsotdesj = jH ! jS 1 1 1 1.43 So the third-order intermodulation distortion is given by: IM = Amplitude of the Third-Order Intermodulation Output Amplitude of the Fundamental Output = 3jH ! ; ! ; ,! jS S 4jH ! jS ;; = 3jH !jH!!,! jS 1.44a 4 j 3 3 2 2 1 2 2 1 1 1 1 3 2 2 1 2 2 1 1 c L. Larson January 7, 1999 16 CHAPTER 1. 1.4. EFFECTS OF FEEDBACK ON VOLTERRA SERIES VOLTERRA SERIES Once again, it is interesting to compare this to the result obtained for the power-series expansion in the previous Chapter IM = Amplitude of the Third-Order Intermodulation Output Amplitude of the Fundamental Output = 3aa S =4 S = 3aaS 1.45a 4 3 3 3 1 1 3 1 2 1 1 So, the two relationships are very similar, with the power series coe cients being replaced by Volterra Series coe cients in this case. 1.4 E ects of Feedback on Volterra Series We saw in the previous Chapter that the use of negative feedback could improve the linearity performance of a communications circuit. In that case, we derived a new set of power series coe cients for the circuit transfer function when the feedback network was itself perfectly linear. Let's examine the e ects of feedback on linearity in the case where the Volterra coe cients are known, and a perfectly linear feedback network is placed around the original circuit. In this case, the circuit is con gured as shown in Figure 1.9 We need to solve for a nal transfer function of the form So = G !a Si!a  +G !a; !b Si !a + !b  +G !a; !b; !c Si !a + !b + !c + : : : 1 2 3 2 3 1.46 We can express the output of the feedback block | Sf | as Sf = f !aG !a  Si!a +f !a + !bG !a; !b Si !a + !b +f !a + !b + !cG !a; !b; !c Si !a + !b + !c + : : : 1.47 1 2 2 3 3 c L. Larson 17 1.4. EFFECTS OF FEEDBACK ONCHAPTER 1. SERIES VOLTERRA VOLTERRA SERIES Notice that the feedback is applied at each of the frequencies generated by the nonlinearity | !a in the case of a rst-order nonlinearity, !a + !b in the case of a second-order nonlinearity, etc. This feedback term is now subtracted from the input, and the resulting signal at the input to the gain stage H is Se = Si , Sf = 1 , f !aG !a Si!a , f !a + !bG !a; !b Si !a + !b , f !a + !b + !cG !a ; !b; !c Si !a + !b + !c + : : : 1.48 1 2 2 3 3 which is applied to the input of the ampli er block, the resulting output being So = H !a  Se!a  + H !a ; !b Se !a + !b + H !a ; !b; !c Se !a + !b + !c + : : : 1 2 3 2 3 1.49 So, the nal output can be expressed as So = G !a  Si!a +G !a; !b Si !a + !b +G !a; !b; !c Si !a + !b + !c + : : : 1 2 3 2 3 1 1 1 1 2 2 1.50 1.51 = H !a 1 , f !aG !a Si!a ,H !a + !b f !a + !bG !a ; !b Si !a + !b ,H !a + !b + !c f !a + !b + !cG !a; !b; !c Si !a + !b + !c + : : : +H !a; !b 1 , f !aG !a 1 , f !bG !b Si !a + !b ,2H !a ; !b + !c 1 , f !aG !a f !b + !cG !b; !c Si !a + !b + !c +H !a; !b; !c 1 , f !aG !a  1 , f !bG !b 1 , f !cG !c Si !a + !b + !c 1.52 3 3 2 1 1 2 2 1 2 3 3 1 1 1 3 c L. Larson January 7, 1999 18 CHAPTER 1. 1.4. EFFECTS OF FEEDBACK ON VOLTERRA SERIES VOLTERRA SERIES So, we can now equate each term of this Equation to produce the resulting Volterra series coe cients for every term in the output sequence. The rst-order term can be determined by G !a = H !a 1 , f !aG !a 1.53 which implies that a 1.54 G !a  = 1 + fH!!H !   a a which is reassuring, as it is the familiar rst-order transfer function. The second-order term can be calculated as follows 1 1 1 1 1 1 G !a; !b Si !a + !b = ,H !a + !b f !a + !bG !a; !b Si !a + !b + H !a; !b 1 , f !aG !a 1 , f !bG ! Si !a + !b 1.55 2 2 1 2 2 2 1 1  2 which allows us to solve for the second-order term 1 ! a G !a; !b = H !a; !b 1 +, f!!+G Ha !1 , f !bG !b  f !  +!  2 2 1 1 a b 1 a b 1.56 The third-order term is given by G !a; !b; !c = ,H !a + !b + !cf !a + !b + !cG !a ; !b; !c ,2H !a; !b + !c 1 , f !aG !a  f !b + !cG !b; !c +H !a; !b; !c 1 , f !aG !a 1 , f !bG !b 1 , f !cG !c 1.57 3 1 3 2 1 2 3 1 1 1 or !b ,f a G !a; !b; !c = ,2H !a ; 1 ++ !c 1 + ! !aGf!! f !b + !cG !b; !c H !a b + !c  a + !b + !c  + H !a ; !b; !c 1 , f !aG !a 1 , f !bG !b 1 , f !cG !c1.58 1 + H !a + !b + !cf !a + !b + !c 3 2 1 2 1  1 1 1 1 c L. Larson 19 1.5. EXAMPLE CALCULATION USING VOLTERRA SERIES SERIES CHAPTER 1. VOLTERRA 1.5 Example Calculation Using Volterra Series The use of Volterra Series for the calculation of even simple circuit responses can be maddeningly complicated. When the complexity of the circuit exceeds a certain level, the use of a commercial circuit simulator is recommended. However, the use of these techniques for hand analysis of simple circuits can be very powerful. It can provide a modicum of intuitive insight into a circuit that appears to be opaque except to a computer. However, so far we have stayed in the idealized world of block diagrams. Let's now apply these techniques to some real-world applications. There are some interesting areas to explore as a result of this technique 1. How do we model simple capacitive or inductive nonlinearities? 2. What happens when multiple nonlinear circuits are interconnected? 3. How can the transfer function of a complex circuit be built up from simpler circuits? There are two cases for circuit analysis that we need to consider initially. The rst is the case of resistive or memoryless nonlinearities. Our general Volterra model of a nonlinear two-terminal device is given by Figure 1.10 In this case, the output current is a function of the input voltage, i.e. in = a vi + a vi + a vi + : : : 1.59 This transfer function can be expressed as an output current as a function of input voltage, or an output voltage as a function of input current through series inversion. An example of an expansion like this would be the transconductance of a bipolar transistor, or the nonlinear I-V characteristics of a diode. The second case of capacitive nonlinearities will be considered next. A nonlinear capacitor say the Cgs of a MOSFET or the Cbc of a bipolar transistor can be modeled as 1.60 i = dQ = dQ d = c   d dt d dt dt Now, if 1 2 2 3 3 c L. Larson January 7, 1999 20 CHAPTER 1. EXAMPLE CALCULATION USING VOLTERRA SERIES 1.5. VOLTERRA SERIES c  = c + c + c 0 1 2 2 and 0 +::: 1.61 i = c d +c d +c d +c d + dt dt dt dt d + c d + c d + = c dt 2 dt 3 dt 1.62 This model of the capacitive nonlinearity is illustrted in Figure 1.11. It immediately presents us with the dilemma of the appropriate Volterra representation of the derivatives. The rst-order derivative can be simpli ed by 1 2 2 3 3 0 1 2 2 3 d H j!  S !  = a ia dt 1 1 S H ! d ej!q1 t q1 q1 dt q1 ,Q 2 Q X 1 = j!q1 Sq1 H !q1 ej!q1 t q1 ,Q 2 = j!a H j!a Si !a 1.63 1 = 1 = 1 Q X So, the basic rule" of this result is that a rst order time derivative of any order Volterra output can be obtained by multiplying the output by j!x where !x is the frequency produced by the nth order output. In other words d hH j!  H j!  S ! + ! i = a b b ia dt j !a + !b H j!a H j!b Si !a + !b 1 1 2 1 1 2 1.64 1.65 and d hH ! ; !  S ! + ! i = j ! + !  H ! ; !  S ! + ! 1.66 ab b a b ab b ia ia dt Inductive nonlinearities, though rarely encountered in microwave electronics, can be modeled in a similar manner. 2 2 2 2 c L. Larson 21 1.5. EXAMPLE CALCULATION USING VOLTERRA SERIES SERIES CHAPTER 1. VOLTERRA Now, we are ready to use these results to calculate a simple" circuit example. We will start with the simplest possible example | a nonlinear RC lowpass lter shown in Figure 1.12, where the resistor is completely linear, but the capacitor is nonlinear. In this case, the nonlinear capacitor might represent the reverse-bias capacitance of an input protection diode at the bond-pad of a high-frequency integrated circuit input. Volterra Series calculations of multiple node circuits are very similar to calculations involving linear circuits, with the caveat that KCL must be maintained for all orders of nonlinearity. The rst-order case is calculated rst, and this generates higher order nonlinearities which can be employed to calculate subsequent nonlinearities. In this case, the steps required to do this are fairly straightforward. We are looking for an output of the form So = H !a  Si!a 1.67 + H !a ; !b Si !a + !b 1.68 + H !a ; !b; !c Si !a + !b + !c + : : : 1.69 and we need to calculate each of the coe cients in turn, starting with the input Now, the KCL calculation at the output node is 1 2 3 2 3 g So , Si + c dSo + c2 dSo + c3 dSo + : : : = 0 dt dt dt which can be expanded into 1 0 1 2 2 3 1 1 2 2 3 3 1.70 0 = g H !a , 1 Si!a + H !a; !b Si !a + !b 1.71a + H !a; !b; !c Si !a + !b + !c + : : : 1.71b 1.71c 1.71d + c d H !a Si !a + H !a; !b Si !a + !b dt + H !a; !b; !c Si !a + !b + !c + : : : 1.71e 1.71f c d H !  S !  + H ! ; !  S ! + !  1.71g + 2 dt a ia ab b ia + H !a; !b; !c Si !a + !b + !c + : : : + : : : 1.71h 0 1 2 2 3 3 1 1 2 2 3 3 2 c L. Larson January 7, 1999 22 CHAPTER 1. EXAMPLE CALCULATION USING VOLTERRA SERIES 1.5. VOLTERRA SERIES Now, it is not too hard to group each of the terms by their order. The rst-order term is 0 = g H !a , 1 Si!a + c d H !a Si !a dt = g H !a , 1 Si!a + c j!a H !a Si!a 1 1 1 1 0 0 1 1 1.72 1.73 which implies that where r = 1=g . This is the rst order transfer function, and its value is just what we would expect from simple linear circuit analysis. Now, we can calculate the second-order transfer function, by equating second order terms. 1 1 1 H !a  = 1 + ! r c 1 a 1 o 0 = g H !a; !b Si !a + !b +c d H !a; !bSi !a + !b dt c d H !  S !  = 0 + a ia 2 dt = g H !a; !b Si !a + !b +c j !a + !bH !a; !bSi !a + !b c j ! + !  H !  H !  S ! + !  1.74 +2 a b a b b ia 1 2 2 0 2 2 1 1 2 1 2 2 0 2 2 2 1 1 1 And the nal second-order transfer function is rc  H j!a; j!b = ,j !a2+1!brc jH!j!a! H j!b 1.75 +  a + b Now, we can use this result to calculate the second-order harmonic distortion H! HD = SijjH ! ;j! j = So jH ! ; ! j jH ! j and the magnitude of the output is given by 2 1 1 1 0 1 2 2 2 1 1 1 2 2 1 1 1 1 1 1 2 c L. Larson 23 1.5. EXAMPLE CALCULATION USING VOLTERRA SERIES SERIES CHAPTER 1. VOLTERRA 1 2So!rc 1.76 1 + r c 4!  Now, let's examine the case of third-order distortion by applying KCL at the output node. jHD j = 2 1 22 0 2 1 2 g H !a; !b; !c Si !a + !b + !c +c j !a + !b + !c H !a; !b; !c Si !a + !b + !c c j ! + ! + ! H !  H !  H !  S ! + ! + !  +3 a b c a b c b c ia c j ! + ! + ! H !  H ! ; !  S ! + ! + !  = 0 +2 a b c a bc b c ia 1 3 3 3 3 0 2 1 1 1 3 1 1 2 3 1.77 1.78 1.79 1.80 So, the output third order transfer function is H= 3 ,j !a + !b + !c h c1 H 2 !aH !b; !c + c2 H !a H !bH !c 1.81 g + c j !a + !b + !c 1 2 3 1 1 1 1 0   i Now, we calculated the intermodulation earlier, oj IM  3 4 sj2HH3!1!2jj;!21;,!1j;2j 1.82 H !2 1 So, we have now expended a great deal of e ort to analyze a simple RC lter, but we have also developed some simple expressions that can be used to analyze the distortion characteristics of an arbitrary nonlinear circuit. We are trying to nd the power-series coe cients of the nal transfer function, i.e. 3     id !a = b !avin!a + b !a vin!a 1.83 Since the feedback coe cients are frequency dependent, the resulting closed-loop power series coe cients must also be frequency dependent. This presents a signi cant complication for circuit analysis, so we will have to treat each intermodulation and harmonic distortion term on an individual basis at each frequency. Now, inserting Equation 1.83 into Equation 1.4 yields 1 2 2 c L. Larson January 7, 1999 24 CHAPTER 1. EXAMPLE CALCULATION USING VOLTERRA SERIES 1.5. VOLTERRA SERIES b !avin!a  + b !avin!a = a vin!a , j!aLb !avin!a + b !a vin!a 1.84 + a vin!a , j!aLb !avin + b !avin  1.85 !a  1 2 2 1 2 1 1 2 2 2 2 2 We can derive the rst-order terms rst from this expression, by equating the linear terms b !a  = a , j!La b !a  1 1 11 1.86 1.87 which implies that Now we move onto the second-order terms, by once again equating coe cients. However, the matter is complicated somewhat by the multiple frequencies present in the second-order term. a b !a = 1 + j!a L 1 1 1 vin = S cos ! t + S cos ! t 1.88a = S 1 + cos! + ! t + cos! , ! t + cos2! t=2 + cos2! t=2 1.88b 2 1 1 1 2 2 2 1 1 2 1 2 1 2 So the second-order term exhibits a total of ve new frequencies, each with its own unique transfer function. Fortunately, in the case of secondorder intermodulation, we are only interested in two of these ! + ! and ! , ! . There are two sources of these frequencies on the right-hand side of Equation ?? The rst is the ,a j!Lb vin term, which yields 1 2 1 2 1 2 2 ,a j ! +! Lb ! +! S cos! +! t,a j ! ,! Lb ! ,! S cos! ,! t 1.89 The second generator of these frequencies is the a 1 , j!Lb vin term, which yields 2 1 2 1 1 2 2 1 2 2 1 1 2 1 1 2 2 1 2 2 1 1 2 a 1 , j! Lb S cos ! t + 1 , j! Lb S cos ! t and the terms at frequencies ! + ! and ! , ! are 2 1 1 1 1 2 1 1 2 1 2 1 2 2 1.90 c L. Larson 25 1.5. EXAMPLE CALCULATION USING VOLTERRA SERIES SERIES CHAPTER 1. VOLTERRA a S 1 , j! Lb 1 , j! Lb ! t + ! cos! t + ! t + 1 , j! Lb 1 , j! Lb ! t , ! 1.91 So, equating all of the second order coe cents at ! + ! yields 2 2 1 1 1 2 1 1 2 1 2 1 1 2 1 1 1 2 2  , 1.92 b ! + !  = a 1 , j!1Lb a ! 1+ !j! Lb !  + j!  a = 1.93 1 + a j ! + ! L1 + a j! L1 + a j! L and similarly equating all of the second order coe cents at ! , ! yields 2 1 2 2 1 1 1 2 1 2 1 1 2 2 1 1 2 1 2 1 1 1 2 Lb ! 1 1.94 b ! , !  = a 1 , j! + aj ! ,, j! Lb !  1 ! L = 1 + a j ! , ! L1a+ a j! L1 + a j! L 1.95 So the amplitude of the second-order terms at ! , ! is 2 1 2 2 1 1 1 2 1 2 1 1 2 2 1 1 2 1 2 1 1 1 2 Amplitude = S 1 + a j ! , ! L1a+ a j! L1 + a j! L 1.96 and amplitude of the second-order terms at ! + ! is 2 1 2 1 1 2 1 2 1 1 1 2 Amplitude = S 1 + a j ! + ! L1a+ a j! L1 + a j! L 1.97 and the amplitude of the fundamental at ! is 2 1 2 1 1 2 1 2 1 1 1 a Amplitude = S 1 + j! a L so the intermodulation at the di erence frequency is 1 1 1 1 1.98 IM = 2 S 2 1 1+ 1  1 a = S a 1 + a j ! , ! 1L1 + a j! L 1 2 1 1 1 2 1 2 a2 a j ! ,!2 L1+a1 j!2 L1+a1 j!1 L a S1 1+j!11a1 L 1.99 1.100 c L. Larson January 7, 1999 26 CHAPTER 1. EXAMPLE CALCULATION USING VOLTERRA SERIES 1.5. VOLTERRA SERIES and the intermodulation at the sum frequency is IM = 2 S 2 2 1 1+ 1  1 + 2  1+ 1 aj! ! L a = S a 1 + a j ! + ! 1L1 + a j! L 1.102 The input intercept point occurs when the value of IM2 is unity so 1 2 1 1 1 2 1 2 a j!2 L1+a1 j!1 L a1 S1 1+j!1a1 L a 1.101 IIP 2sum = a 1 + a j ! + ! L1 + a j! L a 1 2 1 1 2 1 2 1.103 and IIP 2sum = a 1 + a j ! , ! L1 + a j! L a 1 2 1 1 2 1 2 1.104 for the di erence frequency term. Note that at dc, this result converges to the memoryless power-series result from the previous Chapter for the second-order input intercept point. Even more elaborate calculations can be performed for the third-order case. This approach to the development of a power series expansion of a frequency dependent nonlinearity has an ad-hoc" quality to it, and there is clearly a need for a systematic approach for this class of problems. The use of Volterra Series will provide a convenient means for performing this calculation. Problem 1.1 Number of possible frequencies Derive an expression for the possible number of new frequencies created with a fourth order nonlinearity and three input tones. Problem 1.2 Volterra Coe cients Calculate the expression for IM2 using the Volterra Coe cients Problem 1.3 Volterra Coe cients Derive the third-order Volterra transfer function G3 !a ; !b ; !c for the case of linear feedback. c L. Larson 27 1.5. EXAMPLE CALCULATION USING VOLTERRA SERIES SERIES CHAPTER 1. VOLTERRA c L. Larson January 7, 1999 28 Bibliography 1 H.L. Vazquez and H.A. Jardon, Analysis of am-pm conversion in a communication link.," Rome, Italy, 1996, pp. 471 6 vol.2, Univ. Rome La Sapienza'. 2 T. Ishikawa, K. Nakashima, Y. Kajikawa, and Y. Nomura, A consideration on elimination of nonlinear distortion of the loudspeaker system by using digital volterra lter.," Transactions of the Institute of Electronics, Information and Communication Engineers A, vol. vol.J79-A,, no. no.7, pp. 1236 43, July 1996. 3 S. Maas, Nonlinear Microwave Circuits, IEEE Press, 1995. 4 N. Wiener, Nonlinear Problems in Random Theory, MIT Press, 1958. 29 FIGURES FIGURES I V in out L1 Figure 1.1: Long-channel MOSFET with inductive feedback. c L. Larson January 7, 1999 30 FIGURES FIGURES V in( t ) LPF Vout ( t ) a1 a2 a 3 Figure 1.2: Model of linear lter followed by nonlinear circuit. c L. Larson 31 FIGURES FIGURES V in( t ) LPF Vout ( t ) a1 a2 a 3 Figure 1.3: Model of nonlinear circuit followed by a linear lter. c L. Larson January 7, 1999 32 FIGURES FIGURES BPF Desired HD2 HD3 Ampl. (V) Frequency (Hz) Figure 1.4: RC lowpass lter. c L. Larson 33 FIGURES FIGURES H1( ω) H2( ω) Vout V in H3( ω) Hn(ω ) Figure 1.5: Frequency domain model of Volterra Series, as the sum of the contributions from each order of the nonlinear transfer function. c L. Larson January 7, 1999 34 FIGURES FIGURES R V in C Vout Figure 1.6: Linear RC low-pass lter. c L. Larson 35 FIGURES FIGURES R V in C Vout Figure 1.7: Linear RC low-pass lter. c L. Larson January 7, 1999 36 FIGURES FIGURES R V in C Vout Figure 1.8: Linear RC low-pass lter. c L. Larson 37 FIGURES FIGURES Se S in H Sout f Figure 1.9: Feedback operation of Volterra circuit. c L. Larson January 7, 1999 38 FIGURES FIGURES Iin V in a1 a 2 a3 Figure 1.10: Nonlinear model of memoryless nonlinearity. c L. Larson 39 FIGURES FIGURES Iin V in c0 c 1 c2 3 2 Figure 1.11: Nonlinear model of capacitive nonlinearity. c L. Larson January 7, 1999 40 FIGURES FIGURES R V in C Vout Figure 1.12: Nonlinear capacitor LPF example. c L. Larson 41 ...
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