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5.1 Dr. Yuri Panarin, DT021/4, Electronics Introduction Nonlinear operations on continuous-valued analog signals are often required in instrumentation, communication, and control-system design. These operations include • rectification, • Modulation - demodulation, • frequency translation, 7. Gilbert cell & Analog Multipliers • multiplication, and division. In this chapter we analyze the most commonly used techniques for performing multiplication and division within a monolithic integrated circuit In analog-signal processing the need often arises for a circuit that takes two analog inputs and produces an output proportional to their product. Such circuits are termed analog multipliers. In the following sections we examine several analog multipliers that depend on the exponential transfer function of bipolar transistors . Recommended Text: Gray, P.R. & Meyer. R.G., Analysis and Design of Analog Integrated Circuits (3rd Edition), Wiley (1992) pp. 667-681 1 DT021/4 Electronics – 7 Analog Multipliers The Emitter-Coupled Pair The emitter-coupled pair, was shown in to produce output currents that were related to the differential input voltage by : The Emitter-Coupled Pair Q1 Q2 + Vi1 The emitter-coupled pair, was shown in to produce output currents that were related to the differential input voltage by : I c1 = I c 2 exp(Vid / VT ) = ( I EE − I c1 ) exp(Vid / VT ) Ic2 Ic1 Vi1 − Vbe1 + Vbe 2 − Vi 2 = 0 + IEE - - Vi2 V −V V I c1 I S1 exp be1 be 2 = exp id = I c2 I S 2 VT VT + Vid I EE = −( I e1 + I e 2 ) = ( I c1 + I c 2 ) / α F ≈ I c1 + I c 2 Q1 Q1 I EE exp(Vid / VT ) I EE = 1 + exp(Vid / VT ) 1 + exp(−Vid / VT ) + IEE - + Vid I c1 = IEE I EE 1 + exp(−Vid / VT ) Ic2 = I EE 1 + exp(Vid / VT ) - Vi2 Ic2 Ic1 Q2 - Q2 + Vi1 I c1 = Ic2 Ic1 Ic2 Ic1 I c1 (1 + exp(Vid / VT )) = I EE exp(Vid / VT ) Vbe1 = VT ln(I c1 / I S1 ) I c1 = I S1 exp(Vbe1 / VT ) Vbe 2 = VT ln (I c 2 / I S 2 ) I c 2 = I S 2 exp(Vbe 2 / VT ) 2 DT021/4 Electronics – 7 Analog Multipliers Q1 Q2 IEE I c1 = I c 2 exp(Vid / VT ) = ( I EE − I c1 ) exp(Vid / VT ) DT021/4 Electronics – 7 Analog Multipliers 3 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 4 5.2 Dr. Yuri Panarin, DT021/4, Electronics Notes The Emitter-Coupled Pair The emitter-coupled pair, was shown in to produce output currents that were related to the differential input voltage by : Q1 + I EE I EE I c1 = Ic2 = 1 + exp(−Vid / VT ) 1 + exp(Vid / VT ) Vid Ic2 Ic1 Q2 IEE ∆I c = I c1 − I c 2 = I EE tanh(Vid / 2VT ) This relationship is plotted => and shows that the emitter-coupled pair by itself can be used as a primitive multiplier. tanh(x) = x + x 3 / 3 + .... ≅ x or assuming (Vid / 2VT ) << 1, ⇒ ∆I c = I EE (Vid / 2VT ) 5 DT021/4 Electronics – 7 Analog Multipliers DT021/4 Electronics – 7 Analog Multipliers 6 Notes Simple Multiplier The current IEE is actually the bias current for the emitter-coupled pair. With the addition of more circuitry, we can make IEE proportional to a second input signal. Thus we have Ic2 Ic1 + Vid - I EE ≅ K o (Vi 2 − VBE ( on ) ) Q1 Q2 IEE R + The differential output current of the V i2 emitter-coupled pair can be calculated to give ∆I c ≅ Q3 Q4 K oVid (Vi 2 − VBE ( on ) ) 2VT DT021/4 Electronics – 7 Analog Multipliers 7 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 8 5.3 Dr. Yuri Panarin, DT021/4, Electronics Two-Quadrant restriction Notes Thus we have produced a circuit that functions as a multiplier under the assumption that Vid is small, and that Vi2 is greater than VBE(on). The latter restriction means that the multiplier functions in only two quadrants of the Vid - Vi2 plane, and this type of circuit is termed a two-quadrant multiplier. The restriction to two quadrants of operation is a severe one for many communications applications, and most practical multipliers allow four-quadrant operation. The Gilbert multiplier cell, shown, is a modification of the emitter-coupled cell, which allows four-quadrant multiplication. 9 DT021/4 Electronics – 7 Analog Multipliers DT021/4 Electronics – 7 Analog Multipliers Notes Gilbert multiplier cell The Gilbert multiplier cell is the basis for most integratedcircuit balanced multiplier systems. The series connection of an emitter-coupled pair with two cross-coupled, emitter-coupled pairs produces a particularly useful transfer characteristic,. IO =I35 - I46 I35 I3 I4 Q3 Q4 I c5 = I c1 I c1 I = 1 + exp(−V1 / VT ) c 4 1 + exp(V1 / VT ) Ic2 1 + exp(V1 / VT ) Ic6 = I6 Q5 I1 Q6 I2 Q1 I c3 = I46 I5 V1 10 Q2 V2 IEE Ic2 1 + exp(−V1 / VT ) DT021/4 Electronics – 7 Analog Multipliers 11 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 12 5.4 Dr. Yuri Panarin, DT021/4, Electronics Gilbert cell - DC Analysis The two currents Ic1 and Ic2 are related to V2 I3 I4 Q3 Q4 I EE [1 + exp(−V1 / VT )][1 + exp(−V2 / VT )] I c4 = [1 + exp(V1 / VT )][1 + exp(−V2 / VT )] I c5 = [1 + exp(V1 / VT )][1 + exp(V2 / VT )] I c6 = [1 + exp(−V1 / VT )][1 + exp(V2 / VT )] I46 I5 Substituting Ic1 and Ic2 in expressions for Ic3 , Ic4, Ic5 and Ic6 get : V1 I c3 = IO =I35 - I46 I35 I EE I EE I c1 = I c2 = 1 + exp(−V2 / VT ) 1 + exp(V2 / VT ) Notes I6 Q5 Q6 I1 I EE I2 Q1 Q2 V2 I EE IEE I EE 13 DT021/4 Electronics – 7 Analog Multipliers DT021/4 Electronics – 7 Analog Multipliers Gilbert cell First show that 1 1 − = tanh( x / 2) 1 + e−x 1 + e x 14 Notes e− x / 2e x / 2 = 1 e x / 2e x / 2 = e x 1 1 1 + e x −1 − e− x − = = 1 + e− x 1 + e x 1 + e− x 1 + e x ( = e (e x/2 ) −x e −e = + e− x / 2 e x / 2 e− x / 2 + e x / 2 x −x / 2 )( e − x / 2e − x / 2 = e − x ) ( ) a 2 − b 2 = ( a + b)(a − b) e x – e -x = (e x/ 2 + e -x/ 2 ) ⋅ (e x/ 2 - e -x/ 2 ) = (e (e )( )( ) ) + e−x / 2 e x / 2 − e−x / 2 = tanh( x / 2) x/2 + e−x / 2 e−x / 2 + e x / 2 x/2 DT021/4 Electronics – 7 Analog Multipliers tanh (x) = e x - e -x e x + e -x 15 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 16 5.5 Dr. Yuri Panarin, DT021/4, Electronics Gilbert cell Notes The differential output current is then given by ∆I = I c 3−5 − I c 4−6 = I c 3 + I c 5 − (I c 4 + I c 6 ) = (I c 3 − I c 6 ) − (I c 4 − I c 5 ) I c3 − I c 6 = (1 + e I EE = 1 + e −V1 / VT ( I EE I EE − = 1 + e −V2 / VT 1 + e −V1 / VT 1 + eV2 / VT −V1 / VT )( ) ( )( 1 1 − −V2 / VT 1 + eV2 / VT 1+ e )( ) ( ) I EE = tanh(V2 / 2VT ) −V1 / VT 1+ e ) ( ) 1 1 − = tanh( x / 2) 1 + e−x 1 + e x Similar: Ic 4 − Ic4 = (1 + e V1 / VT I EE I EE I EE − = tanh(V2 / 2VT ) 1 + e −V2 / VT 1 + eV1 / VT 1 + eV2 / VT 1 + eV1 / VT )( ) ( )( ) ( ) 17 DT021/4 Electronics – 7 Analog Multipliers DT021/4 Electronics – 7 Analog Multipliers Gilbert cell Notes The differential output current is then given by ∆I = I c 3−5 − I c 4−6 = I c 3 + I c 5 − (I c 4 + I c 6 ) = (I c 3 − I c 6 ) − (I c 4 − I c 5 ) Where I c3 − I c6 = I EE (1 + e −V1 / VT ∆I = (I c 3 − I c 6 ) − (I c 4 − I c 5 ) = ) tanh(V 2 I EE (1 + e −V1 / VT I c 4 − I c5 = / 2VT ) ) tanh(V 2 / 2VT ) − 18 I EE (1 + e V1 / VT I EE (1 + e V1 / VT ) tanh(V 2 ) tanh(V 2 / 2VT ) / 2VT ) = I EE I EE = − tanh(V2 / 2VT ) = I EE tanh(V1 / 2VT ) tanh(V2 / 2VT ) −V1 / VT 1 + e 1 + eV1 /VT ( Finally ) ( ) ∆I = I EE tanh(V1 / 2VT ) tanh(V2 / 2VT ) The dc transfer characteristic, then, is the product of the hyperbolic tangent of the two input voltages. The are three main application of Gilbert cell depending of the V1 an V2 range: tanh(V1, 2 / 2VT ) ≅ V1, 2 / 2VT DT021/4 Electronics – 7 Analog Multipliers 19 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 20 5.6 Dr. Yuri Panarin, DT021/4, Electronics Gilbert cell Applications Notes ∆I = I EE tanh(V1 / 2VT ) tanh(V2 / 2VT ) (1) If V1 < VT and V2 < VT then : and it woks as multiplier (2) If one of the inputs of a signal that is large compared to VT, this effectively multiplies the applied small signal by a square wave, and acts as a modulator. (3) If both inputs are large compared to VT, and all six transistors in the circuit behave as nonsaturating switches. This is useful for the detection of phase differences between two amplitude-limited signals, as is required in phase-locked loops, and is sometimes called the phase-detector mode. DT021/4 Electronics – 7 Analog Multipliers 21 DT021/4 Electronics – 7 Analog Multipliers Gilbert cell as Multiplier 22 Notes (1) If V1 < VT and V2 < VT then : tanh(x) = x + x 3 / 3 + .... ≅ x Thus for small-amplitude signals, the circuit performs an analog multiplication. Unfortunately, the amplitudes of the input signals are often much larger than VT, An alternate approach is to introduce a nonlinearity that predictors the input signals to compensate for the hyperbolic tangent transfer characteristic of the basic cell. The required nonlinearity is an inverse hyperbolic tangent characteristic DT021/4 Electronics – 7 Analog Multipliers 23 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 24 5.7 Dr. Yuri Panarin, DT021/4, Electronics Pre-warping circuit inverse hyperbolic tangent Notes We assume for the time being that the circuitry within the box develops a differential output current that is linearly related to the input voltage 7i. Thus I1 = I o1 + K1V1 and I 2 = I o1 − K1V1 Here Io1 is the dc current that flows in each output lead if V1 is equal to zero, and K1 is the transconductance of the voltage-to-current converter I + K1V1 I − K1V1 I + K1V1 = VT ln o1 - VT ln o1 ∆V = VT ln o1 Is Is I o1 − K1V1 The differential voltage developed across the two diode-connected transistors is tanh -1x = ln((1 + x)/(1 - x) ) /2 Using the identity: We get And finally KV ∆V = 2VT tanh −1 1 1 I o1 KV K V ∆I = I EE 1 1 2 2 I o1 I o 2 DT021/4 Electronics – 7 Analog Multipliers 25 DT021/4 Electronics – 7 Analog Multipliers Complete Analog Multiplier Vout = I EE K 3 26 Notes K1 K 2 V1V2 = 0.1V1V2 I o1 I o 2 DT021/4 Electronics – 7 Analog Multipliers 27 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 28 5.8 Dr. Yuri Panarin, DT021/4, Electronics Notes Balanced Modulator In communications systems, the need frequently arises for the multiplication of a continuously varying signal by a square wave. This is easily accomplished with the multiplier circuit by applying a sufficiently large signal directly to the cross-coupled pair. Vm (t ) = Vm cos ω mt ∞ nπ nπ Vc (t ) = ∑ An cos nω c t , where An = sin / 2 4 n =1 ∞ Vo (t ) = K [Vc (t )Vm (t )] = K ∑ AnVm cos ω mt cos nω c t = n =1 ∞ = K∑ n =1 AnVm cos(nω c t − ω nt ) cos(nω c t + ω nt ) 2 DT021/4 Electronics – 7 Analog Multipliers 29 DT021/4 Electronics – 7 Analog Multipliers Spectra for balanced modulator Notes The spectrum has components located at frequencies ωm above and below each of the harmonics of ωc, but no component at the carrier frequency ωc or its harmonics. The spectrum of the input signals and the resulting output signal is shown below. The lack of an output component at the carrier frequency is a very useful property of balanced modulators. The signal is usually filtered following the modulation process so that only the components near ωc. are retained DT021/4 Electronics – 7 Analog Multipliers 30 31 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 32 5.9 Dr. Yuri Panarin, DT021/4, Electronics Phase Detector Notes If unmodulated signals of identical frequency coo are applied to the two inputs, the circuit behaves as a phase detector and produces an output whose dc component is proportional to the phase difference between the two inputs. DT021/4 Electronics – 7 Analog Multipliers 33 DT021/4 Electronics – 7 Analog Multipliers 34 Notes The output waveform that results is shown in Fig. and consists of a dc component and a component at twice the incoming frequency. The dc component is given by: Vaverage = 1 2π ∫ 2π 0 Vo (t ) d (ω ot ) = −1 [A1 − A2 ] π where areas A1 and A2 are as indicated. Thus π −ϕ ϕ 2ϕ Vaverage = − I EE RC − I EE RC = I EE RC − 1 π π π If input signals are comparable to or smaller than VT, the circuit still acts as a phase detector. However, the output voltage then depends both on the phase difference and on the amplitude of the two input waveforms DT021/4 Electronics – 7 Analog Multipliers 35 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 36 5.10 Dr. Yuri Panarin, DT021/4, Electronics Four-quadrant multiplier AD534 Notes Figure shows the complete multiplier AD534. Four-quadrant operation is achieved by using two transconductance pairs with the bases driven in antiphase and the emitters driven by a second V-I converter. Z1 − Z 2 = K ( X 1 − X 2 )(Y1 − Y2 ) K= Rz R y Rx I x DT021/4 Electronics – 7 Analog Multipliers 37 DT021/4 Electronics – 7 Analog Multipliers AD534 Basic Configuration 38 Notes The basic connection for four-quadrant multiplication, is used in • amplitude modulation, • voltage-controlled amplification, and • instantaneous power measurements. When one of the inputs is zero, the output should also be zero, regardless of the signal at the other input. In practice, a small fraction of the other input will feed through to the output, causing an error. This can be minimized by applying an external voltage to the X2 or Y2 input. This basic configuration has a number of useful variations. • For instance, tying the inputs together yields the squaring function. • Deriving Z1 from Vo via a voltage divider allows for scale factors other than 1/(10 V). • Applying a signal to the Z1 terminal will cause it to be summed to the output DT021/4 Electronics – 7 Analog Multipliers 39 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 40 5.11 Dr. Yuri Panarin, DT021/4, Electronics Notes AD534 Applications Z − Z = K ( X − X )(Y − Y ) 1 2 1 2 1 2 − Vz = (1 / 10) × Vx (−Vo ) Vo = 10× Vz / Vx − Vz = (1 / 10) × Vo (−Vo ) Vo = 10 × Vz DT021/4 Electronics – 7 Analog Multipliers 41 DT021/4 Electronics – 7 Analog Multipliers Test Show that ( ) Vo = Vx2 − V y2 / 10 X 1 = V X and Y1 = V X + VY 2 Notes Z1 − Z 2 = K ( X 1 − X 2 )(Y1 − Y2 ) Z1 = 42 10 kΩ V VO = O 10 kΩ + 30 kΩ 4 VO V + VY VX + VY V − VY VX + VY V 2 − VY2 = 0.1× X × = 0.1× X = 0.1× VX − X × 4 2 2 2 2 4 DT021/4 Electronics – 7 Analog Multipliers 43 DT021/4 Electronics – 7 Analog Multipliers http://www.electronics.dit.ie/staff/ypanarin/DT021-notes.htm 44