Miller Pole Splitting and Zero

Miller Pole Splitting and Zero Objective The step response of a twopole amplifier depends on the ratio of the pole frequencies, with less ringing of the output when the poles are widely separated. However, for any fixed ratio, the speed of the system is determined by the higher pole frequency, becoming faster as the second pole frequency increases. Thus, it becomes important to understand how the frequency of the second pole can be increased. One way to do this is to use Miller compensation. For a simple twostage amplifier we show here how the pole frequencies behave when Miller compensation is used. We also show that the zero introduced by Miller compensation can interfere, limiting the advantages of the higher frequency pole. Schematic R R A V A Sweep V IN A υ V IN V A υ V V O β FB V O V O FIGURE A feedback voltage amplifier with Miller compensation using compensation capacitor Figure shows a twostage feedback voltage amplifier with voltage feedback β FB. The capacitor is inserted between the first and second stage to change the poles of the openloop amplifier (the amplifier with β FB = ). Specifically, moves the lowfrequency pole lower in frequency, and the highfrequency pole higher in frequency (pole splitting). These shifts in the poles make them further apart, which has two effects: making the higherfrequency pole higher in frequency allows a faster step response, while making the poles further apart reduces ringing of the step response. Analysis First, we look at the modification of the poles in the openloop amplifier. We split the openloop amplifier into an input and an output side. On the input side we obtain Figure below, where the Miller capacitance is given by EQ. V = O M, V with V O the output voltage, which is applied to the output side of in Figure. reate Date 4/6/6 by J R Brews Page 4/7/6

R V A A Sweep V IN A υ V IN M V FIGURE Input side of amplifier using the Miller capacitance Based upon Figure we find EQ. V A V = υ jω R. IN ( M) We then move to the output side of the amplifier, as shown in Figure 3. jω ( V V ) O R A υ V V O FIGURE 3 Output side of the circuit with VS representing the current supplied by the compensation capacitor Using Figure 3 we find EQ. 3 VO A jωr = υ. V jω R ( ) which combines with EQ. to provide EQ. 4 V ω ( ) = = O Aυ j R M. V jω R The Miller approximation neglects the frequency dependence of M, but here we don t use this approximation because we want the exact frequency dependence. We now find the overall gain as the product of EQ. and EQ. 3, to find EQ. 5 VO V VO ( A j ωr ) A = = υ υ. V V V j ω R j ω R j ω R A j ω R IN IN ( ( ) )( ) ( ) υ reate Date 4/6/6 by J R Brews Page 4/7/6

Pole splitting We multiply out the denominator and collect powers of ω to find the denominator in the form EQ. 6 Deno min ator ( j ωτ)( j ωτ) = j ω( τ τ) ( j ω) ( ττ = ), with the coefficients determined from EQ. 5 as EQ. 7 τ τ = R ( A ), and EQ. 8 ( ) ( υ ) R ( ) R τ τ = R. Because our amplifier will exhibit good step response only if we make the two time constants far apart, we now assume the longer time is τ and τ >> τ. Then EQ. 7 is very nearly τ, and EQ. 8 provides τ as EQ. 9 ττ ττ ( ) RR τ = =. τ τ τ R ( A ) R ( ) ( υ ) Interpreting EQ. 7 as τ and assuming A υ is large and negative, we see that greatly increases τ. In fact, if the term in dominates, we have approximately EQ. τ R ( A ) R ( A ), ( ) ( υ ) ( υ ) R which increases linearly with. Likewise, using EQ. 9 assuming the terms in dominate, we find EQ. ττ ( ) RR ( ) RR ( ) R τ =, τ τ R ( A ) R R ( A ) R A ( ) ( ) υ ( ) υ ( ) which is independent of and decreases inversely with A υ. Thus the higher frequency pole increases initially in frequency as increases, but becomes locked in at a value independent of when is large enough. Zero The gain expression EQ. 5 also exhibits a zero at the frequency EQ. A Aυ ω = υ =, R R where we used the fact that A υ is negative. This zero can be a problem if the secondstage gain is too low, or if the compensation capacitance is too high, because the zero crowds the second pole. For frequencies where the zero becomes active, a db/decade increase in slope of the gain is added, while the slope of the phase decreases by 45 /decade. Both effects degrade the step response compared to the situation without the zero, because the phase and gain margins of the amplifier are not as good. PSPIE examples We can explore the above equations using PSPIE. The PSPIE version of Figure is shown in Figure 4 below. υ reate Date 4/6/6 by J R Brews Page 3 4/7/6

F {_} A Sweep V_A V E GAIN = {A_v} E6 GAIN = R {R_} {} OUT_ E3 GAIN = {A_v} E GAIN = R {R_} {} OUT VDB STEP {B_FB} E5 GAIN = E4 GAIN = FIRST_NPAIRS = s,.us,,.us,,, TSF = ms VSF = V/V PARAMETERS: T_ = {*T_} T_ = us = uf A_v = {Meg/A_v} A_v = V/V _ = {T_/(A_v)*R_} B_FB = R_ = {T_/} R_ = {T_/} FIGURE 4 PSPIE version of Figure ; the VVS s of Figure are implemented using a GAIN part (triangles) to allow the use of variables for gains A υ and A υ ; so each VVS requires two PSPIE parts E and one GAIN part. Figure 4 shows a PSPIE implementation of Figure. The parameter box allows sweeping of the value of A υ while keeping the overall gain A υ A υ fixed. Also, the value of the compensation capacitor is taken from EQ. to keep τ fixed. By these arrangements, the gain plots all are similar when A υ is varies, making the effect of changing A υ easy to see. Effects of varying the secondstage gain A υ (36.K,87.3) (.3M,5.7) (3.M,3.4) (Av=V/V,38.3K,83.9) (A_v=V/V,7.5M,9.9) (A_V= V/V,.6G,57.4).Hz Hz KHz.MHz MHz GHz VDB(OUT) Frequency FIGURE 5 Gain plot for openloop amplifier with three secondstage gain values; gain at f 8 is marked above the curves Figure 5 shows how the zero encroaches on the gain plot as the secondstage gain diminishes, tending to reduce the gain margin of the amplifier compared to the case with only two poles and no zero, which resembles the high gain case out to about MHz. The frequency f 8 where the reate Date 4/6/6 by J R Brews Page 4 4/7/6

phase flips is marked above the curves, determining the lowest values of /β FB consistent with stability. Frequency f 8 moves closer to the frequency of the zero as the gain goes down. d (Av=V/V,7.5M,4.) d (.3M,8.) (A_v=V/V,.6G,5.) (3.M,8.) d (36.K,8.) (Av=V/V,38.3K,96.4) 3d.Hz Hz KHz.MHz MHz GHz P(V(OUT)) Frequency FIGURE 6 Phase plot for openloop amplifier with three secondstage gain values Figure 6 shows the phase of the openloop amplifier with zerofrequencies from EQ.. At the highest gain (A υ = V/V), the zero lies well above the second pole and hardly affects the phase of the amplifier, which resembles a twopole system out to MHz or so. At A υ = V/V, the second pole and the zero are closer in value, and the phase near the second pole is greatly affected by the zero. The frequencies f 8 for a phase of 8 are marked. 8KV (A_v=V/V,.55u,5.8K) 4KV V (A_v=V/V,.85u,6.9K) B_FB=4.4E5 V/V 4KV s us 4us 6us 8us us V(OUT) Time FIGURE 7 Step response of closed loop amplifier with β FB = 4.4 5 V/V, which is the maximum feedback consistent with stability for the case A υ = V/V Figure 7 shows the step response of the closedloop amplifier. The value of β FB is chosen using Figure 5, which shows the case A υ = V/V has the strongest limitation on β FB (requires the smallest amount of feedback to remain stable). Figure 7 shows that in this case the step response is very slow to settle (if it does at all), while the other cases with A υ = V/V and V/V reach a peak in.85 µs and then remain constant. Summary When Miller compensation is used, a zero is introduced that can limit the design in some cases. An example with low second stage gain illustrates these limitations. reate Date 4/6/6 by J R Brews Page 5 4/7/6

Effects of varying According to EQ. 9 and EQ., as increases, the lower pole moves down and the higher pole moves up in frequency. Figure 8 shows the results for the case A υ = V/V. Phase (deg) 5 5 5 3.E.E.E4.E6.E8.E Frequency (Hz) _=nf _=6uF _=5uF FIGURE 8 Phase of openloop amplifier for various values of compensation capacitance ; A υ = V/V The position of the poles and the zero are marked in Figure 8. The lower pole moves down as increases, as expected. The zero also moves down, as shown by the heavily filled in dots on the curves, and in the case of = 5 µf the zero is actually lower in frequency than the second pole. Even in the case = 6 µf the zero has a big effect on the phase near the higher pole, and tends to make the amplifier less stable than when the zero is high in frequency. The higher pole moves up as increases, as expected, and the increase from = nf to = 6 µf is substantial. However, further increase in to 5 µf does not increase the second pole frequency much at all, as expected from EQ., which predicts a final value for this frequency that does not depend on when is large compared to //. Summary When Miller compensation is used, the upper pole frequency is increased, until reaches a value greater than //. For a given pole frequency ratio, a larger f allows a larger choice for the lower pole frequency f as well, leading to a faster amplifier for a given ratio f /f. A zero is introduced by that can limit the design in some cases. Several examples with large values of illustrate these limitations. omment: getting around the zero problem In more advanced classes you will learn some other feedback compensation techniques that overcome the introduction of a zero, providing more flexible adjustment of the amplifier poles than Miller compensation. Example design: Given the open loop amplifier of Figure with τ = µs and τ = µs, and A υ = V/V, we want to build a closed loop amplifier with a step response corresponding to ζ =.5 (that is, a "stable" design with slope db/decade from the new f all the way to f. We specify a closedloop gain of V/V and use Miller compensation to get the fastest amplifier possible. Find the appropriate Miller compensation capacitor. We begin by finding the lowest pole frequency assuming the higher pole frequency doesn t move. The graphical approach (see document Feedback, Frequency Response and Step Response ) is shown in Figure 9 below. reate Date 4/6/6 by J R Brews Page 6 4/7/6

A υ βfbaυ Aυ βfbaυ βfb βfbaυ ' πτ πτ πτ FIGURE 9 Determination of new pole position f = /( πτ ) We want τ = (β FB A υ ) τ = ( 3 6 ) µs = ms. According to EQ., we want EQ. 3 τ R R = = 9.79 µf. R ( A )R According to EQ. 9, this value of will move the time constant of the higher pole to the value given in EQ. 4 below. EQ. 4 ( ) RR τ = = 9.56 ns. R ( A ) R υ ( ) ( ) This movement of τ will make the pole separation too big. We find τ = (β FB A υ ) τ = ( 3 6 ) 9.56 ns = 9.6 µs. The new τ =.354 µs. We average the estimates of τ and try again. We find after several iterations =.978 µf, τ =.94 ms, τ =.94 µs. The gain and phase plots of the openloop amplifier are shown in Figure and Figure. υ,.) (f_=.46khz,7.) (f_=.46mhz,57.) (/b_fb=6. db) (f_z=63.6mhz,9.).hz Hz Hz.KHz KHz KHz.MHz MHz MHz.GHz GHz VDB(OUT) db(e3) Frequency FIGURE Gain plot for openloop amplifier; pole frequencies from design are labeled reate Date 4/6/6 by J R Brews Page 7 4/7/6

d d (f_=.46khz,45.) (f_=.46mhz,35.5) d (f_z=63.6mhz,4.5) 3d.Hz Hz Hz.KHz KHz KHz.MHz MHz MHz.GHz GHz P(V(OUT)) Frequency FIGURE Phase plot for openloop amplifier; pole frequencies from design are labeled Figure and Figure show that the openloop amplifier with the chosen value for satisfies the requirements that (i) the pole separation is τ /τ = f /f = (β FB A υ ) =, and (ii) the gain drops at db/decade all the way from the new f to the new f, and (iii) the /β FB line hits the openloop gain plot at the new f, making the openloop phase at f, φ (f ) = 35..KV (t_max=396.3ns,.7kv) (.9us,.KV).KV V (t_min=79.3ns,97.3v).kv s.5us.us.5us.us.5us 3.us 3.5us 4.us 4.5us 5.us V(OUT) Time FIGURE Step response of amplifier Figure shows the step response. After. µs it has settled at the final value of gain V/V. The first maximum occurs at t MAX = 396 ns. The theoretical twopole t MAX is (see document Feedback, Frequency Response and Step Response ) EQ. 5 π t MAX= = = 396 ns, / / ω N ζ f 4 showing that the openloop zero has little effect on this design. This simple twopole design gets us close enough to the design we want that we can use simulation to finetune it if needed. reate Date 4/6/6 by J R Brews Page 8 4/7/6