Frequency Compensation

Posted on Edited on In

Miller's approximation

miller-approx.drawio

Right-Half-Plane Zero

\[ \left[(v_i - v_o)sC_c - g_m v_i\right]R_o = v_o \] Then \[ \frac{v_o}{v_i} = -g_mR_o\frac{1-s\frac{C_c}{g_m}}{1+sR_oC_c} \] right-half-plane Zero \(\omega _z = \frac{g_m}{C_c}\)

image-20251024211342562

Equivalent cap

The amplifier gain magnitude \(A_v = g_m R_o\) \[ I_\text{c,in} = (v_i - v_o)sC_c \] Then \[\begin{align} I_\text{c,in} &= (v_i + A_v v_i)sC_c \\ & = v_i s (1+A_v)C_c \end{align}\]

we get \(C_\text{in,eq}= (1+A_v)C_c\simeq A_vC_c\)

Similarly \[\begin{align} I_\text{c,out} &= (v_o - v_i)sC_c \\ & = v_o s (1+\frac{1}{A_v})C_c \end{align}\]

we get \(C_\text{out,eq}= (1+\frac{1}{A_v})C_c\simeq C_c\)

Pole Splitting

Generic circuit in textbook

In addition to lowering the required capacitor value, Miller compensation entails a very important property: it moves the output pole away from the origin. This effect is called pole splitting

image-20230103223438823

The 1st stage is replaced with Thevenin equivalent circuit , \(V_i \cong V_i \cdot g_{m1}R_{o1}\)

\[\begin{align} \frac{V_i-V_{o1}}{R_{o1}} &= V_{o1}\cdot sC_{o1}+(V_{o1}-V_o)\cdot sC_c \\ V_{o1} &= \frac{V_i+sR_{o1}C_cV_o}{1+sR_{o1}(C_{o1}+C_c)} \end{align}\] \[ (V_{o1}-V_o)sC_c=g_{m2}V_{o1}+V_o(\frac{1}{R_{o2}+sC_L}) \] substitute \(V_{o1}\), we get

\[\begin{align} \frac{V_o}{V_i} &= \frac{(sC_c-g_{m2})R_{o2}}{s^2R_{o1}R_{o2}(C_cC_{o1}+C_LC_{o1}+C_LC_c)+s\left\{ R_{o1}C_c\cdot g_{m2}R_{o2}+R_{o2}(C_c+C_L)+R_{o1}(C_{o1}+C_c) \right\} +1} \\ &= \frac{g_{m2}R_{o2}(s\frac{C_c}{g_{m2}}-1)}{s^2R_{o1}R_{o2}(C_cC_{o1}+C_LC_{o1}+C_LC_c)+s\left\{ R_{o1}C_c\cdot g_{m2}R_{o2}+R_{o2}(C_c+C_L)+R_{o1}(C_{o1}+C_c) \right\} +1} \end{align}\]

left hand plane poles

\[\begin{align} \omega_1 &= \frac{1}{R_{o1}C_c\cdot g_{m2}R_{o2}+R_{o2}(C_c+C_L)+R_{o1}(C_{o1}+C_c)} \\ \omega_2 &= \frac{R_{o1}C_c\cdot g_{m2}R_{o2}+R_{o2}(C_c+C_L)+R_{o1}(C_{o1}+C_c)}{R_{o1}R_{o2}(C_cC_{o1}+C_LC_{o1}+C_LC_c)} \end{align}\]

and RHP (right-hand plane) zero \[ \omega_z=\frac{g_{m2}}{C_c} \]

with series switch

image-20230103230122637

replace \(sC_L\) with \(1/(R_{sw}+\frac{1}{sC_L})\) \[\begin{align} \frac{V_{o2}}{V_i} &= \frac{g_{m2}R_{o2}(s\frac{C_c}{g_{m2}}-1)(1+sR_{sw}C_L)}{s^3R_{o1}R_{o2}R_{sw}C_{o1}C_cC_L+s^2\left\{R_{o1}R_{o2}(C_cC_{o1}+C_LC_{o1}+C_LC_c)+ \left[ R_{o1}C_c\cdot g_{m2}R_{o2}+R_{o2}(C_c+0)+R_{o1}(C_{o1}+C_c)\right]R_{sw}C_L \right\}+s\left\{ R_{o1}C_c\cdot g_{m2}R_{o2}+R_{o2}(C_c+C_L)+R_{o1}(C_{o1}+C_c) +R_{sw}C_L\right\} +1} \end{align}\] Due to \[ \frac{V_o}{V_{o2}} = \frac{\frac{1}{sC_L}}{R_{sw}+\frac{1}{sC_L}}=\frac{1}{1+sR_{sw}C_L} \] Then \[\begin{align} \frac{V_o}{V_i} &= \frac{V_{o2}}{V_i} \cdot \frac{V_o}{V_{o2}} \\ &= \frac{g_{m2}R_{o2}(s\frac{C_c}{g_{m2}}-1)}{s^3R_{o1}R_{o2}R_{sw}C_{o1}C_cC_L+s^2\left\{R_{o1}R_{o2}(C_cC_{o1}+C_LC_{o1}+C_LC_c)+ \left[ R_{o1}C_c\cdot g_{m2}R_{o2}+R_{o2}(C_c+0)+R_{o1}(C_{o1}+C_c)\right]R_{sw}C_L \right\}+s\left\{ R_{o1}C_c\cdot g_{m2}R_{o2}+R_{o2}(C_c+C_L)+R_{o1}(C_{o1}+C_c) +R_{sw}C_L\right\} +1} \end{align}\]

\(R_{sw}\) & \(R_c\)

image-20230103232837318

\[\begin{align} \frac{V_{o2}}{V_i} &=-g_{m2}R_{o2}\frac{sC_c(R_c-1/g_{m2})+1}{(1+sR_{o1}C_{o1})sR_{o2}C_c+sR_{o1}\cdot g_{m2}R_{o2}C_c+\frac{s(R_{o2}+R_{sw})C_L+1}{sR_{sw}C_L+1}\left[(1+sR_{o1}C_{o1})(1+sR_cC_c)+sR_{o1}C_c \right]} \\ &=-g_{m2}R_{o2}\frac{sC_c(R_c-1/g_{m2})+1}{s^2R_{o1}R_{o2}C_{o1}C_c+sR_{o2}C_c+sR_{o1}\cdot g_{m2}R_{o2}C_c+\frac{s(R_{o2}+R_{sw})C_L+1}{sR_{sw}C_L+1}\left[(1+sR_{o1}C_{o1})(1+sR_cC_c)+sR_{o1}C_c \right]} \\ &=-g_{m2}R_{o2}\frac{\left[ sC_c(R_c-1/g_{m2})+1 \right](sR_{sw}C_L+1)}{s^2R_{o1}R_{o2}C_{o1}C_c(sR_{sw}C_L+1)+sR_{o2}C_c(sR_{sw}C_L+1)+sR_{o1}\cdot g_{m2}R_{o2}C_c(sR_{sw}C_L+1)+\left[s(R_{o2}+R_{sw})C_L+1\right]\left[(1+sR_{o1}C_{o1})(1+sR_cC_c)+sR_{o1}C_c \right]} \end{align}\]

\(s^3\) terms in denominator \[ H_3 = s^3\cdot(R_{o1}R_{o2}R_c+R_{o1}R_{o2}R_{sw} +R_{o1}R_cR_{sw})\cdot C_{o1}C_cC_L \] \(s^2\) terms in denominator \[\begin{align} H_2 &=s^2\cdot(R_{o1}R_{o2}C_{o1}C_c+R_{o1}R_{o2}C_{o1}C_L+R_{o2}R_cC_cC_L+R_{o1}R_{o2}C_cC_L+R_{o1}R_cC_{o1}C_c\\ &+R_{o2}R_{sw}C_cC_L+R_{o1}R_{sw}C_cC_L\cdot g_{m2}R_{o2}+R_{o1}R_{sw}C_{o1}C_L+R_{sw}R_cC_cC_L+R_{o1}R_{sw}C_cC_L) \end{align}\]

\(s^1\) term in denominator \[ H_1=s(R_{o1}\cdot g_{m2}R_{o2}C_c+R_{o1}C_{o1}+R_cC_c+R_{o1}C_c+R_{o2}C_c+R_{o2}C_L+R_{sw}C_L) \] \(s^0\) term in denominator \[ H_0=1 \] set \(R_c=0\) and \(R_{sw}=0\), the \(H_*\) reduced to \[\begin{align} H_3 &= 0 \\ H_2 &=s^2R_{o1}R_{o2}(C_{o1}C_c+C_{o1}C_L+C_cC_L) \\ H_1&=s(R_{o1}\cdot g_{m2}R_{o2}C_c+R_{o1}C_{o1}+R_{o1}C_c+R_{o2}C_c+R_{o2}C_L) \\ H_0&=1 \end{align}\] That is \[ H=s^2R_{o1}R_{o2}(C_{o1}C_c+C_{o1}C_L+C_cC_L)+s(R_{o1}\cdot g_{m2}R_{o2}C_c+R_{o1}C_{o1}+R_{o1}C_c+R_{o2}C_c+R_{o2}C_L)+1 \]

which is same with our previous analysis of Generic circuit in textbook

And we know \[ \frac{V_o}{V_{o2}}=\frac{1}{1+sR_{sw}C_L} \] Finally, we get \(\frac{V_o}{V_i}\) \[\begin{align} \frac{V_o}{V_i} &= \frac{V_{o2}}{V_i} \cdot \frac{V_o}{V_{o2}} \\ &= -g_{m2}R_{o2}\frac{\left[ sC_c(R_c-1/g_{m2})+1 \right](sR_{sw}C_L+1)}{H_3+H_2+H_1+1} \cdot \frac{1}{1+sR_{sw}C_L} \\ &= -g_{m2}R_{o2}\frac{ sC_c(R_c-1/g_{m2})+1}{H_3+H_2+H_1+1} \end{align}\]

The loop transfer function is \[ \frac{V_o}{V_i} =-g_{m1}R_{o1}g_{m2}R_{o2}\frac{ sC_c(R_c-1/g_{m2})+1}{H_3+H_2+H_1+1} \]

vccs model

image-20230106214027323

simplify the transfer function

  1. omit \(C_{o1}\)

We omit \(C_{o1}\) in frequency range of interest

\[\begin{align} H_3 &= 0 \\ H_2 &=s^2(R_{o2}R_c+R_{o1}R_{o2}+R_{o2}R_{sw}+R_{o1}R_{sw}\cdot g_{m2}R_{o2}+R_{sw}R_c+R_{o1}R_{sw})\cdot C_cC_L \\ H_1 &=s(R_{o1}\cdot g_{m2}R_{o2}C_c+R_cC_c+R_{o1}C_c+R_{o2}C_c+R_{o2}C_L+R_{sw}C_L) \\ H_0 &= 1 \end{align}\]

two poles and 1 zero

  1. more simplification

Then, some terms can be omitted

\[\begin{align} H_2 &=s^2R_{o1}R_{o2}(1+g_{m2}R_{sw})\cdot C_cC_L \\ H_1 &=sR_{o1}\cdot g_{m2}R_{o2}C_c \\ H_0 &= 1 \end{align}\]

The poles can be deduced \[\begin{align} \omega_1 &= \frac{1}{R_{o1}\cdot g_{m2}R_{o2}C_c} \\ \omega_2 &= \frac{1}{1+g_{m2}R_{sw}}\cdot \frac{g_{m2}}{C_L} \\ &= \frac{1}{(gm_2^{-1}+R_{sw})C_L} \end{align}\]

The pole \(\omega_2=\frac{1}{gm_2^{-1}C_L}\) is changed to \(\omega_2=\frac{1}{(gm_2^{-1}+R_{sw})C_L}\)

In order to cancell \(\omega_2\) with \(\omega_z\), \(R_c\) shall be increased

\[ R_{eq}=g_{m2}^{-1}+R_{sw} \]

fndRsw.drawio.svg

image-20251024233028351

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gm1 = 400e-6;
gm2 = 3.2e-3;
Ro1 = 800e3;
Ro2 = 100e3;
Co1 = 100e-15;
CL = 7e-12;
Rc = 15e3;
Cc = 2e-12;
Rsw = 6e3;


s = tf('s');

%%%%%%%%%%%%%%%%%%%%%%%%%%
% org equation
H3 = s^3*(Ro1*Ro2*Rc + Ro1*Ro2*Rsw+Ro1*Rc*Rsw)*Co1*Cc*CL;
H2 = s^2*(Ro1*Ro2*Co1*Cc+Ro1*Ro2*Co1*CL+Ro2*Rc*Cc*CL+Ro1*Ro2*Cc*CL+Ro1*Rc*Co1*Cc+ ...
    Ro2*Rsw*Cc*CL+Ro1*Rsw*Cc*CL*gm2*Ro2+Ro1*Rsw*Co1*CL+Rsw*Rc*Cc*CL+Ro1*Rsw*Cc*CL);
H1 = s*(Ro1*gm2*Ro2*Cc+Ro1*Co1+Rc*Cc+Ro1*Cc+Ro2*Cc+Ro2*CL+Rsw*CL);
Nmt = s*Cc*(Rc-1/gm2)+1;
TF = -gm1*Ro1*gm2*Ro2*Nmt/(H3+H2+H1+1);
[mag, phase, wout] = bode(TF, {100*2*pi, 100*1e9*2*pi});


%%%%%%%%%%%%%%%%%%%%%%%%%%
% omit Co1
H3 = 0;
H2 = s^2*(Ro2*Rc+Ro1*Ro2+Ro2*Rsw+Ro1*Rsw*gm2*Ro2+Rsw*Rc+Ro1*Rsw)*Cc*CL;
H1 = s*(Ro1*gm2*Ro2*Cc+Rc*Cc+Ro1*Cc+Ro2*Cc+Ro2*CL+Rsw*CL);
Nmt = s*Cc*(Rc-1/gm2)+1;
TF = -gm1*Ro1*gm2*Ro2*Nmt/(H3+H2+H1+1);
[mag2, phase2, wout2] = bode(TF, {100*2*pi, 100e9*2*pi});


%%%%%%%%%%%%%%%%%%%%%%%%%%
% more simplification
H3 = 0;
H2 = s^2*Ro1*Ro2*(1+gm2*Rsw)*Cc*CL;
H1 = s*Ro1*gm2*Ro2*Cc;
Nmt = s*Cc*(Rc-1/gm2)+1;
TF = -gm1*Ro1*gm2*Ro2*Nmt/(H3+H2+H1+1);
[mag3, phase3, wout3] = bode(TF, {100*2*pi, 100e9*2*pi});

subplot(2,1,1)
semilogx(wout(:)/2/pi, 20*log10(mag(:)), 'k', 'linewidth', 2);
hold on;
semilogx(wout2(:)/2/pi, 20*log10(mag2(:)), 'ro--', 'linewidth', 2)
semilogx(wout3(:)/2/pi, 20*log10(mag3(:)), 'bs--', 'linewidth', 2)
grid on; title("loop gain (dB)"); xlabel("frequency (Hz)"); legend("original", "Omit C_{o1}", "more simp");

subplot(2,1,2)
semilogx(wout(:)/2/pi, phase(:), 'k', 'linewidth', 2);
hold on;
semilogx(wout2(:)/2/pi, phase2(:), 'ro--', 'linewidth', 2);
semilogx(wout3(:)/2/pi, phase3(:), 'bs--', 'linewidth', 2);
grid on; title("loop phase (Ao)"); xlabel("frequency (Hz)"); legend("original", "Omit C_{o1}", "more simp");

non-dominant pole in Sansen's book

image-20230103231300609

Following demonstrate how derive \(f_{nd}\) from Razavi's equation. We copy \(\omega_2\) here \[ \omega_2 = \frac{R_{o1}C_c\cdot g_{m2}R_{o2}+R_{o2}(C_c+C_L)+R_{o1}(C_{o1}+C_c)}{R_{o1}R_{o2}(C_cC_{o1}+C_LC_{o1}+C_LC_c)} \] which can be reduced as below

\[\begin{align} \omega_2 &= \frac{R_{o1}C_c\cdot g_{m2}R_{o2}+R_{o2}(C_c+C_L)+R_{o1}(C_{o1}+C_c)}{R_{o1}R_{o2}(C_cC_{o1}+C_LC_{o1}+C_LC_c)} \\ &= \frac{R_{o1}C_c\cdot g_{m2}R_{o2}}{R_{o1}R_{o2}(C_cC_{o1}+C_LC_{o1}+C_LC_c)} \\ &= \frac{C_c\cdot g_{m2}}{C_cC_{o1}+C_LC_{o1}+C_LC_c} \\ &= \frac{g_{m2}}{C_{o1}+C_L\frac{C_{o1}}{C_c}+C_L} \\ &= \frac{g_{m2}}{C_L\frac{C_{o1}}{C_c}+C_L} \\ &= \frac{g_{m2}}{C_L} \cdot \frac{1}{1+\frac{C_{o1}}{C_c}} \end{align}\]

image-20230129221424763

Exercise of 2-stage opamp

image-20230105001747174

image-20230105001758050

image-20230105001812304

image-20230105001849554

\(R_{o1}\) and \(R_{o2}\) don't affect stability, if \(f_{nd}>3\text{GBW}\)

DC gain: \(g_{m1}g_{m2}R_{o1}R_{o2}\)

dominant pole: \(\omega_d=\frac{1}{R_{o1}\cdot g_{m2}R_{o2}C_c}\)

image-20230105001923985

image-20230105002653735

\(20log_{10}(1.414^2)=6\text{dB}\)

Cascode Compensation

This cascode compensation topology is popularly known as ahuja compensation

The cause of the positive zero is the feedforward current through \(C_m\).

To abolish this zero, we have to cut the feedforward path and create a unidirectional feedback through \(C_m\).

  1. Adding a resistor(nulling resistor) is one way to mitigate the effect of the feedforward current.

  2. Another approach uses a current buffer cascode to pass the small-signal feedback current but cut the feedforward current

People name this approach after the author Ahuja

The benefits of Ahuja compensation over Miller compensation are severa

  • better PSRR

  • higher unity-gain bandwidth using smaller compensation capacitor

  • ability to cope better with heavy capacitive and resistive loads

image-20240817193513058

image-20240817201727109

Of course, , if the capacitance at the gate of \(M_1\) is taken into account, pole splitting is less pronounced.

including \(r_\text{o2}\)

image-20240819202642809 \[ \frac{V_{out}}{I_{in}} \approx \frac{-g_{m1}R_SR_L(g_{m2}+C_Cs)}{\frac{R_S+r_\text{o2}}{r_\text{o2}}R_LC_LC_Cs^2+g_{m1}g_{m2}R_LR_SC_Cs+g_{m2}} \] The poles as

\[\begin{align} \omega_{p1} &\approx \frac{1}{g_{m1}R_LR_SC_c} \\ \omega_{p2} &\approx \frac{g_{m2}R_Sg_{m1}}{C_L}\frac{r_\text{o2}}{R_S+r_\text{o2}} \end{align}\]

and zero is not affected, which is \(\omega_z =\frac{g_{m2}}{C_C}\)

the above model simulation result is shown below

image-20240819221653262

the zero is located between two poles

take into the capacitance at the gate of \(M_1\) and all other second-order effect

image-20240819222727276

intuitive analysis of zero

miller compensation

  • zero in the right half plane \[ g_\text{m1}V_P = sC_c V_P \]

cascode compensation

  • zero in the left half plane \[ g_\text{m2}V_X = - sC_c V_X \]

zero_loc.drawio

Mitigate Impact of Zero

cascode_compensation

dominant pole \[ \omega_\text{p,d} = \frac {1} {R_\text{eq}g_\text{m9}R_{L}C_{c}} \] first nondominant pole \[ \omega_\text{p,nd} = \frac {g_\text{m4}R_\text{eq}g_\text{m9}} {C_L} \] zero \[ \omega_\text{z} = (g_\text{m4}R_\text{eq})(\frac {g_\text{m9}} {C_c}) \] a much greater magnitude than \(g_\text{m9}/C_C\)

ahuja variations

image-20250609210452373

Pole-Zero Compensation

Pole-Zero Compensation is also known as Lead Compensation, Parallel Compensation

image-20220307234938855

Note: The dominant pole is at output of the first stage, i.e. \(\frac{1}{R_{EQ}C_{EQ}}\).

image-20221006001114562

Pole & Zero in transfer function

Design with operational amplifiers and analog integrated circuits / Sergio Franco, San Francisco State University. – Fourth edition

image-20221005220854411

\[ Y = \frac{1}{R_1} + sC_1+\frac{1}{R_c+1/SC_c} \]

\[\begin{align} Z &= \frac{1}{\frac{1}{R_1} + sC_1+\frac{1}{R_c+1/SC_c}} \\ &= \frac{R_1(1+sR_cC_c)}{s^2R_1C_1R_cC_c+s(R_1C_c+R_1C_1+R_cC_c)+1} \end{align}\] If \(p_{1c} \ll p_{3c}\), two real roots can be found \[\begin{align} p_{1c} &= \frac{1}{R_1C_c+R_1C_1+R_cC_c} \\ p_{3c} &= \frac{R_1C_c+R_1C_1+R_cC_c}{R_1C_1R_cC_c} \end{align}\]

The additional zero is \[ z_c = \frac{1}{R_cC_c} \] Given \(R_c \ll R\) and \(C_c \gg C\) \[\begin{align} p_{1c} &\simeq \frac{1}{R_1(C_c+C_1)} \simeq \frac{1}{R_1C_c}\\ p_{3c} &= \frac{1}{R_cC_1}+\frac{1}{R_cC_c}+\frac{1}{R_1C_1} \simeq \frac{1}{R_cC_1} \end{align}\]

The output pole is unchanged, which is \[ p_2 = \frac{1}{R_LC_L} \] We usually cancel \(p_2\) with \(z_c\), i.e. \[ R_cC_c=R_LC_L \]

Phase margin

unity-gain frequency \(\omega_t\) \[ \omega_t = A_\text{DC}\cdot P_{1c} =\frac{g_{m1}g_{m2}R_L}{C_c} \]

  1. PM=45\(^o\) \[ p_{3c} = \omega_t \] Then, \(C_c\) and \(R_c\) can be obtained

    \[\begin{align} R_c &= \sqrt{\frac{R_1}{C_1\cdot A_{DC}\cdot p_2}}=\sqrt{\frac{R_1\cdot R_LC_L}{C_1\cdot A_{DC}}} \\ C_c &= \sqrt{\frac{A_{DC}\cdot C_1}{R_1\cdot p_2}}=\sqrt{\frac{A_{DC}\cdot C_1 \cdot R_LC_L}{R_1}} \end{align}\]

  2. PM=60\(^o\) \[ p_{3c} = 2\cdot\omega_t \] Then, \(C_c\) and \(R_c\) can be obtained \[\begin{align} R_c &= \sqrt{\frac{R_1}{C_1\cdot 2A_{DC}\cdot p_2}} = \sqrt{\frac{R_1\cdot R_LC_L}{C_1\cdot 2A_{DC}}} \\ &= \sqrt{\frac{C_L}{2g_{m1}g_{m2}C_1}}\\ C_c &= \sqrt{\frac{2A_{DC}\cdot C_1}{R_1\cdot p_2}} = \sqrt{\frac{2A_{DC}\cdot C_1 \cdot R_LC_L}{R_1}} \\ &= R_L\sqrt{2g_{m1}g_{m2}C_1C_L} \end{align}\]

    for the unity-gain frequency \(\omega_t\) we find \[ \omega_t = \sqrt{\frac{1}{2}\cdot \frac{g_{m1}g_{m2}}{C_1C_L}} \] The parallel compensation shows a remarkably good result. The new 0 dB frequency lies only a factor \(\sqrt{2}\) lower than the theoretical maximum

To increase \(\phi_m\), we need to raise \(C_c\) a bit while lowering \(R_c\) in proportion in order to maintain pole-zero cancellation. This causes \(p_{1c}\) and \(p_{3c}\) to split a bit further apart.

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clc;
clear;

fd = 84*1e3;	% dominant freq, unit: Hz
fnd = 3.25*1e6;	% unit: Hz
C = 478*1e-15;
R = 1/fd/(2*pi)/C;
Adc = 10^(80/20);

ri = 2; % PM=45: 1;  PM=60: 2
Rc = (R/C/fnd/2/pi/ri/Adc)^0.5; % compensation resistor
Cc = (ri*Adc*C/fnd/2/pi/R)^0.5; % compensation capacitor

wzc = 1/2/pi/Rc/Cc; % zero frequency

ECEN 457 (ESS). Op-Amps Stability and Frequency Compensation Techniques [https://people.engr.tamu.edu/s-sanchez/457%20Stability%20Final.pdf]

image-20251025011022249

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Pole-Zero Doublet

image-20251103210637155

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The denominator part of \(H_{closed}(s)\) is \[ D(s) = \frac{s^2}{(A_0+1)\omega_{p1}\omega_{p2}}+\frac{\frac{1}{\omega_{p1}} + \frac{1}{\omega_{p2}}+\frac{A_0}{\omega_{z}}}{A_0+1}s+1 \]

image-20251102165416360

suppose \(\omega_{pA} \ll\omega_{pB}\), then \[ D(s) = \left( 1+ \frac{s}{\omega_{pA}}\right)\left( 1+ \frac{s}{\omega_{pB}}\right)\approx \frac{s^2}{\omega_{pA}\omega_{pB}}+\frac{s}{\omega_{pA}} + 1 \]

Thus, the two poles of the closed-loop transfer function of system are \[\begin{align} \omega_{pA} &= \frac{A_0+1}{\frac{1}{\omega_{p1}} + \frac{1}{\omega_{p2}}+\frac{A_0}{\omega_{z}}} = \frac{(A_0+1)\omega_{p1} \omega_{p2}}{\omega_{p1} + \omega_{p2} + \frac{A_0}{\omega_z}\omega_{p1} \omega_{p2}}\\ \omega_{pB} &= \omega_{p1} + \omega_{p2} + \frac{A_0}{\omega_z}\omega_{p1} \omega_{p2} \end{align}\]

image-20251102170244359

non-dominant pole \(\omega_{p2}\) and zero \(\omega_z\) are in UGB

\[\begin{align} \omega_{pA} &\approx \omega_{p2}\\ \omega_{pB} &\approx (1+A_0)\omega_{p1} \end{align}\] Then, closed-loop transfer function is \[ H_{closed}(s) \approx \frac{\frac{A_0}{A_0+1}\left(1+\frac{s}{\omega_z}\right)}{\left(1+\frac{s}{(1+A_0)\omega_{p1}}\right)\left( 1+\frac{s}{\omega_{p2}} \right)} \]

Consider the Laplace transform function of step response, \(X(s)=\frac{1}{s}\) \[ Y(s)=\frac{1}{s}\times H_{closed}(s) \] Thus, the small-signal step response of the closed-loop amplifier is \[ y(t)=\frac{A_0}{A_0+1}\left[1-e^{-(A_0+1)\omega_{p1}t}-\left(1-\frac{\omega_{p2}}{\omega_z}\right)e^{-\omega_{p2}t} \right]u(t) \] Since, \(\omega_{p2}\ll (1+A_0)\omega_{p1}\). rewrite the \(y(t)\) \[ y(t)\approx \frac{A_0}{A_0+1}\left[1-\left(1-\frac{\omega_{p2}}{\omega_z}\right)e^{-\omega_{p2}t} \right]u(t) \] image-20251102173521194

perfect pole-zero cancellation with \(\omega_{p2}=\omega_z\) \[ y(t) \approx \frac{A_0}{A_0+1}\left[1-e^{-(A_0+1)\omega_{p1}t}\right]u(t) \approx \frac{A_0}{A_0+1}u(t) \]

image-20251102194445318

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clear
clc

A0 = 1e6;
wp1= 2*pi*1;   % 1Hz
wpz = 2*pi*2e4; % 20KHz

s=tf('s');

wp2 =0.5*wpz;
wz =2*wpz;
Ho = A0*(1+s/wz)/(1+s/wp1)/(1+s/wp2);
Hc = Ho/(1+Ho);


wi = logspace(-3, 7, 100000)*2*pi;
ti = linspace(0, 100,100000)*1e-6;

[mag_0p1, phase_0p1, wout_0p1] = bode(Ho, wi);
[vo_0p1, to_0p1] = step(Hc, ti);


wp2 =1*wpz;
wz =1*wpz;
Ho = A0*(1+s/wz)/(1+s/wp1)/(1+s/wp2);
Hc = Ho/(1+Ho);
[mag_1p0, phase_1p0, wout_1p0] = bode(Ho, wi);
[vo_1p0, to_1p0] = step(Hc, ti);


wp2 =2*wpz;
wz =0.5*wpz;
Ho = A0*(1+s/wz)/(1+s/wp1)/(1+s/wp2);
Hc = Ho/(1+Ho);
[mag_10p0, phase_10p0, wout_10p0] = bode(Ho, wi);
[vo_10p0, to_10p0] = step(Hc, ti);

subplot(2,2,1)
semilogx(wout_0p1(:)/2/pi, 20*log10(mag_0p1(:)),'b-',LineWidth=2);
hold on
semilogx(wout_1p0(:)/2/pi, 20*log10(mag_1p0(:)),'r-',LineWidth=2);
semilogx(wout_10p0(:)/2/pi, 20*log10(mag_10p0(:)),'g-',LineWidth=2);
grid on; xlabel('Hz'); ylabel('Mag (dB)');
legend('\omega_{p2}<\omega_{z}', '\omega_{p2}=\omega_{z}', '\omega_{p2}>\omega_{z}')

subplot(2,2,3)
semilogx(wout_0p1(:)/2/pi, phase_0p1(:),'b-',LineWidth=2);
hold on
semilogx(wout_1p0(:)/2/pi, phase_1p0(:),'r-',LineWidth=2);
semilogx(wout_10p0(:)/2/pi, phase_10p0(:),'g-',LineWidth=2);
grid on; xlabel('Hz'); ylabel('Phase')
legend('\omega_{p2}<\omega_{z}', '\omega_{p2}=\omega_{z}', '\omega_{p2}>\omega_{z}')

subplot(2,2,[2 4])
plot(to_0p1(:), vo_0p1(:),'b-',LineWidth=2);
hold on
plot(to_1p0(:), vo_1p0(:),'r-',LineWidth=2);
plot(to_10p0(:), vo_10p0(:),'g-',LineWidth=2);
grid on; xlabel('time'); ylabel('V')
legend('\omega_{p2}<\omega_{z}', '\omega_{p2}=\omega_{z}', '\omega_{p2}>\omega_{z}')

image-20230108233523345

image-20230108234123707

The zero comes from the mirror node

Thanks to unity gain buffer, zero is alleviated for \(C_c\)

image-20251103220259992

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A0 = 1e6;
wp1= 2*pi*1;   % 1Hz
wpz = 2*pi*2e4; % 20KHz
wp2 = 5*A0*wp1;

s=tf('s');

rio = 0.8;
wpx =rio*wpz;
wzx =wpz/rio;
wp1x = 1.5*wp1;   % keep UGB constant by changing dominant pole wp1x
Ho = A0*(1+s/wzx)/(1+s/wp1x)/(1+s/wp2)/(1+s/wpx);  % 2 poles + pole-zero doublet pair
Hc = Ho/(1+Ho);


wi = logspace(-3, 7, 100000)*2*pi;
ti = linspace(0, 100,100000)*1e-6;

[mag_0p1, phase_0p1, wout_0p1] = bode(Ho, wi);
[vo_0p1, to_0p1] = step(Hc, ti);


rio = 1;
wpx =rio*wpz;
wzx =wpz/rio;
wp1x = 1*wp1;
Ho = A0*(1+s/wzx)/(1+s/wp1x)/(1+s/wp2)/(1+s/wpx);
Hc = Ho/(1+Ho);
[mag_1p0, phase_1p0, wout_1p0] = bode(Ho, wi);
[vo_1p0, to_1p0] = step(Hc, ti);


rio = 1.25;
wpx =rio*wpz;
wzx =wpz/rio;
wp1x = 0.65*wp1;
Ho = A0*(1+s/wzx)/(1+s/wp1x)/(1+s/wp2)/(1+s/wpx);
Hc = Ho/(1+Ho);
[mag_10p0, phase_10p0, wout_10p0] = bode(Ho, wi);
[vo_10p0, to_10p0] = step(Hc, ti);

subplot(2,2,1)
semilogx(wout_0p1(:)/2/pi, 20*log10(mag_0p1(:)),'b-',LineWidth=2);
hold on
semilogx(wout_1p0(:)/2/pi, 20*log10(mag_1p0(:)),'r-',LineWidth=2);
semilogx(wout_10p0(:)/2/pi, 20*log10(mag_10p0(:)),'g-',LineWidth=2);
grid on; xlabel('Hz'); ylabel('Mag (dB)');
legend('\omega_{p2}<\omega_{z}', '\omega_{p2}=\omega_{z}', '\omega_{p2}>\omega_{z}')

subplot(2,2,3)
semilogx(wout_0p1(:)/2/pi, phase_0p1(:),'b-',LineWidth=2);
hold on
semilogx(wout_1p0(:)/2/pi, phase_1p0(:),'r-',LineWidth=2);
semilogx(wout_10p0(:)/2/pi, phase_10p0(:),'g-',LineWidth=2);
grid on; xlabel('Hz'); ylabel('Phase')
legend('\omega_{p2}<\omega_{z}', '\omega_{p2}=\omega_{z}', '\omega_{p2}>\omega_{z}')

subplot(2,2,[2 4])
plot(to_0p1(:), vo_0p1(:),'b-',LineWidth=2);
hold on
plot(to_1p0(:), vo_1p0(:),'r-',LineWidth=2);
plot(to_10p0(:), vo_10p0(:),'g-',LineWidth=2);
grid on; xlabel('time'); ylabel('V')
legend('\omega_{p2}<\omega_{z}', '\omega_{p2}=\omega_{z}', '\omega_{p2}>\omega_{z}')

reference

Viola Schaffer, ISSCC 2021 Tutorials Designing Amplifiers for Stability [pdf]

J. H. Huijsing, "Operational Amplifiers, Theory and Design, 3rd ed. New York: Springer, 2017"

Razavi, Behzad. Design of Analog CMOS Integrated Circuits. India: McGraw-Hill, 2017. [pdf]

Sansen, Willy M. Analog Design Essentials. Germany: Springer US, 2006.

Gray, P. R., Hurst, P. J., Lewis, S. H., & Meyer, R. G. (2024). Analysis and design of analog integrated circuits (Sixth edition.). John Wiley & Sons, Inc..

Ahuja Compensation

B. K. Ahuja, "An Improved Frequency Compensation Technique for CMOS Operational Amplifiers," IEEE 1. Solid-State Circuits, vol. 18, no. 6, pp. 629-633, Dec. 1983. [https://sci-hub.se/10.1109/JSSC.1983.1052012]

U. Dasgupta, "Issues in "Ahuja" frequency compensation technique", IEEE International Symposium on Radio-Frequency Integration Technology, 2009. [https://sci-hub.se/10.1109/RFIT.2009.5383679]

R. J. Reay and G. T. A. Kovacs, "An unconditionally stable two-stage CMOS amplifier," in IEEE Journal of Solid-State Circuits, vol. 30, no. 5, pp. 591-594, May 1995 [https://sci-hub.se/10.1109/4.384174]

A. Garimella and P. M. Furth, "Frequency compensation techniques for op-amps and LDOs: A tutorial overview," 2011 IEEE 54th International Midwest Symposium on Circuits and Systems (MWSCAS), 2011 [https://sci-hub.se/10.1109/MWSCAS.2011.6026315]

H. Aminzadeh, R. Lotfi and S. Rahimian, "Design Guidelines for Two-Stage Cascode-Compensated Operational Amplifiers," 2006 13th IEEE International Conference on Electronics, Circuits and Systems, 2006 [https://sci-hub.se/10.1109/ICECS.2006.379776]

H. Aminzadeh and K. Mafinezhad, "On the power efficiency of cascode compensation over Miller compensation in two-stage operational amplifiers," Proceeding of the 13th international symposium on Low power electronics and design (ISLPED '08), Bangalore, India, 2008 [https://sci-hub.se/10.1145/1393921.1393995]

Stabilizing a 2-Stage Amplifier URL:https://a2d2ic.wordpress.com/2016/11/10/stabilizing-a-2-stage-amplifier/

EE 240B: Advanced Analog Circuit Design, Prof. Bernhard E. Boser [OTA II, Multi-Stage]

Parallel Compensation

R.Eschauzier "Wide Bandwidth Low Power Operational Amplifiers", Delft University Press, 1994.

Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. 2018. Feedback Control of Dynamic Systems (8th Edition) (8th. ed.). Pearson. 6.7 Compensation

Application Note AN-1286 Compensation for the LM3478 Boost Controller

ECEN 607 Advanced Analog Circuit Design Techniques Spring 2017 [Lect 1D Op-Amps Stability and Frequency Compensation Techniques]

Sergio Franco, San Francisco State University, Design with Operational Amplifiers and Analog Integrated Circuits, 4/e [pdf]

Pole-Zero Doublet

Elad Alon, Lecture 10: Settling-Limited Amplifier Design Methodology, EE 240B – Spring 2018, Advanced Analog Integrated Circuits [https://inst.eecs.berkeley.edu/~ee240b/sp18/lectures/Lecture10_Settling_Design_2up.pdf]

Eric Chang, Prof. Elad Alon EE240B HW3 [https://inst.eecs.berkeley.edu/~ee240b/sp18/homeworks/hw3.pdf and https://inst.eecs.berkeley.edu/~ee240b/sp18/homeworks/hw3_soln.pdf]

Prof. Tai-Haur Kuo, Analog IC Design ( 類比積體電路設計 ), Operational Amplifiers http://msic.ee.ncku.edu.tw/course/aic/201809/chapter5.pdf

SERGIO FRANCO, Demystifying pole-zero doublets [https://www.edn.com/demystifying-pole-zero-doublets/]

B. Y. T. Kamath, R. G. Meyer and P. R. Gray, "Relationship between frequency response and settling time of operational amplifiers," in IEEE Journal of Solid-State Circuits, vol. 9, no. 6, pp. 347-352, Dec. 1974, [https://sci-hub.se/10.1109/JSSC.1974.1050527]

P. R. Gray and R. G. Meyer, "MOS operational amplifier design-a tutorial overview," in IEEE Journal of Solid-State Circuits, vol. 17, no. 6, pp. 969-982, Dec. 1982, [https://sci-hub.se/10.1109/JSSC.1982.1051851]