Second-Order System

Posted on Edited on In

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show qualitatively how changing pole locations in the s-plane affect impulse responses

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\(\omega_d\) called damped natural frequency \[ \omega_n^2 = \sigma^2 + \omega_d^2 \] image-20251025093918419

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Resonant Frequency

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Phase Margin & Damping Factor

  • Phase Margin is defined for open loop system

  • Damping Factor (\(\zeta\)) is defined for close loop system

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We can analyze open loop system in a better perspective because it is simpler. So, we always use the loop gain analysis to find the phase margin and see whether the system is stable or not.

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zta = 0:.001:1;
PM = atan(2*zta ./ sqrt(sqrt(1+4*zta.^4) - 2*zta.^2))/pi*180;

[~, idx45] = min(abs(PM - 45));
[~, idx60] = min(abs(PM - 60));
[~, idx70] = min(abs(PM - 70));

plot(PM, zta, 'b',  LineWidth=3);
hold on
plot(PM(idx45), zta(idx45), 'ro', MarkerFaceColor='r', LineWidth=3, MarkerSize=8)
text(PM(idx45)+2, zta(idx45), ['\zeta=' num2str(zta(idx45)) ' @PM=45^o'])
plot(PM(idx60), zta(idx60), 'ro', MarkerFaceColor='r', LineWidth=3, MarkerSize=8)
text(PM(idx60)+2, zta(idx60), ['\zeta=' num2str(zta(idx60)) ' @PM=60^o'])
plot(PM(idx70), zta(idx70), 'ro', MarkerFaceColor='r', LineWidth=3, MarkerSize=8)
text(PM(idx70)-10, zta(idx70), ['\zeta=' num2str(zta(idx70)) ' @PM=70^o'])
grid on; xlabel('Phase Margin, ^o'); ylabel('Damping ratio, \zeta')

Zeros & Non-Minimum Phase Zeros

Influence of Zeros and Non-Minimum Phase Zeros of Transfer Functions on Dynamical System Behavior [https://aleksandarhaber.com/effect-of-zeros-of-transfer-functions-on-dynamical-system-behavior/]

The zeros in the right half of the complex plane are called nonminimum phase zeros. Systems with nonminimum phase zeros are called nonminimum phase systems

Zero close to the real pole attenuates the effect of that pole on the system response

Zeros Tend to Increase the Overshoot of the System

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Nonminimum Phase Zeros – Effect on the Transient Response

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\[\begin{align} TF &= \frac{s +\omega_z}{s^2+2\zeta \omega_ns+\omega_n^2} \\ &= \frac{\omega _z}{\omega _n^2}\cdot \frac{1+s/\omega _z}{1+s^2/\omega_n^2+2\zeta s/\omega_n} \end{align}\]

Let \(s=j\omega\) and omit factor, \[ A_\text{dB}(\omega) = 10\log[1+(\frac{\omega}{\omega _z})^2] - 10\log[1+\frac{\omega^4}{\omega_n^4}+\frac{2\omega^2(2\zeta ^2 -1)}{\omega_n^2}] \] peaking frequency \(\omega_\text{peak}\) can be obtained via \(\frac{d A_\text{dB}(\omega)}{d\omega} = 0\) \[ \omega_\text{peak} = \omega_z \sqrt{\sqrt{(\frac{\omega_n}{\omega_z})^4 - 2(\frac{\omega_n}{\omega_z})^2(2\zeta ^2-1)+1} - 1} \]

Settling Time

One Pole

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we have \[ \tau \approx \left(1 + \frac{R_1}{R_2}\right)\frac{1}{A_0\omega_0}= \frac{1}{\beta \omega_\text{ugb}} \]

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Two Poles

with open-loop transfer function \(A_{OL}=\frac{A_0}{(1+s/\omega_1)(1+s/\omega_2)}\) and assuming \(\omega_1\) is dominant pole, then yield closed-loop transfer function

\[\begin{align} A_{CL}(s) &= \frac{\frac{A_0}{(1+s/\omega_1)(1+s/\omega_2)}}{1+\beta \frac{A_0} {(1+s/\omega_1)(1+s/\omega_2)}} \\ &= \frac{A_0}{1+A_0 \beta}\frac{1}{\frac{s^2}{\omega_1\omega_2(1+A_0\beta)}+\frac{1/\omega_1+1/\omega_2}{1+A_0\beta}s+1} \\ &\approx \frac{A_0}{1+A_0 \beta}\frac{1}{\frac{s^2}{\omega_u\omega_2}+\frac{1}{\omega_u}s+1} \\ &= \frac{A_0}{1+A_0 \beta}\frac{\omega_u\omega_2}{s^2+\omega_2s+\omega_u\omega_2} \end{align}\]

That is \(\omega_n = \sqrt{\omega_u\omega_2}\), \(\zeta = \frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\) , where \(\omega_u\approx \beta A_0 \omega_1\) is the unity gain bandwidth

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Rise Time (0% to 100% )

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\[ t_r = \frac{\pi - \beta}{\omega_d}=\frac{\pi - \arctan\frac{\omega_n\sqrt{1-\zeta^2}}{\zeta\omega_n}}{\omega_n\sqrt{1-\zeta^2}}\approx\frac{\pi - \arctan\frac{\sqrt{1-\zeta^2}}{\zeta}}{\sqrt{\omega_u\omega_2}\sqrt{1-\zeta^2}}=\frac{\pi - \arctan\frac{\sqrt{1-\zeta^2}}{\zeta}}{\omega_u\sqrt{k(1-\zeta^2)}} \] where \(k = \frac{\omega_2}{\omega_u}\), is the function of PM

PM \(45^o\) \(60^o\) \(70^o\)
\(k\) 1 1.73 2.75
\(\zeta\) 0.42 0.61 0.80
\(t_r\) \(\frac{2.2}{\omega_u}\) \(\frac{2.14}{\omega_u}\) \(\frac{2.53}{\omega_u}\) 
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PM = 70;
ztap = 0.80;
k = 1/tan((90-PM)/180*pi)
tr = (pi - atan(sqrt(1-ztap^2)/ztap))/sqrt(k*(1-ztap^2))

Settling Time

Gene F. Franklin, Feedback Control of Dynamic Systems, 8th Edition

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As we know \[ \zeta \omega_n=\frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\cdot \sqrt{\omega_u\omega_2}=\frac{1}{2}\omega_2 \]

Then \[ t_s = \frac{9.2}{\omega_2} \]

For \(\text{PM}=70^o\), \(\omega_2 = 2.75\omega_u\), that is \[ t_s \approx \frac{3.35}{\omega_u} \]

For \(\text{PM}=45^o\), \(\omega_2 = \omega_u\), that is \[ t_s \approx \frac{9.2}{\omega_u} \]

Above equation is valid only for underdamped, \(\zeta=\frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\lt 1\), that is \(\omega_2\lt 4\omega_u\)

2 Stage RC filter

High Pass Filter

image-20251025021242772

Since \(1/sC_1+R_1 \gg R_0\) \[ \frac{V_m}{V_i}(s) \approx \frac{R_0}{R_0 + 1/sC_0} = \frac{sR_0C_0}{1+sR_0C_0} \] step response of \(V_m\) \[ V_m(t) = e^{-t/R_0C_0} \] where \(\tau = R_0C_0\)

And \(V_o(s)\) can be expressed as \[\begin{align} \frac{V_o}{V_i}(s) & \approx \frac{sR_0C_0}{1+sR_0C_0} \cdot \frac{sR_1C_1}{1+sR_1C_1} \\ &= \frac{sR_0C_0R_1C_1}{R_0C_0-R_1C_1}\left(\frac{1}{1+sR_1C_1} - \frac{1}{1+sR_0C_0}\right) \end{align}\]

Then step response of \(Vo\) \[\begin{align} Vo(t) &= \frac{R_0C_0R_1C_1}{R_0C_0-R_1C_1} \left(\frac{1}{R_1C_1}e^{-t/R_1C_1} - \frac{1}{R_0C_0}e^{-t/R_0C_0}\right) \\ &= \frac{1}{R_0C_0-R_1C_1}\left(R_0C_0e^{-t/R_1C_1} - R_1C_1e^{-t/R_0C_0}\right) \\ &\approx \frac{1}{R_0C_0-R_1C_1}\left(R_0C_0e^{-t/R_1C_1} - R_1C_1\right) \end{align}\]

where \(\tau=R_1C_1\)

Partial-fraction Expansion

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syms C0
syms R0
syms C1
syms R1
syms s

Z0 = 1/s/C1 + R1;
Z1 = R0*Z0/(R0+Z0);
vm = Z1 / (Z1 + 1/s/C0);
vo = R1/Z0 * vm;
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>> partfrac(vm, s)
 
ans =
 
1 - (s*(C1*R0 + C1*R1) + 1)/(C0*C1*R0*R1*s^2 + (C0*R0 + C1*R0 + C1*R1)*s + 1)
 
>> partfrac(vo, s)
 
ans =
 
1 - (s*(C0*R0 + C1*R0 + C1*R1) + 1)/(C0*C1*R0*R1*s^2 + (C0*R0 + C1*R0 + C1*R1)*s + 1)

\[\begin{align} V_m(s) &= 1 - \frac{s(C_1R_0+C_1R_1)+1}{C_0C_1R_0R_1s^2+(C_0R_0+C_1R_0+C_1R_1)s+1} \\ V_o(s) &= 1 - \frac{s(C_0R_0+C_1R_0+C_1R_1)+1}{C_0C_1R_0R_1s^2+(C_0R_0+C_1R_0+C_1R_1)s+1} \end{align}\]

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C0 = 200e-9;
R0 = 50;
C1 = 400e-15;
R1 = 200e3;

s = tf("s");

Z0 = 1/s/C1 + R1;
Z1 = R0*Z0/(R0+Z0);
vm = Z1 / (Z1 + 1/s/C0);
vo = R1/Z0 * vm;

vm_exp = 1 - (s*(C1*R0 + C1*R1) + 1)/(C0*C1*R0*R1*s^2 + (C0*R0 + C1*R0 + C1*R1)*s + 1);
vo_exp = 1 - (s*(C0*R0 + C1*R0 + C1*R1) + 1)/(C0*C1*R0*R1*s^2 + (C0*R0 + C1*R0 + C1*R1)*s + 1);

figure(1)
subplot(1,2,1)
step(vm, 500e-9, 'k-o');
hold on;
step(vm_exp, 500e-9, 'r-^')
title('vm step response')
grid on;
legend()


subplot(1,2,2)
step(vo, 500e-9, 'k-o');
hold on;
step(vo_exp, 500e-9, 'r-^')
title('vo step response')
grid on;
legend()


% with approximation
figure(2)
vm_exp2 = s*R0*C0/(1+s*R0*C0);
vo_exp2 = s*R0*C0/(1+s*R0*C0) * s*R1*C1/(1+s*R1*C1);

subplot(1,2,1)
step(vm, 500e-9, 'k-o');
hold on;
step(vm_exp2, 500e-9, 'r-^')
title('vm step response')
grid on;
legend()

subplot(1,2,2)
step(vo, 500e-9, 'k-o');
hold on;
step(vo_exp2, 500e-9, 'r-^')
title('vo step response')
grid on;
legend()

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spectre vs matlab

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C0 = 200e-9;
R0 = 50;
C1 = 400e-15;
R1 = 200e3;

s = tf("s");

Z0 = 1/s/C1 + R1;
Z1 = R0*Z0/(R0+Z0);
vm = Z1 / (Z1 + 1/s/C0);
vo = R1/Z0 * vm;

step(vm, 500e-9);
hold on;
step(vo, 500e-9);

grid on
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>> vm

vm =
 
          1.024e-44 s^5 + 2.56e-37 s^4 + 1.6e-30 s^3
  -----------------------------------------------------------
  1.024e-44 s^5 + 2.571e-37 s^4 + 1.626e-30 s^3 + 1.6e-25 s^2
 
Continuous-time transfer function.

>> vo

vo =
 
                 8.194e-52 s^6 + 2.048e-44 s^5 + 1.28e-37 s^4
  ---------------------------------------------------------------------------
  8.194e-52 s^6 + 3.081e-44 s^5 + 3.871e-37 s^4 + 1.638e-30 s^3 + 1.6e-25 s^2
 
Continuous-time transfer function.

Low Pass Filter

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\[ \left\{ \begin{array}{cl} \frac{V_i - V_m}{R_0} &= C_0\frac{dV_m}{dt} + C_1\frac{dV_o}{dt} \\ \frac{V_m - V_o}{R_1} &= C_1\frac{dV_o}{dt} \\ V_m(t=0) &= 1 \\ V_o(t=0) &= 0 \end{array} \right. \]

Take into initial condition account \[ \frac{dV_m}{dt} \overset{\mathcal{L}}{\longrightarrow} sV_M-V_{m0} \] where \(V_{m0} = 1\)

\[\begin{align} V_O(s) &= \frac{V_I}{s^2R_0C_0R_1C_1+s(R_0C_0+R_1C_1+R_0C_1)+1} + \frac{V_{m0}R_0C_0}{s^2R_0C_0R_1C_1+s(R_0C_0+R_1C_1+R_0C_1)+1} \\ &\approx \frac{V_I}{s^2R_0C_0R_1C_1+sR_0(C_0+C_1)+1} + \frac{V_{m0}R_0C_0}{s^2R_0C_0R_1C_1+sR_0(C_0+C_1)+1} \\ &= \frac{V_I}{R_0(C_0+C_1)}\left(\frac{R_0(C_0+C_1)}{sR_0(C_0+C_1)+1} - \frac{R_1\frac{C_0C_1}{C_0+C_1}}{sR_1\frac{C_0C_1}{C_0+C_1}+1}\right) \\ &+ \frac{C_0}{C_0+C_1}\left(\frac{R_0(C_0+C_1)}{sR_0(C_0+C_1)+1} - \frac{R_1\frac{C_0C_1}{C_0+C_1}}{sR_1\frac{C_0C_1}{C_0+C_1}+1}\right) \end{align}\]

with \(V_I = \frac{1}{s}\), using inverse Laplace transform \[\begin{align} V_o(t) &= 1 - \frac{C_1}{C_0+C_1}e^{-t/\tau_0}-\frac{C_0}{C_0+C_1}e^{-t/\tau_1} - \frac{R_1C_0C_1}{R_0(C_0+C_1)^2} \tag{Eq.0} \\ &= 1 - \frac{C_1}{C_0+C_1}e^{-t/\tau_0}-\frac{C_0}{C_0+C_1}e^{-t/\tau_1} \tag{Eq.1} \end{align}\]

where

\[\begin{align} \tau_0 &= R_0(C_0+C_1) \\ \tau_1 &= R_1C_1\cdot \frac{C_0}{C_0+C_1} \end{align}\]

and \(\tau_0 \gg \tau_1\)

image-20251025022237093

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R0 = 100e3;
C0 = 200e-15;
R1 = 10e3;
C1 = 100e-15;

s = tf('s');

VI = 1/s;
Vm0 = 1;

deno = (s^2*R0*C0*R1*C1 + s*(R0*C0 + R1*C1 + R0*C1) + 1);
VO = VI/deno + (Vm0*R0*C0)/deno;
VM = (1+s*R1*C1)*VO;

t = linspace(0,40,1e3);  % ns
[vot, ~] = impulse(VO, t*1e-9);
[vmt, ~] = impulse(VM, t*1e-9);

tau0 = R0*(C0+C1)*1e9; %ns
tau1 = R1*C0*C1/(C0+C1)*1e9; %ns
y0 = 1 - C1/(C0+C1)*exp(-t/tau0) - C0/(C0+C1)*exp(-t/tau1) - R1*C0*C1/R0/(C0+C1)^2;  % Eq.0
y1 = 1 - C1/(C0+C1)*exp(-t/tau0) - C0/(C0+C1)*exp(-t/tau1);  % Eq.1

plot(t, vot, t, vmt,LineWidth=2);
hold on
plot(t,y0, t,y1, LineWidth=2, LineStyle="--" );
xlim([-10, 40])
legend('V_o(t)', 'V_m(t)','y_0(t)', 'y_1(t)', fontsize=12)
xlabel('t (ns)')
ylabel('mag (V)')
grid on;

reference

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

Katsuhiko Ogata, Modern Control Engineering, 5th edition [pdf]

C. T. Chuang, "Analysis of the settling behavior of an operational amplifier," in IEEE Journal of Solid-State Circuits, vol. 17, no. 1, pp. 74-80, Feb. 1982 [https://sci-hub.se/10.1109/JSSC.1982.1051689]