OTA based Type-II compensator transfer function derivation

Type-II compensator - OTA based

Fig. 1 shows a Type-II compensator, where an OTA is being used.

Fig. 1Type-II compensator block diagram^[1]

The transfer function of Type-II compensator is defined as:

The small signal AC model is as shown in Fig. 2, where V_ref is set to be AC ground.

Fig. 2Type-II compensator block diagram - small signal model

The Matlab script used to derive the transfer function of Eq. 1 is as shown below.

clc; clear; close all;

syms Vea Vfb H
syms ro R1 C1 C2 gm s

eqn1 = -gm*Vfb == Vea/ro + Vea*s*C2 + Vea/(R1+1/s/C1);
eqn2 = H == Vea/Vfb;

results = solve(eqn1,eqn2,[Vea,H]);

H = simplify(results.H)

pretty(H)

The Matlab derived results are as shown in Fig. 3.

Fig. 3Matlab derived results for Type-II compensator transfer function

Here, let us assume that capacitor C₁ is much larger than capacitor C₂ and OTA output impedance r_o is much larger than resistor R₁. Then, we can use wxMaxima to further simplify the result as shown in Fig. 4.

Fig. 4wxMaxima simplified results for Type-II compensator transfer function

From Fig. 4, we have:

The zero location can be easily calculated to be:

Next, let us derive the pole locations. The denominator can be written as:

Assuming the two poles are far away separated from each other,

Therefore, we have:

By comparing the s and s² coefficient between Eq. 6 and Eq. 4, we have:

As a result, the pole locations can be calculated as:

As long as OTA output impedance r_o is much larger than resistor R₁, Eq. 8 result shall be valid.

Simplis verification

The Type-II compensator test bench in Simplis is as shown in Fig. 5.

Fig. 5Type-II compensator test bench in Simplis

The AC simulation results from Simplis is as shown in Fig. 6.

Fig. 6Type-II compensator AC simulation results in Simplis

The following Matlab script is used to compare the Bode plot between Eq. 2 and Simplis simulatio results.

clc; clear; close all;

gm = 100e-6;
ro = 100e6;
R1 = 100e3;
C1 = 100e-12;
C2 = 200e-15;

z = 1/R1/C1;
p1 = 1/ro/C1;
p2 = 1/R1/C2;

s = tf('s');

H = gm*ro*(1+s/z)/(1+s/p1)/(1+s/p2);

h = bodeplot(H);                    
setoptions(h, 'FreqUnits', 'Hz');   % change frequency scale from rad/sec to Hz
set(findall(gcf,'type','line'),'linewidth',2)
grid on;
hold on;

% read simplis simulation results from csv file
data = csvread("simplis.csv", 1, 0);
freq = data(:,1);
mag = data(:,2);
phase = data(:,3);

ax = findobj(gcf, 'type', 'axes');
phase_ax = ax(1);
mag_ax = ax(2);


% append simplis plot to bode plot
plot(phase_ax, freq, phase, 'r--', 'LineWidth', 2);
plot(mag_ax, freq, mag, 'r--', 'LineWidth', 2);
legend('Math', 'Simplis')

The comparison between Mathematical derivation and Simplis AC simulation results are as shown in Fig. 7.

Fig. 7Type-II compensator small signal transfer function comparison

Summary

Fig. 8 visually check the components which generate poles and zero.

Fig. 8Type-II compensator small signal pole and zero demonstration

• At low frequency, r_o and C₁ forms the dominant pole. This pole behaves sort of as the integrator. The high DC gain guarantees the DC regulation accracy of switching regulators.
• At mid-band frequency, R₁ and C₁ generates a zero, which is going to provide the phase boost in mid-band frequency.
• When frequency becomes higher, capacitor C₁ becomes short. Therefore, the mid-band DC gain can be calculated as g_mR₁. Also, resistor R₁ and C₂ generates a high frequency pole, which can be used to damping the ripple around switching frequency.

The Bode plot illustration is as shown in Fig. 9.

Fig. 9Type-II compensator Bode plot illustration

References and downloads

[1] Demystifying Type II and Type III Compensators Using OpAmp and OTA for DC/DC Converters

[2] Opamp based Type-II compensator test bench in Simplis - pdf

[3] Opamp based Type-II compensator test bench in Simplis - download