Gvd derivation of inverted Buck-Boost (IBB) converter

Step 1 - construct small-signal equations

Fig. 11Inverted Buck-Boost (IBB) block diagram^[1]

Voltage-second balance equation

Fig. 1 shows a synchronous inverted Buck-Boost (IBB) power stage, where it contains a low side switch S₁ and a high side switch S₂.

For the inductor, we can write the voltage-second balance as^[1]:

Where, I is the inductor current, V_g is the Boost converter's input voltage, and V is Boost converter's output voltage V_out. Next, let us perturb and linearize Eq. 1 by introducting the small signal perturbation as:

Here we are trying to derive the transfer function of G_vd. As a result, we can assume V_g is constant. Removing the DC terms from and second order small signal terms Eq. 2 , we have:

Eq. 3 can be written in s domain as:

charge balance equation

For the capacitor, we can write the charge balance as^[1]:

Next, let us perturb and linearize Eq. 5 by introducting the small signal perturbation as:

Removing the DC terms and second order small signal terms from Eq. 6, we have:

Eq. 7 can be written in s domain as:

Step 2 - solve the G_vd in Matlab

The Matlab script used to derive the G_vd transfer function is as shown below:

clc; clear; close all;

syms s
syms v i d
syms R L C V Dp I Vg D
syms Gvd Gid

eqn1 = s*L*i == d*Vg + Dp*v - d*V;
eqn2 = s*C*v == -Dp*i + d*I - v/R;
eqn3 = I*Dp == -V/R;
eqn4 = D*Vg + Dp*V == 0;
eqn5 = Gvd == v/d;
eqn6 = Gid == i/d;

results = solve(eqn1, eqn2, eqn3, eqn4, eqn5, eqn6, [v i I Vg Gvd Gid]);

Gvd = simplify(results.Gvd)
Gid = simplify(results.Gid)

Fig. 2 shows the G_vd derived result from Matlab.

Fig. 2Gvd derived result from Matlab

Next, we can use wxMaxima to simplify the results as shown in Fig. 3.

Fig. 3Gvd and Gid transfer function simplified results from wxMaxima

From Fig. 3, we have:

Eq. 9 can be rewritten as:

Simplis for verification of Gvd transfer function

To simulate G_vd transfer function in Simplis, the open loop Boost converter model is as shown in Fig. 4.

Fig. 4Open-loop IBB converter model for Gvd simulation

To set the property of the Laplace Transfer Function block, Eq. 10 can be re-written as:

We can plug in the inductor, capacitor, resistor, V and D' values into Eq. 11. We have:

Based on Eq. 11, the property of the 2nd-order Laplace Transfer Function is as shown in Fig. 5.

Fig. 52nd-order Laplace Transfer Function block property

The Simplis simulation results are as shown in Fig. 6. From Fig. 6, we can see that the mathematical Laplace transfer function matches with the AC simulation results of G_vd very well at low frequency range.

Fig. 6Simulation results comparison between mathematical derivation and AC analysis

But the difference is small enough and the it will not impact the actual design when we use the mathematical equations since in most of the design the closed loop cross over frequency is much less than the switching frequency f_sw.

Gid verification

From Fig. 2, the G_id transfer function can be written as:

To verify the G_id transfer function, the Simplis test bench can be modified as shown in Fig. 7.

Fig. 7Gid test bench in Simplis for IBB converter

To set the property of the Laplace Transfer Function block, Eq. 12 can be re-written as:

We can plug in the inductor, capacitor, resistor and Vg values into Eq. 13. We have:

Based on Eq. 14, the property of the 2nd-order Laplace Transfer Function is as shown in Fig. 8.

Fig. 82nd-order Laplace Transfer Function block property for Gid simulation

The Simplis simulation results are as shown in Fig. 9. From Fig. 9, we can see that the mathematical Laplace transfer function matches with the AC simulation results of G_id.

Fig. 9Gid comparison between mathematical derivation and AC analysis

References and downloads

[1] Fundamentals of power electronics - Chapter 2

[2] Open-loop IBB converter model for Gvd simulation in Simplis - pdf

[3] Open-loop IBB converter model for Gvd simulation in Simplis - download

[4] Open-loop IBB converter model for Gid simulation in Simplis - pdf

[5] Open-loop IBB converter model for Gid simulation in Simplis - download