Gvd derivation of Boost converter

Ming Sun

Ming Sun / November 29, 2022

15 min read––– views

Step 1 - construct small-signal equations

Boost power stage block diagram
Fig. 1Boost power stage block diagram[1]

Voltage-second balance equation

Fig. 1 shows a synchronous Boost 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]:

`L{dI}/{dt} = D*V_g + D^'*(V_g-V) = V_g - D^'*V`
(1)

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

`L{d(I+hat(i))}/{dt} = V_g - (D^'-hat(d))*(V+hat(v))`
(2)

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

`L{dhat(i)}/{dt} = -D^'*hat(v)+hat(d)*V`
(3)

Eq. 3 can be written in s domain as:

`sL*hat(i) = -D^'*hat(v)+hat(d)*V`
(4)

charge balance equation

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

`C{dV}/{dt} = D^'*I-V/R`
(5)

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

`C{d(V+hat(v))}/{dt} = (D^'-hat(d))*(I+hat(i))-(V+hat(v))/R`
(6)

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

`C{dhat(v)}/{dt} = D^'*hat(i) - hat(d)*I - hat(v)/R`
(7)

Eq. 7 can be written in s domain as:

`sC*hat(v) = D^'*hat(i) - hat(d)*I - hat(v)/R`
(8)

Step 2 - solve the Gvd in Matlab

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

Gvd_boost.m
clc; clear; close all;

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

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

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

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

Fig. 2 shows the Gvd derived result from Matlab.

Gvd derived result from Matlab
Fig. 2Gvd derived result from Matlab

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

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

From Fig. 3, we have:

`G_{vd} = V/D^' * (1-s*L/(RD^('2)))/{1+s*L/(RD^('2)) + (LC)/D^('2)*s^2}`
(9)

Simplis for verification of Gvg transfer function

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

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

To set the property of the Laplace Transfer Function block, Eq. 9 can be re-written as:

`G_{vd} = V/D^' * (1-s*L/(RD^('2)))/{1+s*L/(RD^('2)) + (LC)/D^('2)*s^2} = (V*D^')/(LC) * (1-s*L/(RD^('2)))/{D^('2)/(LC)+s/(RC) + s^2}`
(10)

We can plug in the inductor, capacitor, resistor, V and D' values into Eq. 10. We have:

`G_{vd} = (V*D^')/(LC) * (1-s*L/(RD^('2)))/{D^('2)/(LC)+s/(RC) + s^2} = 5T* (1-s*250n)/(640G + 160k*s + s^2)`
(11)

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

2nd-order Laplace Transfer Function block property
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 Gvd very well at low frequency range. Magnitude and phase in the Bode plot starts to shown some differences when the frequency is close to the switching frequency. At half of the switching frequency, the magnitude between mathematical derivation and Simplis simulation results still match with each other very well while there is about 7º phase difference between the two.

Simulation results comparison between mathematical derivation and AC analysis
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 fsw.

Gid verification

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

`G_{id} = V/(RD^('2)) * (2+sRC)/(1+s*L/(RD^('2)) + s^2*(LC)/(D^('2)))`
(12)

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

Gid test bench in Simplis for Boost converter
Fig. 7Gid test bench in Simplis for Boost converter

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

`G_{id} = V/(RD^('2)) * (2+sRC)/(1+s*L/(RD^('2)) + s^2*(LC)/(D^('2))) = V/(RLC) * (2+sRC)/(D^('2)/(LC)+s/(RC) + s^2)`
(13)

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

`G_{id} = V/(RLC) * (2+sRC)/(D^('2)/(LC)+s/(RC) + s^2) = 2T*(1+s*3.125µ)/(640G+s*160k+s^2)`
(14)

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

2nd-order Laplace Transfer Function block property for Gid simulation
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 Gid.

Gid comparison between mathematical derivation and AC analysis
Fig. 9Gid comparison between mathematical derivation and AC analysis

References and downloads

[1] Fundamentals of power electronics - Chapter 2

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

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

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

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


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