Finding the Small-Signal Model Using Perturbed Equations

by Brad Suppanz

After finding the State-Space Averaged (SSA) equations for a converter, we generally find that they are non-linear equations. If we want to do hand calculations to study the AC operation of the converter, we have to find the small-signal model. (However, if we plan to use SPICE to simulate the SSA model, it can extract the small-signal model for us! That's no fun though, because then we don't really know what is going on.)

I will discuss two methods for extracting the small-signal AC model from the non-linear SSA equations:

Perturbed Equations
Partial Derivatives

Method 1 is the industry standard, but I like method 2 because I can apply it by inspection in most cases and it can handle cases that method 1 can not. Let's take Bello's SSA switch model equations and apply both methods.

Recall Bellow's SSA switch model equations:

v_cp = d*v_ap
i_a = d*i_c

Note that these equations are non-linear because of the product between two variables that appears on the right hand side of each. In the linearized small-signal equations, each variable can only be multiplied by a constant - not another variable. The following sections show how to apply each method.

Perturbed Equations Method

In this method, we make use of the idea that there is a DC operating point about which some small AC signal are applied.

Outline of Steps:

Break each variable into DC and AC components.
Substitute both DC and AC components into the non-linear equation.
Eliminate all pure DC and cross-products of AC components.

Step 1. Break each variable into DC and AC components:

v_cp = V_cp + Dv_cp
v_ap = V_ap + Dv_ap
d = D + Dd
i_a = I_a + Di_a
i_c = I_c + Di_c

Note: The DC component is represented by capitol letters and the AC component by the variable preceded by a D (delta). It is common to represent the AC by putting a "hat" over the variable, but I use the delta because it is easier to type and it ties into calculus notation.

Step 2. Substitute both DC and AC components into the non-linear equation:

(V_cp + Dv_cp) = (D + Dd)*(V_ap + Dv_ap)
(I_a + Di_a) = (D + Dd)*(I_c + Di_c)

Step 3. Eliminate all pure DC and cross-products of AC components:

Dv_cp = V_ap*Dd + D*Dv_ap
Di_a = I_c*Dd + D*Di_c

And this is our linearized small-signal model. Each AC signal is multiplied by a constant "gain" coefficient which is generally one of the DC operating point values. Thus, we will need to know the DC operating point before doing the AC analysis. The DC operating point is found simply by solving the equations at DC. Further simplifications are often made such as ignoring the D*Dv_ap term if applicable.

Partial Derivative Method

In this method, we also make use of the idea that there is a DC operating point about which some small AC signal are applied.

Outline of Steps:

Substitute DC variables (capitol letters) into the non-linear equation.
Take partial derivatives with respect to each variable. These become the constant gain coefficients.
Multiply each AC variable by it's gain coefficient and sum. This is "the chain rule".

Step 1. Substitute DC variables (capitol letters) into the non-linear equation:

Bellow's SSA switch model equations become:

V_cp = D*V_ap
I_a = D*I_c

Step 2. Take partial derivatives with respect to each variable. These become the constant gain coefficients:

dV_cp/dD = V_ap
dV_cp/dV_ap = D
dI_a/dD = I_c
dI_a/dI_c = D

Step 3. Apply the chain rule. (Multiply each AC variable by it's gain coefficient and sum.)

Dv_cp = (dV_cp/dD)*Dd + (dV_cp/dV_ap)*Dv_ap
Di_a = (dI_a/dD)*Dd + (dI_a/dI_c)*Di_c

Substituting, we get:

Dv_cp = V_ap*Dd + D*Dv_ap
Di_a = I_c*Dd + D*Di_c

And again, this is our linearized small-signal model. It is the same thing we got using the Perturbed Equation Method. The Partial Derivative / Chain Rule Method even works in some cases where the Perturbed Equation Method fails, such as when we have a transcendental expression.

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Written 8/29/97
Last updated 7/20/04
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