Cramer's Rule Helps Solve DC-DC Converter Transfer Functions

A switching dc-dc converter converts an input voltage to an output of a different magnitude and/or polarity with high efficiency. Typically, such a converter must be regulated; that is, able to produce a fixed output voltage from a variable input voltage. To provide this regulation, it is necessary to feedback-stabilize the converter.

Fig. 1 shows examples of regulated buck and boost converters. In both cases, an error amplifier is used to compare a fixed reference voltage with the output voltage, or a fraction of it. The error amplifier is usually part of a power management IC that controls most, if not all the power supply's operational functions. The error amplifier's output voltage is proportional to the error between the actual output voltage and its desired value set by the reference voltage. The error voltage is used to generate a switching signal for the MOSFETs (or other controlled switches) that forces the supply's output voltage to stay at the desired value despite system perturbations.

The proper behavior of the converter under feedback regulation depends on the correct design of the compensation network around the error amplifier. A proper understanding of the desired properties of the compensation network requires the precise determination of the small-signal transfer functions of the converter. Deriving these transfer functions in a consistent and relatively error-free way is a problem for many designers.

There is a logical and structured general method for solving the transfer functions. Based on Cramer's Rule, it is essentially a simplified version of the method for solving the transfer functions in state-space-averaged dc-dc converters.

Typically, for reasons of compactness and elegance of expression, the simultaneous equations describing a dc-dc converter are formulated in matrix form. The complete description of a power converter system usually contains two excitations or perturbations: the duty ratio, d̂, and the input voltage, V̂_in, and a great deal of computational complexity is created because these excitations are solved for simultaneously.

The method presented here exploits the superposition principle to solve for the effect of each perturbation separately. You can then use Cramer's Rule to simplify the derivation of the transfer functions for each excitation. Once mastered, the method allows the engineer to easily write down the analytical expressions for the transfer functions of common dc-dc converters. Though Cramer's Rule is inefficient for numerical computation, the equations encountered in dc-dc converters are of low enough order — typically two to four — that this is not a major hindrance, and the method can serve as an excellent base for engineering design spreadsheets based on programs like Mathcad.

For second-order systems of equations, which are encountered in the analysis of buck, boost, and buck-boost converters and certain of their derivatives, Cramer's Rule can be easily evaluated by hand.

Cramer's rule is a simple way of solving systems of linear equations in matrix form. To illustrate this rule, let the system of n variables be described by the matrix relationship: (view article equations)

or, more compactly:

A⃑x⃑ = B⃑

where, as shown, A⃑ is an nxn matrix of constant coefficients, x̄ is an nx1 column vector of the variables to be solved for, and B⃑ is an nx1 column vector of constants. By Cramer's Rule the ith variable x_i is solved for by the expression:

where det(A) is the determinant of the A matrix, i.e.,

The determinant is a scalar, i.e. a simple number, and the formula for computing it can be found in all textbooks on linear algebra. In the particular case of a 2×2 system of equations, the determinant is given by the simple formula:

The numerator of Eq. 3 is the determinant corresponding to the ith variable x_i, which is computed after replacing the ith column of det(A) by the B⃑ column vector, i.e.,

As an example, a 2×2 system of equations can be expressed in matrix form as:

The variables are computed by Cramer's rule to be:

The formulas in Eq. 8 will be used to compute the transfer functions of the converter.

Before applying Cramer's Rule, the equations of the dc-dc converter must be derived. But switching converters are nonlinear systems, and there are no easy methods that can be used to analyze them directly.

This problem is typically surmounted by developing linear models of the converters, to which Kirchhoff's voltage and current laws can then be applied to derive the transfer functions. The first step in the process is to average out the effects of switching, a step that is justified by the fact that the feedback loop operates at frequencies that are typically low relative to the switching frequency. The averaged model is then linearized around the operating point of the converter, and the transfer function of the linearized model is determined.

Various such models have been developed ^[1] and all give essentially the same results. Here, we will use the small-signal linear pulse-width-modulator (PWM) switch model of the converter developed by Vorperian ^[2]. This model is used to make a terminal-for-terminal substitution for the two switches found in all basic converters, yielding a linear circuit. The transfer functions of interest in this model are from excitations or perturbations in the duty ratio, d̂(s), and in the input voltage, V̂_in(s), to the state variables, i.e. the capacitor voltages and inductor currents, and to the system's output voltage.

Fig. 2a shows the PWM switch for a converter in continuous-conduction mode with parasitic resistances ignored, while Fig. 2b shows its linearized terminal-for-terminal small-signal model for continuous conduction. There is a separate model for discontinuous conduction but we will deal only with the continuous conduction case, partly because this is where stability issues are typically more severe.

The use of this model is illustrated with a buck converter (Fig. 3a). When the MOSFET and diode are replaced by the PWM switch model we get the linearized converter model in Fig 3b. Note: in the buck converter, V _AP = V_IN and I_C = I_LOAD. Also, the output capacitance is generally modeled with r_c, its ESR.

Now, the following steps can be followed to formulate the equations of this or any other converter circuit.

1. Write the differential equations that describe the system. The number of equations will always be equal to the number of independent capacitors plus the number of independent inductors. In this case there will be two equations because there is one capacitor and one inductor. If a SEPIC or Cuk converter (each with two inductors and two capacitors) were being analyzed there would be four equations. If the buck converter had a second inductor-capacitor filter on the output there would also be four equations.
2. The equations should be written only for the state variables of the system, i.e., for the inductor currents and the capacitor voltages. Mesh or loop equations (Kirchhoff's voltage law) should be written for the inductor currents, and node equations (Kirchhoff's current law) should be written for the capacitor voltages.
3. The derivative of any state variable should appear in only one equation. The mesh equations will contain the derivatives of the current variables and the node equations will have the derivatives of the voltage variables.
4. Eliminate all variables other than the state variables from the equations and then arrange the equations in matrix form.
5. Take Laplace transforms of the equations.
6. Treating the frequency variable s as a constant, rearrange the equations into a form suitable for using Cramer's rule.
7. Apply Cramer's rule to obtain the transfer functions from the perturbation ď(s) or v̌_in(s) to the state variables.
8. Derive further transfer functions from the state variables to any other voltages and currents of interest. In particular, if the output capacitor has ESR, derive a transfer function from the capacitor voltage to the output voltage.

The exact same procedure is followed if the switch model is refined ^[2] by the addition of parasitic resistances.

Let us apply this procedure to the buck converter. We will solve for the small-signal transfer functions from the duty ratio, d̂(s), to the inductor current,î₁, and to the capacitor voltage, v̂_co(s), and hence to the output voltage, v̂_o(s). The duty-ratio-to-inductor-current transfer function, î₁(s)/d̂(s), is needed to close the current loop in current-programmed converters. The duty-ratio-to-output-capacitor-voltage transfer function, v̂_co(s)/d̂(s), is needed to close the voltage loop in both voltage- and current-programmed converters. When solving for these transfer functions the input voltage is held constant, i.e., v̂_in(s)=0. This is permissible because a linear system obeys the superposition principle, so the effect of one excitation can be determined while others are suppressed.

The resulting circuit to be analyzed is shown in Fig. 3c. The quantities V_IN, D and V_OUT are known. The quantities V_p, V_s, î₁ and V_co and V_o are unknowns, while d̂ is the perturbation. The quantities v̂_p and v̂_s must be eliminated from the equations that are derived.

For the sake of completeness the differential equations will be formulated in the frequency domain, though with practice this step can be omitted, and the frequency domain relationships written down directly. With reference to Fig. 3c, writing a mesh equation for the inductor current around the loop encompassing the inductor, the output capacitor and the secondary of the transformer gives us:

To eliminate v̂_s(t) we solve for the transformer voltages, to get:

To eliminate v̂_o(t) and leave only the state variables, we note that two ways in which the capacitor current can be expressed are:

and

Equating the two expressing and solving for v̂_o(t) we get:

Substituting for v̂_s(t) and v̂_o(t) in Eq. 9 we get:

We now write a node equation at the output to get:

Using Eq. 13 to substitute for v̂_o)t) in Eq. 15, we get:

Eq. 14 and 16 are the two differential equations required and can be arranged in matrix form as:
(See Eq. 17)

Now we can take Laplace transforms of both sides of the equations. Noting that the Laplace transform of a derivative is equal to s times the Laplace transform of the variable we get:
(See Eq. 18)

We note that matrices obey the following relationship:

Therefore we can rearrange Eq. 18 to be:
(See Eq. 20)

This can be simplified to:

Now we divide both sides by d̂(s) to get:

The matrices are now in a form that allows for solution using Cramer's Rule, and the quantities î₁(s)/d̂(s) and v̂_co(s)/d̂(s) are the desired transfer functions.

The solution for the duty-ratio-to-inductor current transfer function is:

This transfer function can be used to analyze and design the current loop in current mode controlled converter systems.

The solution for the duty-ratio-to-capacitor-voltage transfer function, which is needed to design feedback compensators in both voltage-mode and controlled-mode controlled converters, is:

The determinants in the equations above can be evaluated and simplified using the symbolic equation manipulators in programs such as Mathcad. The capacitor voltage differs from the output voltage because of the capacitor's ESR. A further transfer function is needed from the capacitor voltage to the output voltage. This is obtained by noting that two other ways in which the current in the output capacitor can be expressed are:

and

Eliminating the current î_c(t) from Eq. 25 and Eq. 26 gives us:

Laplace-transforming this equation and rearranging it gives us the desired transfer function:

Therefore, the complete transfer function from the duty ratio to the output voltage is:
(See Eq. 29 on p. 16)

We note that if the capacitor has no ESR, i.e., r_c = 0, this transfer function reduces, as expected, to:

The transfer functions just derived can now be used to design a feedback compensator to regulate the converter's output voltage. The same derivation procedure will work to determine the transfer functions for the other basic converters. The procedure will work perfectly when the model is refined, and thus complicated by the addition of parasitic resistances to the inductors and MOSFETs. It will also work in all general cases, for example, when capacitors of arbitrary value and arbitrary ESR are paralleled at the output.

It is evident that the final expressions for the transfer functions become very complex-looking when parasitics are included in the component models. Relative to straight solution of the simultaneous equations the method presented here reduces errors caused by the presence of a large number of terms in the equations. The method is an excellent vehicle for deriving the exact transfer functions in the absence of parasitics, when the small number of terms in the equations makes hand-calculation of the determinants straightforward.