Developing Digital System Transfer Functions for a Power Converter

Evolution of the Power Converter Block Diagram

Figure 1 shows the functional block diagram of the power conversion system.

From a control point of view, the input to the system is the difference between the reference signal and the output signal, y, which is processed by the transducer block. This difference is called error, e. The overall system will operate so that the error is zero at any time. When this happens, the real output signal matches the reference signal, which is the desired output value. Figure 2 was re-drawn from the above diagram, showing that all the block transfer functions from the input signal (e) to the output  are lumped together. The complete transfer function (obtained by multiplying the transfer functions of the three blocks) is then called G(s).

The feedback path, in this case, is the transducer, which is used to read the output voltage, for instance. The transfer function of the transducer can simply be again implemented in the circuit as a resistor divider.

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For further information on digital power implementation, please visit "Digital Block with a dsPIC® Digital Signal Controller (DSC)" page.

System Transfer Functions

Figure 3 shows the general aspect of a feedback system based on the above diagram. The controller and process can be mathematically represented by transfer functions. It is possible to graphically represent the transfer functions as shown in Figure 3 as bode plot, G(s) and H(s), which are the approximated representations of the real transfer function.

The transfer functions depend on variables in the Laplace domain. Without going through too many details on the Laplace transform, it's important to note that using the transformation from the frequency domain to the Laplace domain greatly simplifies the computation.

In the figure, G(s) is the process transfer function and H(s) is the controller transfer function. With simple computation, it can be shown that the input/output transfer function is G_cl(s) = [G(s)] / [1 + H(s) G(s)], where G_cl means closed-loop. This is the closed-loop gain of the system (i.e., the transfer function from input to output).
 

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Open Loop Function Approximation

The open-loop transfer function is simply the product of the two functions, G(s) and H(s).

It is convenient to express the open-loop transfer function in dB. The basic reasons for this is that:

• Products become additions
• Divisions become subtractions

It is easy to build a graphical representation of the system that allows us to evaluate the most important aspects of the behavior of the system in the frequency domain using a bode plot.

From the bode plot, it is possible and extremely easy to evaluate the overall system stability.

For the open-loop gain, it can be computed as the sum of the transfer function in the forward path (G(s)) and the transfer function in the backward path (H(s)). Using the properties of logarithmic functions, the sum can be rewritten as the subtraction of the reverse of H(s).

Using bode plots, we can perform an approximation of the real function behavior to evaluate stability and other data of the system

Let’s start plotting the various components of the open-loop transfer function. The first one is G(s). In the plot, we suppose it is flat from 0 Hz up to a certain frequency, where a single pole is located. The shape of G(s) is not essential to the discussion we are going to do. The single pole expression only makes the description easier, but the results we are going to obtain are valid for any number of singularities in G(s) (singularities are poles and/or zeroes).

H(s), in Figure 7, has been plotted with the vertical axis in dB: 20log(1) = 0.

At this point, if we carefully look at the equation that defines the open-loop transfer function and at the plot, we recognize that G_ol(s) is simply obtained by summing the bode plot of G(s) and that of H(s). Since the bode plot is an approximation of the functions using only straight lines, this addition can be done very easily. D1 represents the modulus of the open-loop function. In this case |G(s)| equals |G_ol(s)|.

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Closed-Loop Function Approximation

Until now, we have been looking at the open-loop gain G_ol(s) only. However, we are interested in the transfer function of the overall system (i.e., the closed-loop transfer function, G_cl(s)).

If we look at the closed-loop transfer function equation, we see that we can distinguish two regions of the frequency axis. The frequency interval where G_ol(s) is greater than one: in this interval G_cl(s) = 0 dB, and the interval where it is smaller than one. In this interval, the closed-loop transfer function equals G(s).

The point where G(s) equals 1 (0 dB) is the frequency where G_ol(s) crosses the x-axis.

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Stability Conditions

Let us now investigate the stability condition. For a system to be stable, given a limited input perturbation, the output will be limited and decay to zero. The problem we have right now is, how can we determine if the system at hand is stable? The answer comes quite easily if we consider the closed-loop transfer function.

We know that for this equation to be meaningful, the denominator must be different from zero. Should it be zero, we would have an infinite gain (instability). At this point, what are the conditions that make the denominator equal to zero? They are shown in Figure 12: at the point (frequency) where |G_ol(s)| = |G(s)H(s)| = 1, its phase must be different from 180 degrees. In fact, if this is not satisfied, G(s)H(s) equals -1 and the denominator is zero. We have transformed the problem of determining the stability of a closed-loop system into the solution of two sets of equations that involve the open-loop transfer function.

Let us consider again the bode plot we have already seen before, where we have right now is only the |G(s)| and |[H(s)]| functions. The first equation simply identifies the point where the two plots cross each other. This frequency point is called Cross-Over Frequency (FC). The second equation simply states that the phase of the open-loop gain, at this point, must be different from 180 degrees.

The plot in Figure 14 also presents the phase of G_ol(s). Since we only have one pole, the phase will start from 0 degrees at 0 Hz and reach -90 degrees after the pole itself. In the plot, the phase margin is also shown. Phase margin indicates how far from the critical value (180 degrees) we are. The larger the phase margin the better in terms of stability. However, in practical situations, a phase margin of 45 to 60 degrees corresponds to a relatively stable system.

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