System Transfer Functions

Evolution of the Power Converter Block Diagram

The figure below shows the functional block diagram of the power conversion system.

power-conversion-block-diagram.png

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. The figure below was re-drawn from the above diagram, showing that all the block transfer functions from the input signal (e) to the output (y) are lumped together. The complete transfer function (obtained 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, for instance, the output voltage. The transfer function of the transducer can simply be a gain implemented in the circuit as a resistor divider.

power-conversion-block-lumped.png

For further information on digital power implementation, please visit Digital Block with a dsPIC® DSC page.

System Transfer Functions

The figure below 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 below as bode plot, G(s) and H(s), which are the approximated representations of the real transfer function.

system-transfer-function.png

The transfer functions depend on variable “s” in the Laplace domain. Without going through too many details on Laplace transform, it's important to note that using the transformation from the frequency domain to the Laplace domain greatly simplify 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: Gcl(s) = [G(s)] / [1 + H(s) G(s)], where Gcl means closed-loop. This is the closed-loop gain of the system, i.e., the transfer function from input to output.

Open-Loop Function Approximation

Open Loop transfer function is simply the product of the two functions G(s) and H(s).

system-transfer-function-gs-hs.png

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
open-loop-transfer-function.png

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 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 really 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).

open-loop-gs.png

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

system-transfer-function-hs.png

At this point if we carefully look at the equation that defines the open loop transfer function and at the plot, we recognize that Gol(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 |Gol(s)|.

system-transfer-function-gol.png

Closed-Loop Function Approximation

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

system-transfer-function-gcl.png

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 Gol(s) is greater than one: in this interval Gcl(s) = 0 dB, and the interval where it is smaller than one. In this interval the closed-loop transfer function equals G(s).

system-transfer-function-gol-1.png

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

system-transfer-function-gol-less-1.png

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 in order 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 the figure below: at the point (frequency) where |Gol(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) = -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.

stability-conditions.png

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 point in frequency 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.

stability-conditions-bode.png

The plot in the figure below also presents the phase of Gol(s) . Since we only have one pole, the phase will start from 0 degree at 0 Hz and reach -90 degrees after the pole itself. In the plot, phase margin is also shown. Phase margin indicates how far from the critical value (180 degrees) we are. Obviously, 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 relative stable system.

stability-conditions-phase.png

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