Unwind di Power Electronics - 2 di 5

The purpose of averaging in signal processing

The purpose of averaging is to filter out the switching ripple from the waveform and focus on the low-frequency content of the signal, also defined as the low-frequency dynamics. The low-frequency dynamics corresponds to this waveform. In synthesis, making an averaging operation means performing a removing operation of the high-frequency content of a signal. In the case of v(t), everything related to the switching harmonics is removed. Clearly, this is an approximation, but the interesting part of the signal is concentrated at low frequencies.

Approximation and frequency domain limitations

The approximation considered on v(t) is valid only if the switching period is much smaller than the period of the motion. This also means that the model of the controller has a limit of validity in the frequency domain because the frequency of the motion must be much smaller than the switching frequency, or the model becomes invalid.

Spectrum of v(t)

The spectrum of v(t) is something like the following: f f f f_s m_s m_s f_s.t.fm_s.fm The spectrum of v(t) is V(f) and is composed of three terms:

• DC component
• Modulation frequency
• Switching frequency and side bands that are repeated infinite times with decreasing amplitude

In steady state, there is only one switching frequency, but out of steady state, there are spectral groups instead of spectral lines located at multiples of the fundamental switching frequency. In these spectral groups, there are multiples of the fundamental switching frequency and side bands that come from the non-linear interaction between the switching frequency f_s and the modulation frequency f_m. In fact, the side bands are at: n*f_s+f_m, or at: n*f_s-f_m.

Averaging as a lowpass filter

The goal of the averaging operation is to eliminate all these high-frequency components so the spectrum of the averaged v(t) becomes lower frequency. Due to the fact that the moving average operation removes the high-frequency content of v(t), this operation can be seen as a lowpass filter.

Averaging operation on an inductor

Let’s consider now the averaging operation applied to an ideal inductor remembering that quantities are not periodic, so it cannot be invoked the steady state principle. The inductor remains an inductor, but after the moving average operation, only the averaged quantities are considered (averaged current and averaged voltage). After the average operation, the fundamental formula of the inductor becomes:

After the averaging operation, the formula is the same, but instead of instantaneous quantities, averaged quantities are considered.

Averaging operation on a capacitor

The same results can be obtained on an ideal capacitor: The capacitor remains an inductor, but after the moving average operation, only the averaged quantities are considered (averaged current and averaged voltage). After the average operation, the fundamental formula of the capacitor becomes:

After the averaging operation, the formula is the same, but instead of instantaneous quantities, averaged quantities are considered.

Steady state and transformation

The resolution of these two models in a steady state case transforms the capacitor into an open circuit and the inductor into a shortcut because the average current and the average voltage are zero.

Periodic steady-state and DC steady-state

After the operation of averaging, there is the operation of linearization. The situation now considers only the averaged v(t) that can be seen as a superposition of a constant DC value and a small signal AC component.

Linearization and Taylor expansion

The idea behind the operation of linearization is to consider the AC component of the averaged v(t) small such that the averaged v(t) can be expanded in a Taylor expansion.

Averaging operation on a buck converter

Let’s apply now the averaging operation on a buck converter. In this case, the input voltage must be considered as a function of time because it must be assumed that it can change in time. About the capacitor and the inductor, the voltages and the currents just become the averaged currents and voltages. The fundamental difference now is on the quantities related to the switches.

The circuit is as follows:

The output voltage of the switches is vx(t) and is defined as follows:

The average value of vx(t) is:

In general, if there are no small signal components superimposed, v_g(t) is equal to its average value, and the average vx(t) is equal to:

The approximation used on v_g(t) corresponds to the Small Ripple Approximation (SRA). In particular, the high-frequency content is negligible so v_g(t) is assumed as a base band signal (the important harmonics are concentrated at very low frequencies). In particular, in the case of steady state, the averaged value of v_g(t) corresponds to V_g.

The average value of c(t) corresponds to the time-varying duty cycle, indicated with d(t):

So the averaged vx(t) is:

This relation is a non-linear relation because the two signals are multiplied.

Current of the primary switch

Let’s consider now an expression for the current of the primary switch, i_s(t):

Using this expression, the moving average of i_s(t) becomes:

In steady state, the current is:

In general, if i_L(t) can be written as an averaged quantity superimposed by a linear ripple (i_L(t) is the superposition of a constant value and a small ripple like a triangle waveform with a small value), then the averaged value of i_s(t) becomes:

In general, the average value of the product of two signals is not equal to the product of the averaged value of the two signals. In the CCM analysis, sometimes happens that the average value of the product of two signals is equal to the product of the averaged value of the two signals, but only due to the particular shape of the waveforms. For example, this is not the case in DCM analysis in which the averaged value of Is is different from the averaged value of IL.

Using the obtained expressions of the averaged i_s(t) and vx(t), the circuit becomes:

In this model, the current of the primary switch and the voltage at the output of the two switches are controlled, modeled as controlled generators. Putting together these two controlled generators, they can be seen as an ideal transformer that is a mathematical definition of an electrical device defined by these equations:

The two generators can be seen as a transformer because the averaged voltage of vx(t) is related to the averaged voltage of v_g(t) (the voltage at the secondary side is related to the voltage at the primary side) and the averaged current of i_s(t) is related to the averaged current of i_L(t) (the current at the primary side is related to the current at the secondary side).

A fundamental property of the transformer is that the instantaneous power can pass through the transformer ideally without losses. In fact:

Substituting the transformer in the circuit of the CCM buck converter, the model becomes:

The circuit is now time-invariant because it doesn’t change its topology but is still non-linear because there are quantities that depend on time. For example, vx(t) depends on d(t) and v_g(t) (in average), so the function is not linear.

Averaged behavior of the switching cell

The averaged behavior of the CCM switching cell is that of an ideal, non-linear transformer with a transformation ratio 1:d(t). This is a general result. In a very general way, it can be assumed that the averaged switching cell is a transformer with a transform ratio 1:d(t).

If the circuit is solved at DC, all quantities become constants, the duty cycle becomes the constant D, the inductor becomes a shortcut and the capacitor becomes an open circuit. The DC model of the averaged buck-converter becomes:

The average circuit at DC corresponds to the steady-state solution; in fact, applying average operation in steady-state conditions (which means “periodicity”), transforms the quantities in the circuit into DC quantities because the average of steady-state quantities are constant quantities. In other words, the averaged model of the converter must contain the results of the steady-state analysis.

The model of the buck converter with a transformer is useful also for an analysis that considers some non-idealities. For example, assuming a non-ideal inductor (the inductor is composed of the series of an ideal inductor and a resistor) and a DC analysis in steady-state, the voltage conversion ratio corresponds to:

Let’s consider now the operation of linearization. After the operation of averaging, the considered model of the buck converter becomes:

In order to linearize the system, it must be linearized around an operating point which is the DC operating point of the converter. So the inputs of the converter must be considered as superimposition of two quantities that are the DC operating point and the small perturbation. The inputs of the buck converter are two, and they are the duty cycle and the input voltage:

The operating point of the buck converter corresponds to: