Unwind di Power Electronics - 3 di 5

c(w)Frequency Response of the Naturally Sampled PWML. Corradini, a.a. 2020−2021 6

The model that realises the task of calculating the relation between the variation of the modulating signal and the output signal is the following:

Conceptually, the idea is the following: the portion of the spectrum of the output around wm is isolated using an ideal band-pass filter to generate a phasor with frequency wm and at the inputs it’s done the same thing, so the portion of the spectrum of the input around wm is isolated using an ideal band-pass filter to generate a phasor with frequency wm. At least it’s made the ratio between the phasor related to the output and the phasor related to the input but this ratio is done in the limit that the input perturbation goes to 0:

Gianni CwmUwm 0UwmNaturally-Sampled Trailing-Edge ModulationL. Corradini, a.a. 2020−2021 7

If Gpwm is properly calculated, it leads a simple result:

JWGrown µSo the small signal model of the pulse width

modulator is a constant. This type of modulation is called trailing-edge modulation because it’s the trailing edge that is modulated while the leading edge no.

There are other type of modulators that estates the same result of Gpwm. Naturally-Sampled Leading-Edge Modulation L. Corradini, a.a. 2020−2021 8 For example, another type of modulator that presents the same Gpwm is the naturally sampled leading-edge modulator in which it’s the leading edge that is modulated while the trailing edge no: It’s clear that the leading edge is modulated while the trailing edge no.

Naturally-Sampled Symmetrical Modulation L. Corradini, a.a. 2020−2021 9 There is a third tipology of modulation that is the triangular modulation, also calle symmetrical modulation: In this case, both leading and trailing edge are modulated because if the value of the modulating signal is changed, then both edges changes their position. The transfer function is still the same: JWGrown µ Naturally

Sampled PWM: Summary

Naturally sampled modulations all have the same frequency response:

• No small-signal phase shift is introduced by the modulator. This is a consequence of c(t) being
• modulated by the instantaneous value of u(t).

Previous considerations justify the practice, common in analog control design, to model the

• PWM modulator as a simple gain:L. Corradini, a.a. 2020-2021 10

So basically the general results of the trailing edge, leading edge and symmetrical modulators is that they

presents the same Gpwm and it's a constant. Due to the fact that Gpwm is a constant, there is no small-signal

phase shift. In other words, the modulator doesn't introduce phase shift in the loop. This property comes from

the fact that the modulated edge of the PWM is generated by the sampled value of the modulating signal at the

same instant. Uniformly Sampled PWM

Consider now a PWM modulator in which the modulating signal u[k] is discrete-time, e.g.

because it is

produced by a digital compensator.Suppose also that sampling and switching frequencies coincide, and that u[k] is updated at the• beginning of each switching interval, as in the Figure above.c(t) can be thought of as being obtained from the comparison between the carrier r(t) and a• fictitious signal u (t), derived from u[k] via zero-order hold (ZOH) interpolation.hL. Corradini, a.a. 2020−2021 11

Let’s consider now uniformly sampled PWM which is a pulse width modulator in which the modulating signal isa discrete-time signal produced by a discrete-time compensator. The modulating signal is updated once atbeginning of each switching period and a zero-order hold block (ZOH) takes this value constant over an entireswitching period. So the ZOH block generates a continuous-time signal called uh(t) that is confronted with thecarrier signal. In general the ZOH block generates a piece wise constant signal using a discrete-time signal. Thisprocess is called ZOH interpolation.

Uniformly Sampled PWM

The duty cycle is determined by the modulating signal at the beginning of the switching interval.

• A delay now exists between the update of u[k] and that of c(t). Such delay originates from the
• discrete-time nature of u[k].

L. Corradini, a.a. 2020-2021 12

Note that the continuous-time signal uh(t) is simply a replica of u[k] so there is a delay between the update of

u[k] and the update of c(t) that comes from the discrete-time nature of u[k]. From another point of view, there is

a delay between the instant in which u[k] is calculated and the instant in which the modulation occurs. This

delay is very important because modify the frequency response.

The duty cycle is:

Where Nr is the peak of the carrier signal.

Frequency Response of the Uniformly Sampled PWM

L. Corradini, a.a. 2020-2021 13

To evalute the frequency of a uniformly sampled PWM it's used the same procedure as before, so at the output

and at the input the spectrums of the output and