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The digital PWM models in the previous sections are derived for the output voltage of single switch. As power inverters are usually implemented by using H bridges, describing the output voltage of H bridges as a function of the mod- ulation signal is required. The transfer function of PWM model varies when different carriers and modulation techniques are used.

The typical circuit diagram of an H bridge is shown in Fig. 3.9. The output of the H bridge is the filter input voltagevin. The switching output is defined as

y=vin/Vdc. The switching output varies significantly when different modulation

strategies are used. In this section we only provide examples with end-of-on-time carriers and symmetric-on-time carriers. Both bipolar switched and unipolar switched PWMs are studied.

3.3.1

Bipolar switched H bridges

If there are two voltage levels produced on the switch voltage vin, i.e., Vdc and

−Vdc, the H bridge is bipolar switched. In order to provide the model for a

single-update-mode bipolar switched H bridge, we assume that the duty-ratio is updated at each sampling instant. Therefore, the DSP delay is one sampling cycle. When the sampling frequency is equal to the switching frequency, the key waveforms of bipolar switched H bridge are shown in Fig. 3.10.

S1 S3 ON OFF ON OFF S1 S3 ON OFF ON OFF (a) (b)

Figure 3.10: Key waveforms of single-update-mode uniformly-sampled bipolar switched H bridge. (a) End-of-on-time modulator. (b) Symmetric-on-time mod- ulator.

It can be seen from Fig. 3.10 that the filter input voltage frequency of the bipolar switched H bridge is equivalent to the switching frequency. The duty- ratio can be updated only once when using sawtooth carriers. However, for triangle carriers, the duty-ratio can be updated twice a switching cycle. As the DSP delay fromx∗ tou∗ is Ts, the small-signal transfer function describing ybas

a function ofx_b∗ for end-of-on-time modulator is written as [25]

G∗_{P W M}(s) =Tse−s(1+D)Ts. (3.16)

function G∗_{P W M}(s) can be expressed as [25]

G∗_{P W M}(s) = Ts 2(e

−s(3−D₂)Ts

+e−s(3+D2)Ts). (3.17)

For double-update-mode PWM where the sampling frequency and updating rate is as twice as the switching frequency, the triangle carriers are usually used. To provide the double-update-mode PWM model, the symmetric-on-time modu- lator is used as the example. The key waveforms of double-update-mode bipolar switched H bridge are shown in Fig. 3.11.

S1 S3 ON OFF ON OFF

Figure 3.11: Key waveforms of double-update-mode uniformly-sampled bipolar switched H bridge.

For bipolar switched H bridges, each switching cycle contains two updated samples with two relevant switching actions. If the sample is updated at the upper peak of the carrier, the delay from u∗ to y_bis (1−D)Ts

2 . On the other hand, if the sample is updated at the lower peak of the carrier, the delay from u∗ to y_b

becomes DTs

2 . During each switching cycle, the possibilities of the two situations are equal. As the exact analytical expression of the double-update-mode PWM model is not easy to obtain, the approximation can be applied by averaging the two delay effects. With half switching cycle DSP delay fromx∗ tou∗, the double- update-mode PWM model of the bipolar switched H bridge is given by [25]

G∗_{P W M}(s) = Ts 4(e

3.3.2

Unipolar switched H bridges

If there are three voltage levels produced on the switch voltage vin, i.e., Vdc, 0

and −Vdc, the H bridge is unipolar switched. When the H bridge is unipolar

switched, the key waveforms with single-update are shown in Fig. 3.12. For end- of-on-time modulator, the duty-ratio can be updated twice a switching cycle. The filter input voltage frequency of the unipolar switched H bridge is equivalent to the switching frequency. However, for symmetric-on-time modulator, the duty- ratio can be updated quadruply a switching cycle since the filter input voltage frequency is as twice as the switching frequency. Hence, the modulation method of using unipolar switched H bridge inverter with symmetric triangle carriers is a good way to reduce the electromagnetic interference.

S1 S3 ON OFF ON OFF S1 S3 ON OFF ON OFF (a) (b)

Figure 3.12: Key waveforms of uniformly-sampled single-update-mode unipolar switched H bridge. (a) End-of-on-time modulator. (b) Symmetric-on-time mod- ulator.

Similarly, the small-signal transfer function describing_byas a function of_bx∗for single-update-mode unipolar switched H bridge with end-of-on-time modulator is

written as [25]

G∗_{P W M}(s) = Ts 2 (e

−s(1+D)Ts _+e−s(2−D)Ts_{). (3.19)}

For single-update-mode unipolar switched H bridge with symmetric-on-time mod- ulator, the transfer function is given by [25]

G∗_{P W M}(s) = Ts 4(e −s(2+D₂)Ts +e−s(3−D2)Ts +e−s (3+D)Ts 2 +e−s (4−D)Ts 2 ). (3.20)

The key waveforms of double-update-mode unipolar switched H bridge is shown in Fig. 3.13. S1 S3 ON OFF ON OFF

Figure 3.13: Key waveforms of uniformly-sampled unipolar switched H bridge with double-update-mode.

For unipolar switched H bridges containing two updated samples in each switching cycle, four relevant switching transients are generated (see Fig. 3.13). Two situation are discussed to obtain the PWM model. If the sample is updated at the upper peak of the carrier, the delay terms fromu∗ toy_bare represented by

τd1 = DT₂s and τd3 = (1 −D)Ts

2 . On the other hand, if the sample is updated at the lower peak of the carrier, the delay terms becomes τd2 =

(1−D)Ts

2 and τd4 =

DTs

Therefore, no matter whether the sampling starts at the upper peak or lower peak of the carrier, the delay effect does not change. As a result, the small-signal double-update-mode PWM model for the unipolar switched H bridge is [25]

G∗_{P W M}(s) = Ts 4(e

−s(2−D₂)Ts

+e−s(1+D2)Ts). (3.21)

Comparing double-update-mode PWMs to the uniformly-sampled PWMs, it can be seen that the double-update-mode PWMs result in a minimum delay time. Hence, the double-update-mode is usually the recommended PWM strategy. In practice, more sampling methods rather than uniform-sampling may be used, such as asynchronous sampling, multisampling and hybrid sampling. In those cases, the small-signal PWM model should be modified to accommodate the sampling methods. However, the strategy of developing the transfer functions in this section can be used in other cases.

3.4

Block diagrams of digitally controlled switch-