2. AC switch controllers

#### 2.1. Single-phase AC switch controllers

In the case of these static converters, the control angle α of the thyristor is defined as the angle determined from the time of zero crossing of the voltage up to input in conduction of the thyristor [2, 5].

The AC switch controllers are AC to AC static converters. The converter output voltage is chopped so that the RMS value of AC output voltage is modified with change of the switching period of the power semiconductors.

Figure 1 shows a single-phase AC switch controller scheme and the voltage waveforms. The switching angle α can be modified between 0 and π. If the switching angle is α ¼ 0, the output voltage is us ¼ usmax [3].

The instantaneous value of output current is given by Eqs. (1–3) [2, 5]:

• For resistive load:

$$\mathbf{i} = \begin{cases} \mathbf{U}\_{\text{m}} \sin \omega \cdot \mathbf{t} & \text{for} \\ \mathbf{0} & \text{for} \end{cases} \quad \omega \cdot \mathbf{t} \in [\alpha, \pi] \cup [\pi + \alpha, 2\pi] \tag{1}$$

$$\boldsymbol{\omega} \cdot \mathbf{t} \in [0, \alpha] \cup [\pi, \pi + \alpha]$$

• For inductive load:

$$\dot{a} = \begin{cases} \mathcal{U}\_m \Big[ \sin \left( \omega \cdot t - \frac{\pi}{2} \right) - \sin \left( \alpha - \frac{\pi}{2} \right) \Big] \\ \quad \text{for} \quad \omega \cdot t \in [a, 2\pi - \alpha] \cup [\pi + a, 3\pi - \alpha] \\ \quad \text{0} \quad \text{for} \quad \omega \cdot t \in [0, \alpha] \cup [2\pi - \alpha, 2\pi] \end{cases} \tag{2}$$

• For resistive-inductive load:

$$\dot{\mathbf{u}} = \frac{\mathbf{U\_m}}{\sqrt{R^2 - (\boldsymbol{\omega} \cdot \mathbf{L})^2}} \left[ \sin \left( \boldsymbol{\omega} \cdot \mathbf{t} - \boldsymbol{\varphi} \right) - e^{-\frac{\mathbf{E}}{\boldsymbol{\omega} \cdot \mathbf{L}} (\boldsymbol{\omega} \cdot \mathbf{t} - \boldsymbol{\alpha})} \sin \left( \boldsymbol{\alpha} - \boldsymbol{\varphi} \right) \right] \tag{3}$$

The output current average value can be determined with Eqs. (4) and (5):

• For resistive load:

$$\mathbf{I}\_{\text{med}} = \frac{\mathbf{I}\_{\text{m}}}{2\pi} (1 + |\cos \alpha|) \ \alpha \in [0, \pi] \tag{4}$$

• For inductive load:

$$\mathbf{I}\_{\text{med}} = \frac{2\mathbf{I}\_{\text{m}}}{2\pi} [\sin \alpha + (\pi - \alpha) \cos \alpha] \cdot \alpha \in \left[\frac{\pi}{2}, \pi\right] \tag{5}$$

These values are useful for dimensioning of power semiconductor devices of the switch controllers.

Figure 1. Single-phase AC switch controller electric scheme and the voltage waveforms.

The RMS output current is given by Eqs. (6) and (7) [2]:

• For resistive load:

2. AC switch controllers

120 Science Education - Research and New Technologies

thyristor [2, 5].

2.1. Single-phase AC switch controllers

period of the power semiconductors.

i ¼

i ¼

8 >><

>>:

<sup>i</sup> <sup>¼</sup> Um ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>R</sup><sup>2</sup> � ð Þ <sup>ω</sup> � <sup>L</sup> <sup>2</sup>

Um

(

voltage is us ¼ usmax [3].

• For resistive load:

• For inductive load:

• For resistive load:

• For inductive load:

controllers.

• For resistive-inductive load:

In the case of these static converters, the control angle α of the thyristor is defined as the angle determined from the time of zero crossing of the voltage up to input in conduction of the

The AC switch controllers are AC to AC static converters. The converter output voltage is chopped so that the RMS value of AC output voltage is modified with change of the switching

Figure 1 shows a single-phase AC switch controller scheme and the voltage waveforms. The switching angle α can be modified between 0 and π. If the switching angle is α ¼ 0, the output

> <sup>R</sup> sin <sup>ω</sup> � t for <sup>ω</sup> � <sup>t</sup><sup>∈</sup> ½ � <sup>α</sup>,<sup>π</sup> <sup>∪</sup>½ � <sup>π</sup> <sup>þ</sup> <sup>α</sup>, <sup>2</sup><sup>π</sup> 0 for ω � t ∈½ � 0, α ∪½ � π,π þ α

> > � sin <sup>α</sup> � <sup>π</sup>

� <sup>R</sup>

h i

2

<sup>ω</sup>�<sup>L</sup>ð Þ <sup>ω</sup> � <sup>t</sup> � <sup>α</sup> sin ð Þ <sup>α</sup> � <sup>ϕ</sup>

<sup>2</sup><sup>π</sup> <sup>ð</sup><sup>1</sup> <sup>þ</sup> cos <sup>α</sup><sup>Þ</sup> <sup>α</sup>∈½ � <sup>0</sup>, <sup>π</sup> <sup>ð</sup>4<sup>Þ</sup>

π <sup>2</sup> ,<sup>π</sup> h i ð1Þ

ð2Þ

ð3Þ

ð5Þ

The instantaneous value of output current is given by Eqs. (1–3) [2, 5]:

Um sin <sup>ω</sup> � <sup>t</sup> � <sup>π</sup>

<sup>q</sup> sin ð Þ� <sup>ω</sup> � <sup>t</sup> � <sup>ϕ</sup> <sup>e</sup>

The output current average value can be determined with Eqs. (4) and (5):

Imed <sup>¼</sup> Im

Imed <sup>¼</sup> 2Im

2 � �

h i � �

<sup>2</sup><sup>π</sup> <sup>½</sup> sin <sup>α</sup> þ ð<sup>π</sup> � <sup>α</sup><sup>Þ</sup> cos <sup>α</sup>� <sup>α</sup><sup>∈</sup>

These values are useful for dimensioning of power semiconductor devices of the switch

f or ω � t ∈½α, 2π � α�∪½π þ α, 3π � α� 0 f or ω � t∈ ½0, α�∪½2π � α, 2π�

$$\mathbf{I}\_{\text{ef}} = \mathbf{I}\_{\text{m}} \sqrt{\frac{1}{\pi} \left[ \frac{1}{2} (\pi - \alpha) + \frac{1}{4} \sin 2\alpha \right]} \tag{6}$$

• For inductive load:

$$\mathbf{I}\_{\rm ef} = \mathbf{I}\_{\rm m} \sqrt{\frac{8}{\pi} \left[ (\pi - \alpha) \left( \cos^2 \alpha + \frac{1}{2} \right) + \frac{3}{4} \sin^2 \alpha \right]} \tag{7}$$

Figure 2 presents the output current waveforms of the AC switch controller in the case of using different kinds of loads, and for different values of the switching angle α [5, 8].

Figure 2. Current waveforms for a resistive load (a), inductive load (b), and a resistive-inductive one (c).

The AC switch controllers can be designed using only one thyristor. Figure 3 presents the basic schemes and the output waveforms of these AC switch controller types.

#### 2.2. Three-phase AC switch controllers

The three-phase AC switch controllers are designed using three single-phase AC switch controllers (kR, kS, kT), one for each phase (Figure 4). The switching angle for each of the single-phase AC switch controllers is the same, but they must be phase angle with 2π/3 [3].

By changing the control angle, α, of the thyristors from each phase, changes the power absorbed by load between the maximum value and zero. Order the thyristors is performed using of the control device grid (DC), which must ensure a phase shift of the control pulses of 2π/3 between the phases [2, 5].

Variation the ignition angle of the voltages and of the currents depends on the load nature.

The voltage waveforms are determined from the vector diagram shown in Figure 5.

So, the voltage UKR on a single-phase of R phase is zero on the period while one of two thyristors leads. The length of time while the thyristors are blocked, the load neutral point moves from 0 to 0' and the voltage thyristor will be 3/2 UR corresponding phasor U0'R. The voltage values are as follows:

• single-phase AC switch controller voltage is given by Eq. (8):

$$\mathbf{u}\_{\mathbf{k}\mathbf{R}} = \begin{cases} \frac{3}{2} \mathbf{u}\_{\mathbf{R}} & \text{if} \quad \mathbf{k}\_{\mathbf{R}} - \text{switch} - \text{off} \\\\ 0 & \text{if} \quad \mathbf{k}\_{\mathbf{R}} - \text{switch} - \text{on} \end{cases} \tag{8}$$

output line-to-line voltage is fit Eq. (9):

The AC switch controllers can be designed using only one thyristor. Figure 3 presents the basic

The three-phase AC switch controllers are designed using three single-phase AC switch controllers (kR, kS, kT), one for each phase (Figure 4). The switching angle for each of the single-phase AC switch controllers is the same, but they must be phase angle with 2π/3 [3]. By changing the control angle, α, of the thyristors from each phase, changes the power absorbed by load between the maximum value and zero. Order the thyristors is performed using of the control device grid (DC), which must ensure a phase shift of the control pulses of

Variation the ignition angle of the voltages and of the currents depends on the load nature.

So, the voltage UKR on a single-phase of R phase is zero on the period while one of two thyristors leads. The length of time while the thyristors are blocked, the load neutral point moves from 0 to 0' and the voltage thyristor will be 3/2 UR corresponding phasor U0'R. The

if kR � switch � off

ð8Þ

if kR � switch � on

The voltage waveforms are determined from the vector diagram shown in Figure 5.

• single-phase AC switch controller voltage is given by Eq. (8):

3 2 uR

8 ><

>:

0

ukR ¼

schemes and the output waveforms of these AC switch controller types.

Figure 2. Current waveforms for a resistive load (a), inductive load (b), and a resistive-inductive one (c).

2.2. Three-phase AC switch controllers

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2π/3 between the phases [2, 5].

voltage values are as follows:

$$\mathbf{u}'\_{\rm RS} = \begin{cases} \mathbf{u}\_{\rm RS} & \text{if} \\ -\frac{1}{2}\mathbf{u}\_{\rm SF} & \text{if} \\ -\frac{1}{2}\mathbf{u}\_{\rm TR} & \text{if} \\ -\frac{1}{2}\mathbf{u}\_{\rm TR} & \text{if} \end{cases} \quad \mathbf{k}\_{\rm R} - \text{switch} - \text{off} \tag{9}$$

output phase voltage (depending of the single phase switch converters which are in conduction) is given by Eq. (10):

$$\mathbf{u}'\_{\rm R} = \begin{cases} 0 & \text{if } & \mathbf{k\_{R}} - \text{switch} - \text{off} \\ \mathbf{u}\_{\rm R} & \text{if } & \mathbf{k\_{R}}, \mathbf{k\_{S}}, \mathbf{k\_{T}} - \text{switch} - \text{on} \\ \frac{1}{2}\mathbf{u}\_{\rm RS} & \text{if } & \mathbf{k\_{T}} - \text{switch} - \text{off} \\ -\frac{1}{2}\mathbf{u}\_{\rm TR} & \text{if } & \mathbf{k\_{S}} - \text{switch} - \text{off} \end{cases} \tag{10}$$

Figure 6 presents the voltage waveforms for a resistive load of an AC switch controller.

Figure 3. Basic schemes and output waveforms of different AC switch controllers: a) with one thyristor, b) with one thyristor and a diode, c) with one thyristor in a diode bridge.

Figure 4. Three-phase AC switch controllers electric scheme.

Figure 5. Vector diagram.

Because the load is resistive, the current waveform through the load is the same as the voltage phase raised on another scale.

In the case of an inductive load, the voltage waveforms are obtained similarly like in case of resistive load, but the thyristor ignition angle (α) is between φ and π, where ϕ is the delay angle between voltage and current due to the load. The current and voltage waveforms for an inductive are present in Figure 7 [2, 3, 5].

If the three-phase AC switch controller supplies AC motors, the supply system must have the possibility to change the phase sequence to obtain a reversible drive system.

Figure 8 presents two schemes of AC switch controllers for reversible AC drive systems.

Figure 6. Voltage waveforms for a resistive load.

Because the load is resistive, the current waveform through the load is the same as the voltage

In the case of an inductive load, the voltage waveforms are obtained similarly like in case of resistive load, but the thyristor ignition angle (α) is between φ and π, where ϕ is the delay angle between voltage and current due to the load. The current and voltage waveforms for an

If the three-phase AC switch controller supplies AC motors, the supply system must have the

Figure 8 presents two schemes of AC switch controllers for reversible AC drive systems.

possibility to change the phase sequence to obtain a reversible drive system.

phase raised on another scale.

Figure 5. Vector diagram.

inductive are present in Figure 7 [2, 3, 5].

Figure 4. Three-phase AC switch controllers electric scheme.

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Figure 7. Current and voltage waveforms for an inductive load.

Figure 8. Reversible AC switch controllers: a) Symmetrical scheme, b) Non-symmetrical scheme.
