## 3 Single-Phase AC Voltage Controllers
### Phase-Angle Control
#### 3.1.1 Voltage Regulation with Unipolar Voltage Control Scheme  
It is employed in a low power rating load as it contains a DC component. In this output voltage control approach, only one-half cycle of the output voltage is controlled. The other half cycle remains uncontrolled, resulting in the output voltage asymmetric along the vertical (y) axis that causes the generation of the DC component and even harmonics. The output has low-frequency harmonics as the switching devices are operated at low line frequency. Figure 2 depicts the output voltage with respect to the input voltage, where '\(\alpha\)' is the firing delay that controls the output voltage only during the positive half cycle of the input voltage. Here '\(v_{s}\)' and '\(v_{o}\)' are the instantaneous input and output voltage respectively.  
With the assumption of sinusoidal input voltage, that is to say  
\[v_{s}=V_{m}\sin(\omega t) \tag{1}\]  
As the instantaneous output voltage waveform is asymmetric along the \(y\)-axis, the average value of the waveform is non-zero. This average or DC value is computed in the form of harmonic coefficient \(a_{0}\) as  
\[a_{0}\ =\ \frac{-V_{m}}{2\pi}(1-\cos(\alpha)) \tag{2}\]  
This value is varied from '0' to \(-V_{m}/\pi\) once the firing delay is regulated from '0' to '\(\pi\)'. The presence of the asymmetry of the output waveform also results in all even and odd harmonics that can be viewed from the harmonics coefficients \(a_{n}\) and \(b_{n}\) respectively, calculated by  
\[a_{n}[n\
eq 1]\ =\ \left\{\begin{array}{l}\frac{V_{m}}{2\pi(n-1)}[1-\cos(n \alpha-\alpha)]-\frac{V_{m}}{2\pi(n+1)}[1-\cos(n\alpha+\alpha)]\\ \text{for }n\ =\ 2,3,4,5,\dots\end{array}\right. \tag{3}\]  
\[b_{n}[n\
eq 1]\ =\ \left\{\begin{array}{l}\frac{V_{m}}{2\pi(n+1)}\sin(n \alpha+\alpha)-\frac{V_{m}}{2\pi(n-1)}\sin(n\alpha-\alpha)\\ \text{for }n\ =\ 2,3,4,5,\dots\end{array}\right. \tag{4}\]  
These harmonics coefficients are calculated by selecting the period of \(2\pi\) as the period of the output and input the waves are the same and they are equal to \(2\pi\). As  
Figure 2: Output RMS voltage control through unipolar phase-angle control.  
the coefficients \(a_{n}\) and \(b_{n}\) become undefined for \(n=1\), they are separately computed in Equations (5) and (6).  
\[a_{n}[n=1]\ =\ \frac{V_{m}}{4\pi}[-1+\cos(2\alpha)] \tag{5}\]  
\[b_{n}[n=1]\ =\ \frac{V_{m}}{2\pi}\bigg{[}2\pi-\alpha+\frac{1}{2}\sin(2\alpha) \bigg{]} \tag{6}\]  
The closed-form of the instantaneous output voltage in terms of various harmonics contents is realized in Equation (7). This depicts that output has all harmonics as the waveform is asymmetric.  
\[v_{o}(\omega t) =\ \frac{-V_{m}}{2\pi}[1-\cos(\alpha)]+\ \frac{V_{m}}{2\pi}(2\pi- \alpha)\sin(\omega t)\ -\frac{V_{m}}{4\pi}[\cos(\omega t)-\cos(\omega t-2\alpha)]\] \[+\frac{V_{m}}{2\pi(n-1)}[\cos(n\omega t)-\cos(n\omega t-n\alpha+ \alpha)]\ -\frac{V_{m}}{2\pi(n+1)}[\cos(n\omega t)-\cos(n\omega t-n\alpha- \alpha)]\] (7) for \(n\ =\ 3,5,7,\dots\)  
The DC (average) component in the output voltage generates a DC component in the input source current that may lead to the core saturation of the transformer connected at the input side. This problem is tackled with a bipolar voltage control scheme.