# Experimental Investigation of Controlled and Uncontrolled Rectifiers for Low-Power Wind Turbines
## 4 Single-Phase Half-Wave Uncontrolled Rectifier  
A single-switch half-uncontrolled wave rectifier is the simplest way to convert an AC into a pulsating DC, as shown in Figure 5. During the positive AC cycle, the diode is forward-biased because the diode's anode has a higher voltage potential than the cathode; thus, the input sinusoidal waveform is allowed to pass with a minor voltage drop due to the diode's forward resistance [11]. However, during the negative AC cycle, the diode is reverse-biased because the diode's anode is at a lower voltage potential than the cathode, thus blocking the flow of current to the output. As a result, the output load will experience a pulsating DC. Figure 4: Three\(-\)phase half-wave rectification. Figure 5: Single-phase half-wave rectification. The output is always fixed for uncontrolled rectifiers for a given AC supply, but the output ripple, average value, efficiency, and AC harmonics are analyzed to benchmark the rectifier circuits. The AC harmonics are superimposed on the output DC and can be measured in terms of the ripple factor [11]. The ripple factor can be related in Equation (1). \[RF_{v}=\frac{V_{ac}}{V_{dc}}=\sqrt{\frac{V_{rms}^{2}-V_{dc}^{2}}{V_{dc}^{2}}}= \sqrt{\frac{V_{rms}^{2}}{V_{dc}^{2}}-1}=\sqrt{Form\ Factor^{2}-1}=\sqrt{\frac{1 }{\eta}-1} \tag{1}\]  
The form factor (FF) is the ratio of the RMS value to the DC average value, and \(\eta\) is the efficiency of the rectifier circuit. The lower ripple factor shows better DC conversion [13]. The average voltage of a single-phase half-wave rectifier can be calculated by integrating the area under the curve of the pulsating output waveform. The average voltage can be expressed as Equation (2). \[V_{avg}=\frac{1}{T}\int\limits_{0}^{T}v(t)dt \tag{2}\]  
where \(T\) is the time period of a single AC input cycle, and \(v(t)\) is the instantaneous value of the voltage. The average voltage can also be calculated by integrating the area under the curve of the current waveform having an instantaneous value of current as \(i(t)\). The average current value is depicted in Equation (3) [13]. \[I_{avg}=\frac{1}{T}\int\limits_{0}^{T}i(t)dt \tag{3}\]  
The RMS voltage and current value can be calculated as in Equations (4) and (5). \[V_{rms}=\sqrt{\frac{1}{T}\int\limits_{0}^{T}v(t)^{2}dt} \tag{4}\]  
\[I_{rms}=\sqrt{\frac{1}{T}\int\limits_{0}^{T}i(t)^{2}dt} \tag{5}\]  
The output voltage and current significantly depend on the type of load. Ideally, for a purely resistive load, the power factor reaches unity, degrading progressively as the load becomes inductive. For any AC input, the ripple factor and the power factor for a resistive load are 1.21 and 0.707, respectively [20]. The high ripples in the output of the rectifier are undesirable. Connecting capacitive filters in parallel with the load is the most straightforward technique to smooth out the pulsating DC output of the single-phase half-wave rectifier. Since the rectifier is reverse-biased for half the period, a large capacitor is required to support the load for at least half the time period. During the positive AC cycle, the current is supplied to the load, and the capacitor is also charged. When the diode is reverse-biased, the capacitor supplies the power to the load. The output voltage during positive and negative AC cycles can be expressed as a piecewise expression considering a purely resistive load with capacitive filtering, as shown in Equation (6). \[V_{o}=\left\{ \begin{array}{ll}V_{m}\sin(\omega t)&(\it{Diode\ Forward\ Biased})\\ \sqrt{2}\ V_{in}\ \sin(\theta)e^{-\frac{\omega t-\theta}{\omega\kappa C}}&(\it{Diode\ Reversed\ Biased})\end{array} \right. \tag{6}\]  
where \(V_{o}\) is the output voltage on the load, \(\omega=2\pi f\) is the angular frequency (f = 60 Hz), Vin is the input AC RMS value, \(R\) is the resistive load, \(C\) is the filter capacitor, and \(\theta\) is the phase angle between the capacitor voltage \(V_{o}\) and the input voltage \(V_{in}\).