Controlled Rectifiers Cheatsheet

Cheatsheet Content

Single-Phase Half-Wave Controlled Rectifier with R-L Load Operation: During the positive half cycle, the SCR (Thyristor T1) is triggered at firing angle $\alpha$. It conducts, and voltage is applied across the R-L load. The inductor stores energy during conduction. In the negative half cycle, the inductor's stored energy can maintain conduction for a period, forward biasing the SCR. Conduction is discontinuous if the current falls to zero before the next positive half cycle. (Refer to "Single Phase Half Wave Controlled Rectifier with 'RL' load:" on page 5, and "Figure above shows the single phase half wave rectifier with RL Load." on page 6 of the notes). Circuit Diagram: (Refer to "Figure: 2.4 Single phase half wave rectifier with RL load with waveforms" on page 5 of the notes). Waveforms: (Refer to "Figure: 2.4 Single phase half wave rectifier with RL load with waveforms" on page 5 of the notes). Operation with Freewheeling Diode (RLE Load) Purpose: A freewheeling diode (FD) provides an alternative path for the inductive load current when the SCR turns off, preventing a sudden drop in current and reducing the negative voltage across the load. This helps in maintaining continuous conduction for longer periods and improving the power factor. (Refer to "Single phase half controlled converter with RLE load" on page 7 and "Single phase half controlled converter with RLE load and freewheeling diode" on page 8 of the notes). Circuit Diagram: (Refer to "Figure: 2.5 single phase half controlled converter with RLE load" on page 7 and "Figure: 2.6 single phase half controlled converter with RLE load and freewheeling diode" on page 8 of the notes). Waveforms: (Refer to "Figure: 2.5 single phase half controlled converter with RLE load" on page 7 and "Figure: 2.6 single phase half controlled converter with RLE load and freewheeling diode" on page 8 of the notes). Three-Phase Half-Wave Rectifier Schematic: A three-phase half-wave rectifier typically uses three thyristors (or diodes for uncontrolled) and a three-phase AC supply. Each thyristor is connected to one phase and shares a common neutral. The load is connected between the common cathode point and the neutral. (Refer to "Operation of three phase half wave rectifier with R and RL loads" and "Figure: 2.16 circuit diagram three phase half wave rectifier" on page 22 of the notes). Operation: Each thyristor conducts for $120^\circ$ of the cycle. The output voltage is formed by the segments of the phase voltages when the corresponding thyristor is conducting. The firing angle $\alpha$ controls the start of conduction for each thyristor, thereby regulating the output DC voltage. (Refer to "The 3-phase half wave converter combines three single phase half wave controlled rectifiers..." on page 23 of the notes). Three-Phase Half-Wave Rectifier with Resistive Load (R) Average Output Voltage ($V_{o,avg}$): $$ V_{o,avg} = \frac{3V_m}{2\pi} (1 + \cos\alpha) $$ (Refer to "The average output voltage $V_{avg}$ = ..." on page 26 of the notes). RMS Output Voltage ($V_{o,rms}$): $$ V_{o,rms} = \sqrt{\frac{1}{2\pi} \left[ \int_{\alpha}^{\pi}\left(V_m \sin(\omega t)\right)^2 d(\omega t) + \int_{\pi}^{2\pi+\alpha}\left(V_m \sin(\omega t)\right)^2 d(\omega t) \right]} $$ For resistive load, the output voltage waveform will follow the input voltage during conduction. The formula is complex and depends on the conduction angle. For continuous conduction, refer to the formula on page 26. Form Factor (FF): Ratio of RMS output voltage to average output voltage. $$ FF = \frac{V_{o,rms}}{V_{o,avg}} $$ (Refer to "Form factor: Form factor is defined as the ratio of RMS voltage to the average DC voltage" on page 16 of the notes). Voltage Ripple Factor ($R_f$): $$ R_f = \sqrt{FF^2 - 1} $$ (Refer to "Ripple factor (Rf)" on page 16 of the notes). Transformer Utilization Factor (TUF): Ratio of output DC power to the VA rating of the transformer. $$ TUF = \frac{P_{dc}}{VA_{rating}} $$ (Refer to "Transformer Utilization Factor (TUF):" on page 16 of the notes). Peak Inverse Voltage (PIV): For a three-phase half-wave rectifier, PIV is typically $\sqrt{3}V_m$. (Notes primarily discuss single-phase PIV on page 16, but in 3-phase, it's line-to-line peak). Single-Phase Full-Wave Bridge Converter with R-L Load Operation: Consists of four SCRs in a bridge configuration. During the positive half cycle, T1 and T2 are triggered. During the negative half cycle, T3 and T4 are triggered. The firing angle $\alpha$ controls the output voltage. The inductive load current can be continuous or discontinuous depending on $\alpha$ and load parameters. (Refer to "Single phase fully controlled converters with RLE load" and "The circuit diagram of a full wave bridge rectifier..." on page 13 of the notes). Circuit Diagram: (Refer to "Figure: 2.11 single phase full converter circuit with RLE load" on page 13 of the notes). Output Voltage and Current Waveforms: (Refer to "Figure: 2.12 single phase full converter circuit with RLE load input and output waveforms" on page 13 of the notes). Average Output Voltage ($V_{o,avg}$): $$ V_{o,avg} = \frac{2V_m}{\pi} \cos\alpha $$ (Refer to "The average output voltage is $V_o = \frac{2V_m}{\pi} \cos\alpha$" on page 14 of the notes). RMS Output Voltage ($V_{o,rms}$): $$ V_{o,rms} = V_m \sqrt{\frac{1}{\pi} \left[ \frac{\pi - \alpha}{2} + \frac{\sin(2\alpha)}{4} \right]} $$ (Refer to "The rms value of voltage is $V_{or} = V_m \sqrt{\frac{1}{2\pi} \left[ (\pi-\alpha) + \frac{1}{2}\sin(2\alpha) \right]}$" on page 9, which is for half-wave, but for full-wave it's a bit different. The notes show $V_{rms} = V_m \sqrt{\frac{1}{2}\left[1+\frac{\alpha}{\pi} - \frac{\sin(2\alpha)}{2\pi}\right]}$ for half-wave, and for full-wave, it's $V_{rms} = V_m \sqrt{\frac{1}{\pi}\left[\frac{\pi-\alpha}{2}+\frac{\sin(2\alpha)}{4}\right]}$ from external sources for continuous conduction). Single-Phase Mid-Point Type Step-Down Cycloconverter Explanation: A cycloconverter converts AC power at one frequency to AC power at another (lower) frequency without an intermediate DC link. A mid-point type cycloconverter uses two converters (e.g., two full-wave rectifiers) connected back-to-back with a common load. For step-down operation, the output frequency is lower than the input frequency. (Not explicitly detailed in the provided notes, but understanding of cycloconverters and mid-point rectifiers can be inferred). Single-Phase Mid-Point Type Step-Up Cycloconverter Explanation: Similar to the step-down cycloconverter, but designed to produce an output AC frequency higher than the input frequency. This is generally more complex and less common than step-down cycloconverters. (Not explicitly detailed in the provided notes). Three-Phase Full-Wave Converter (Fully Controlled Bridge) Operation: Uses six thyristors in a bridge configuration. Each pair of thyristors conducts for $120^\circ$. The output voltage is controlled by the firing angle $\alpha$. It can operate in both rectifier and inverter modes. (Refer to "Operation of three phase fully controlled rectifier with R and RL loads" and "The three phase full converter is extensively used..." on page 28 of the notes). Circuit Diagram: (Refer to "Figure: 2.20 circuit diagram three phase fully controlled rectifier with R and RL load" on page 28 of the notes). Waveforms: (Refer to "Figure: 2.21 Input and output waveforms of three phase fully controlled rectifier" on page 30 of the notes). Three-Phase Full-Wave Rectifier with Resistive Load (R) Average Output Voltage ($V_{o,avg}$): $$ V_{o,avg} = \frac{3\sqrt{3}V_m}{\pi} \cos\alpha $$ (Refer to "The average output voltage is calculated as $V_{avg} = \frac{3\sqrt{3}V_m}{\pi} \cos\alpha$" on page 29 of the notes). RMS Output Voltage ($V_{o,rms}$): $$ V_{o,rms} = \sqrt{\frac{3V_m^2}{2\pi} \left[ \frac{\pi}{3} + \frac{\sin(2\alpha)}{2} \right]} $$ (Refer to "The RMS value of the output voltage is found from $V_{orms} = \sqrt{\frac{3V_m^2}{4\pi} (\frac{2\pi}{3} + \sin(2\alpha))}$" on page 30 of the notes, simplified for resistive load). Form Factor (FF): Ratio of RMS output voltage to average output voltage. $$ FF = \frac{V_{o,rms}}{V_{o,avg}} $$ (Refer to "Form factor: Form factor is defined as the ratio of RMS voltage to the average DC voltage" on page 16 of the notes). Voltage Ripple Factor ($R_f$): $$ R_f = \sqrt{FF^2 - 1} $$ (Refer to "Ripple factor (Rf)" on page 16 of the notes). Transformer Utilization Factor (TUF): Ratio of output DC power to the VA rating of the transformer. $$ TUF = \frac{P_{dc}}{VA_{rating}} $$ (Refer to "Transformer Utilization Factor (TUF):" on page 16 of the notes). Peak Inverse Voltage (PIV): For a three-phase full-wave bridge, PIV is $V_{LL(peak)}$, which is $\sqrt{3}V_m$. (This is a standard value, not explicitly stated for 3-phase full-wave in these notes but derivable). Three-Phase Full-Wave Rectifier with R-L Load Operation: Similar to the resistive load case, but the inductive load influences the current waveform, potentially leading to continuous conduction. The output voltage remains controlled by $\alpha$. (Refer to "Operation of three phase fully controlled rectifier with RLE loads" on page 31 of the notes). Average Output Voltage ($V_{o,avg}$): $$ V_{o,avg} = \frac{3\sqrt{3}V_m}{\pi} \cos\alpha $$ (Same as resistive load for continuous conduction). RMS Output Voltage ($V_{o,rms}$): For R-L load, the RMS calculations become more involved due to current ripple and conduction modes. The notes provide a general form on page 30. Form Factor (FF), Voltage Ripple Factor ($R_f$), Transformer Utilization Factor (TUF), PIV: The definitions remain the same as for resistive loads. The values will change due to the presence of inductance affecting the voltage and current waveforms. Single-Phase Half-Wave Diode Rectifier with Inductive Load (L) Output Voltage and Current Waveforms: In a half-wave diode rectifier with a purely inductive load, the diode conducts when the input voltage is positive. However, the current through the inductor lags the voltage. When the input voltage becomes negative, the inductor's stored energy tries to maintain the current flow, causing the diode to remain forward biased until the current decays to zero. The voltage across the load during conduction is the input voltage. Once the current becomes zero, the diode blocks, and the output voltage is zero until the next positive half cycle. (Refer to "Single Phase Half Wave Controlled Rectifier with 'RL' load:" on page 5, which discusses R-L, for pure L, the current lags by 90 degrees). Single-Phase Semiconductor with RLE Load Explanation: This refers to a single-phase controlled rectifier (using SCRs) with a load containing resistance (R), inductance (L), and a back EMF (E), such as a DC motor. The SCR is triggered at angle $\alpha$. Conduction occurs when the supply voltage is greater than the back EMF. The inductor smooths the current. A freewheeling diode is often used to provide a path for inductive current when the SCR is off, preventing negative voltage spikes and improving performance. (Refer to "Single phase half controlled converter with RLE load" on page 7 and "Operation of Phase Controlled Rectifier" on page 2).