Electrical Machines Theory Cheatsheet

Cheatsheet Content

Part 2: Diode Applications - Rectifiers 1. Rectification: Purpose and Necessity Rectification is the process of converting an alternating current (AC) voltage or current, which periodically reverses direction, into a pulsating direct current (DC) voltage or current, which flows in only one direction. This process is fundamental to almost all electronic devices and power supplies. The primary element used for rectification is the diode, leveraging its inherent property of allowing current flow easily in one direction (forward bias) and blocking it in the opposite direction (reverse bias). Rectification is required for several critical reasons: Powering DC Electronic Devices: The vast majority of electronic circuits and devices, from microprocessors in computers to LEDs in mobile phones, operate on direct current (DC) voltage. However, the electrical power delivered by utility companies is universally alternating current (AC). Rectifiers bridge this gap by converting the available AC power into the DC power required by these devices. Battery Charging: Batteries store energy in DC form. To charge a battery from an AC mains supply, rectification is essential to convert the AC voltage into a pulsating DC voltage that can then be used to replenish the battery's charge. Motor Control: While some motors operate on AC, many precision control systems and variable-speed drives utilize DC motors or require DC power for their control circuitry. Rectification provides the necessary DC supply. Signal Demodulation: In communication systems, rectification is used to extract the information-carrying signal (which is typically low-frequency or DC) from a high-frequency modulated AC carrier wave during the demodulation process. Industrial Applications: Many industrial processes, such as electroplating, welding, and some heavy machinery, require large amounts of DC power. Rectifiers are integral components in providing this DC supply. Without rectification, modern electronic devices and systems that rely on DC power could not function directly from the AC mains supply, highlighting its indispensable role in electrical and electronic engineering. 2. Half-Wave Rectifier: Circuit and Waveforms A half-wave rectifier is the simplest type of rectifier circuit, designed to convert AC voltage into pulsating DC voltage by allowing only one half-cycle of the AC input to pass through to the load. It consists of a single diode connected in series with a load resistor ($R_L$), often preceded by a step-down transformer to provide the desired AC input voltage. Circuit Diagram: The circuit typically includes an AC source (often through a transformer secondary winding), followed by a single P-N junction diode, and then the load resistor ($R_L$) connected across which the output voltage ($V_{out}$) is measured. The diode's anode is connected to the AC source, and its cathode is connected to one end of $R_L$, with the other end of $R_L$ connected back to the AC source. Working Principle and Waveforms: Positive Half-Cycle of Input AC Voltage: During the positive half-cycle of the input AC voltage, the anode of the diode becomes positive with respect to its cathode. This forward-biases the diode. If the input voltage exceeds the diode's knee voltage ($V_K$, typically 0.7V for silicon), the diode acts like a closed switch (or a small voltage drop). Current flows through the diode and the load resistor $R_L$. The output voltage across $R_L$ will be approximately equal to the input voltage minus the diode's forward voltage drop ($V_{out} = V_{in} - V_K$). Thus, the positive half-cycle of the input AC voltage appears across the load. Negative Half-Cycle of Input AC Voltage: During the negative half-cycle of the input AC voltage, the anode of the diode becomes negative with respect to its cathode. This reverse-biases the diode. The diode acts like an open switch, blocking the flow of current. Consequently, no current flows through $R_L$, and the output voltage across $R_L$ is approximately zero volts. The resulting output waveform across the load resistor is a series of positive half-cycles, with the negative half-cycles completely blocked. This output is a pulsating DC voltage, as it flows in only one direction but is not constant; it rises from zero to a peak and then falls back to zero. The frequency of the output ripple is the same as the input AC frequency. While simple, the half-wave rectifier is inefficient, as it utilizes only half of the input AC waveform, resulting in low rectification efficiency and a high ripple factor, making it generally unsuitable for applications requiring smooth DC power. 3. Full-Wave Rectifier with Centre-Tapped Transformer A full-wave rectifier with a centre-tapped transformer is a common rectifier configuration that converts both half-cycles of the AC input voltage into pulsating DC, leading to a smoother output and higher efficiency than a half-wave rectifier. This circuit requires a special transformer with a centre-tapped secondary winding and two diodes. Circuit Diagram: The circuit consists of a transformer with a secondary winding that has a tap exactly at its electrical center. This effectively divides the secondary winding into two equal parts. Two diodes are used: the anode of the first diode ($D_1$) is connected to one end of the secondary winding (say, point A), and the anode of the second diode ($D_2$) is connected to the other end (say, point B). The cathodes of both diodes are joined together, and the load resistor ($R_L$) is connected between this common cathode point and the centre tap of the transformer secondary. Working Principle: Positive Half-Cycle of Input AC Voltage: During the positive half-cycle of the input AC voltage, the top end of the secondary winding (point A) becomes positive with respect to the centre tap, while the bottom end (point B) becomes negative with respect to the centre tap. This forward-biases diode $D_1$ (anode positive, cathode common) and reverse-biases diode $D_2$ (anode negative, cathode common). Therefore, $D_1$ conducts, and $D_2$ acts as an open circuit. Current flows from point A, through $D_1$, through $R_L$ (from top to bottom), and back to the centre tap. The output voltage across $R_L$ is approximately the voltage from A to centre tap, minus $V_K$ of $D_1$. Negative Half-Cycle of Input AC Voltage: During the negative half-cycle of the input AC voltage, the polarity reverses. The top end of the secondary winding (point A) becomes negative with respect to the centre tap, and the bottom end (point B) becomes positive with respect to the centre tap. This now reverse-biases diode $D_1$ and forward-biases diode $D_2$. Therefore, $D_2$ conducts, and $D_1$ acts as an open circuit. Current flows from point B, through $D_2$, through $R_L$ (again, from top to bottom, maintaining the same output polarity), and back to the centre tap. The output voltage across $R_L$ is approximately the voltage from B to centre tap, minus $V_K$ of $D_2$. In both half-cycles, current flows through the load resistor in the same direction, resulting in a pulsating DC output. The output waveform consists of a series of positive half-cycles, as both positive and negative input half-cycles are rectified into positive output pulses. The frequency of the output ripple is twice the input AC frequency, which makes it easier to filter into a smoother DC voltage compared to a half-wave rectifier. While efficient, a drawback is the requirement for a centre-tapped transformer, which can be more expensive and bulky. 4. Full-Wave Bridge Rectifier: Circuit and Waveforms The full-wave bridge rectifier is the most popular and efficient rectifier circuit, capable of converting both positive and negative half-cycles of the AC input into a pulsating DC output. It utilizes four diodes arranged in a bridge configuration, eliminating the need for a costly and bulky centre-tapped transformer, making it suitable for a wide range of applications. Circuit Diagram: The circuit consists of four diodes ($D_1, D_2, D_3, D_4$) connected in a bridge configuration. The AC input voltage (from a transformer secondary or directly from the mains) is applied across two opposite corners of the bridge. The DC output is taken across the other two opposite corners, with the load resistor ($R_L$) connected between these points. Typically, two diodes ($D_1$ and $D_2$) are arranged such that their cathodes point towards the positive output terminal, and two diodes ($D_3$ and $D_4$) have their anodes pointing towards the negative output terminal. Working Principle: Positive Half-Cycle of Input AC Voltage: During the positive half-cycle, let's assume the top terminal of the AC input becomes positive and the bottom terminal becomes negative. Diode $D_1$ becomes forward-biased (anode positive), and $D_2$ becomes reverse-biased (cathode positive). Simultaneously, $D_3$ becomes reverse-biased (anode negative), and $D_4$ becomes forward-biased (cathode negative). Therefore, $D_1$ and $D_4$ conduct, while $D_2$ and $D_3$ are open circuits. Current flows from the positive AC terminal, through $D_1$, through the load resistor $R_L$ (from top to bottom), through $D_4$, and back to the negative AC terminal. The output voltage across $R_L$ is approximately $V_{in} - 2V_K$ (since two diodes are in series). Negative Half-Cycle of Input AC Voltage: During the negative half-cycle, the polarity of the AC input reverses. The top terminal becomes negative, and the bottom terminal becomes positive. Now, diode $D_2$ becomes forward-biased (anode positive), and $D_1$ becomes reverse-biased. Simultaneously, $D_3$ becomes forward-biased (anode positive), and $D_4$ becomes reverse-biased. Therefore, $D_2$ and $D_3$ conduct, while $D_1$ and $D_4$ are open circuits. Current flows from the positive AC terminal (now the bottom one), through $D_3$, through the load resistor $R_L$ (critically, still from top to bottom, maintaining the same output polarity), through $D_2$, and back to the negative AC terminal (now the top one). The output voltage across $R_L$ is again approximately $V_{in} - 2V_K$. In both half-cycles, current flows through the load resistor in the same direction, producing a pulsating DC output. The output waveform consists of a series of positive half-cycles, similar to the centre-tapped full-wave rectifier, with a ripple frequency twice that of the input AC frequency. Its advantages include not needing a centre-tapped transformer, better transformer utilization, and a higher output voltage than a centre-tapped configuration for the same transformer secondary voltage. Its main drawback is the slightly higher voltage drop due to two diodes conducting in series during each half-cycle. 5. Comparison of Half-Wave and Full-Wave Rectifiers Comparing half-wave and full-wave rectifiers (both centre-tapped and bridge) highlights their respective advantages and disadvantages, guiding their selection for specific applications. Output Waveform and Ripple: Half-wave: Produces a pulsating DC output consisting only of positive half-cycles of the input AC. The output contains significant ripple, and the ripple frequency is equal to the input AC frequency ($f_{ripple} = f_{in}$). Full-wave (Centre-tapped & Bridge): Produces a pulsating DC output by rectifying both positive and negative half-cycles. The output is much smoother than half-wave, and the ripple frequency is twice the input AC frequency ($f_{ripple} = 2 f_{in}$). A higher ripple frequency is easier to filter into a smooth DC. Number of Diodes: Half-wave: Requires only one diode. Full-wave Centre-tapped: Requires two diodes. Full-wave Bridge: Requires four diodes. Transformer Requirement: Half-wave & Full-wave Bridge: Can use a standard transformer with a single secondary winding. Full-wave Centre-tapped: Requires a centre-tapped transformer, which is often more expensive and bulky. Peak Inverse Voltage (PIV): This is the maximum voltage a diode must withstand when it is reverse-biased. Half-wave: PIV = $V_m$ (peak input voltage). Full-wave Centre-tapped: PIV = $2V_m$ (where $V_m$ is the peak voltage from center tap to either end). This is a significant disadvantage as diodes must be rated for higher voltages. Full-wave Bridge: PIV = $V_m$. (Each diode withstands the full peak input voltage, similar to half-wave). Output Voltage: Half-wave: Average DC output voltage $V_{dc} = V_m/\pi$. Full-wave (Centre-tapped & Bridge): Average DC output voltage $V_{dc} = 2V_m/\pi$. The DC output is twice that of a half-wave rectifier for the same peak transformer secondary voltage. Rectification Efficiency: This measures how effectively the AC input power is converted to DC output power. Half-wave: Maximum efficiency $\approx 40.6\%$. Full-wave (Centre-tapped & Bridge): Maximum efficiency $\approx 81.2\%$. Full-wave rectifiers are twice as efficient as half-wave. Applications: Half-wave: Used in very low-power, non-critical applications where ripple is acceptable or filtering is simple (e.g., simple battery chargers). Full-wave: Widely used in power supplies for most electronic devices, providing smoother DC output and higher efficiency. The bridge rectifier is generally preferred due to its lower PIV requirement per diode and no need for a centre-tapped transformer, making it more cost-effective and compact for most applications. 6. Ripple Factor and Rectification Efficiency Ripple Factor ($\gamma$): The ripple factor is a quantitative measure of the pulsating nature of the rectified DC output voltage. It indicates how much AC component (ripple) is present in the DC output. A lower ripple factor implies a smoother and purer DC output, which is generally desired for powering electronic circuits. It is defined as the ratio of the RMS value of the AC component of the output voltage to the average (DC) value of the output voltage: $\gamma = \frac{V_{ac,rms}}{V_{dc}}$. Alternatively, it can be expressed as $\gamma = \sqrt{(\frac{V_{rms}}{V_{dc}})^2 - 1}$, where $V_{rms}$ is the RMS value of the total pulsating output voltage. For a half-wave rectifier, the ripple factor is approximately 1.21 (or 121%), indicating a very high ripple content. For full-wave rectifiers (both centre-tapped and bridge), the ripple factor is approximately 0.482 (or 48.2%), which is significantly lower than that of a half-wave rectifier, signifying a much smoother output. The goal of a power supply design is often to minimize the ripple factor through the use of filters. Rectification Efficiency ($\eta$): Rectification efficiency, also known as DC conversion efficiency, quantifies how effectively an AC input power is converted into useful DC output power by the rectifier. It is defined as the ratio of the DC power delivered to the load to the total AC input power supplied to the rectifier: $\eta = \frac{P_{dc}}{P_{ac}} \times 100\%$. The AC input power includes the power dissipated in the diodes and transformer losses. For an ideal rectifier (no diode voltage drop, no transformer losses), the maximum theoretical efficiency for a half-wave rectifier is 40.6%. This relatively low value is because only half of the input AC cycle is utilized. For full-wave rectifiers (both centre-tapped and bridge), the maximum theoretical efficiency is 81.2%. This higher efficiency is achieved because both half-cycles of the input AC are utilized to deliver power to the load. A higher rectification efficiency means less power is wasted as heat within the rectifier circuit, leading to cooler operation and better energy utilization. In practical rectifiers, the efficiency will be slightly lower than these theoretical maximums due to diode voltage drops and other losses. 7. Role of Filters in Rectifiers Rectifier circuits, whether half-wave or full-wave, produce a pulsating DC output voltage rather than a pure, smooth DC voltage. This pulsating output, characterized by its AC component (ripple), is generally unsuitable for powering most sensitive electronic circuits, which require a stable DC supply. Filters are electronic circuits connected at the output of a rectifier to reduce or smooth out these pulsations, thereby converting the pulsating DC into a much purer, almost constant DC voltage. The primary reason for using filters is to reduce the ripple factor of the rectified output. The ripple can cause undesirable effects such as hum in audio circuits, noise in digital circuits, and instability in sensitive analog systems. Filters work by storing energy during the peaks of the rectified voltage and releasing it during the troughs, effectively averaging out the voltage fluctuations. Common types of filters include capacitor filters, inductor filters, and LC or CLC filters. Capacitor filters are the most common and simplest, acting as a reservoir that charges during the positive peaks and discharges through the load when the rectifier output falls. Inductor filters, on the other hand, oppose changes in current, helping to smooth out the current flow. More complex LC filters combine inductors and capacitors for superior ripple reduction. By significantly reducing the ripple, filters ensure that the electronic devices connected to the power supply receive a stable and clean DC voltage, crucial for their proper and reliable operation. 8. Capacitor Filter in Rectifier Circuits A capacitor filter, often referred to as a shunt capacitor filter, is the simplest and most widely used type of filter in rectifier circuits due to its effectiveness and low cost. It consists of a large-value electrolytic capacitor connected in parallel (shunt) with the load resistor at the output of the rectifier. Working Principle: The operation of the capacitor filter relies on the capacitor's ability to store charge and resist sudden changes in voltage. Charging Phase: When the rectified voltage waveform from the rectifier output starts to rise (during the conduction period of the diodes), the capacitor charges rapidly to the peak value of the rectified voltage. During this phase, the diode(s) in the rectifier are forward-biases and conduct current to both the load and the capacitor. Discharging Phase: As the rectified voltage waveform starts to fall (when the instantaneous input voltage drops below the capacitor voltage, causing the diode(s) to become reverse-biased and stop conducting), the capacitor begins to discharge slowly through the load resistor. The capacitor acts as an energy reservoir, supplying current to the load and maintaining the output voltage during the periods when the rectifier is not supplying current (or its voltage is lower than the capacitor's voltage). This charge-and-discharge cycle results in the output voltage fluctuating much less than the unfiltered pulsating DC. The capacitor effectively "fills in" the valleys of the pulsating DC, smoothing out the waveform and reducing the ripple. The larger the capacitance value and the larger the load resistance, the slower the discharge, and thus the smaller the ripple. The ripple frequency of a full-wave rectifier is twice that of a half-wave rectifier, making it easier for a capacitor filter to smooth the output of a full-wave rectifier more effectively. While highly effective, a drawback is that the capacitor charges only during the peaks, meaning the diodes conduct for a shorter duration but with higher peak currents, which must be considered in diode selection.