Semiconductor Diodes, Rectifiers, Filters, and Zener Regulators Study Notes

Semiconductor Diode Fundamentals

A $pn$ junction is known as a semiconductor diode or crystal diode, so named because the $pn$ junction is grown out of a crystal. It possesses a unilateral conduction characteristic, allowing current to flow easily when forward biased while blocking current when reverse biased.

Comparison Between Crystal Diodes and Vacuum Diodes

Advantages of Crystal Diodes

• Smaller in size, more rugged, and have a significantly longer operational life.

• Simpler construction and inherently cheaper to manufacture.

• Require no filament power, thereby generating less heat during operation compared to vacuum tubes.

• Higher operating efficiency in low-voltage applications.

Disadvantages of Crystal Diodes

• Extremely heat-sensitive. A slight rise in temperature increases current appreciably. If the temperature exceeds the rated limit, the junction can be permanently ruined. Germanium diodes must not operate above $80,^\text{o}\text{C}$, while silicon diodes can operate safely up to approximately $200,^\text{o}\text{C}$.

• Lower current-handling capacities and lower peak inverse voltage ratings compared to vacuum diodes.

• Inability to withstand overloads. Even transient overload pulses can permanently destroy a crystal diode, whereas vacuum diodes generally recover once an overload is removed.

Symbol and Terminal Identification

In schematic representations, the arrowhead indicates the direction of conventional current flow under forward bias:

• Forward Bias Condition: The arrowhead is positive with respect to the bar (external circuit attempts to push conventional current in the direction of the arrowhead).

• Reverse Bias Condition: The arrowhead is negative with respect to the bar (external circuit attempts to push current opposite to the arrowhead).

Methods for Identifying Diode Terminals

• Painted Schematic Symbol: Manufacturers paint the diode symbol directly onto the device body (e.g., BY127 and BY114 manufactured by BEL).

• Color Coding Marks: Color bands on the diode casing indicate polarity, where a red mark denotes the arrowhead (anode) and a blue mark denotes the bar (cathode) (e.g., OA80).

Diode Resistance and Equivalent Circuit Models

Diode Resistance

Forward Resistance

Forward resistance is the opposition offered by the diode to forward current. It is categorized into D.C. and A.C. resistances:

• D.C. Forward Resistance ($R_f$): The ratio of d.c. voltage across the diode to the resulting d.c. current through it.

$R_f = \frac{V_{\text{dc}}}{I_{\text{dc}}} = \frac{OA}{OB}$

• A.C. Forward Resistance ($r_f$): The ratio of a small change in voltage across the diode to the corresponding change in current through it. It is defined around an operating point $P$:

$r_f = \frac{\Delta V_F}{\Delta I_F} = \frac{oc - oa}{of - od} = \frac{ac}{df}$

Typical forward resistance values range from $1,\Omega$ to $25,\Omega$.

Reverse Resistance

Reverse resistance is the opposition offered by the diode under reverse bias. Due to minority charge carriers, a small leakage current exists under normal reverse voltages. The ratio of reverse resistance to forward resistance is approximately:

• Germanium Diodes: $40000 : 1$

• Silicon Diodes: $1000000 : 1$

Diode Equivalent Circuits

When conducting, a diode develops an internal voltage drop due to its potential barrier ($V_0$) and internal forward resistance ($r_f$):

$V_F = V_0 + I_f r_f$

Where $V_0 = 0.7,\text{V}$ for silicon and $V_0 = 0.3,\text{V}$ for germanium.

Important Diode Operational Parameters

• Forward Current Rating: The maximum continuous current a forward-biased diode can carry without suffering thermal destruction.

• Peak Inverse Voltage (PIV): The maximum reverse voltage a diode can withstand without experiencing junction breakdown. Typical PIV ratings range from $10,\text{V}$ to $10,\text{kV}$.

• Reverse Leakage Current: The current flowing through a reverse-biased diode caused by minority charge carriers. It is extremely small ($< 1,\mu\text{A}$ for silicon, $\approx 100,\mu\text{A}$ for germanium) under standard operating voltages.

Rectifiers and Circuit Analysis

Rectification is the process of converting alternating current (A.C.) to direct current (D.C.).

Half-Wave Rectifier

In half-wave rectification, the diode conducts only during the positive half-cycles of the input A.C. supply. Negative half-cycles are suppressed.

• Transformer Function: Step-up or step-down voltage transformation, and electrical isolation from the power lines to prevent shock hazards.

• Output Characteristics: Pulsating D.C. output.

• Output Frequency: Equal to input frequency ($f_{\text{out}} = f_{\text{in}}$).

Mathematical Analysis of Half-Wave Rectifier

Given input voltage $v = V_m \sin(\theta)$:

$I_{\text{dc}} = I_{\text{av}} = \frac{1}{2\pi} \int_0^{\pi} I_m \sin(\theta) \, d\theta = \frac{I_m}{\pi}$

$P_{\text{dc}} = I_{\text{dc}}^2 R_L = \left(\frac{I_m}{\pi}\right)^2 R_L$

$I_{\text{rms}} = \frac{I_m}{2}$

$P_{\text{ac}} = I_{\text{rms}}^2 (r_f + R_L) = \left(\frac{I_m}{2}\right)^2 (r_f + R_L)$

$\eta = \frac{P_{\text{dc}}}{P_{\text{ac}}} = \frac{\left(\frac{I_m}{\pi}\right)^2 R_L}{\left(\frac{I_m}{2}\right)^2 (r_f + R_L)} = \frac{4}{\pi^2} \frac{R_L}{r_f + R_L} = \frac{0.406}{1 + \frac{r_f}{R_L}}$

Maximum efficiency occurs when $r_f \ll R_L$:

$\eta_{\max} = 40.6\%$

• Disadvantages: High output ripple requiring extensive filtering, power delivered only during half of the cycle.

Full-Wave Rectifiers

Full-wave rectifiers convert both half-cycles of the input A.C. voltage into D.C. output.

Centre-Tap Full-Wave Rectifier

Utilizes a centre-tapped secondary transformer winding and two diodes ($D_1$ and $D_2$) conducting in alternate half-cycles.

• Peak Inverse Voltage: $PIV = 2 V_m$

• Disadvantages: Difficulty in locating the exact center-tap, lower output voltage relative to total secondary winding, high PIV requirements for diodes.

Full-Wave Bridge Rectifier

Utilizes four diodes ($D_1, D_2, D_3, D_4$) connected in a bridge configuration across a standard transformer secondary winding.

• Conduction Path: Positive half-cycle conducts through $D_1$ and $D_3$ in series with $R_L$; negative half-cycle conducts through $D_2$ and $D_4$ in series with $R_L$.

• Peak Inverse Voltage: $PIV = V_m$

• Advantages: Eliminates center-tapped transformer, twice the D.C. output voltage compared to centre-tap for same secondary voltage, half the PIV rating required.

• Disadvantages: Requires four diodes; two diode forward drops ($2 r_f$) occur in series during conduction.

Mathematical Analysis of Full-Wave Rectifier

$I_{\text{dc}} = \frac{2 I_m}{\pi}$

$P_{\text{dc}} = I_{\text{dc}}^2 R_L = \left(\frac{2 I_m}{\pi}\right)^2 R_L$

$I_{\text{rms}} = \frac{I_m}{\sqrt{2}}$

$P_{\text{ac}} = I_{\text{rms}}^2 (r_f + R_L) = \left(\frac{I_m}{\sqrt{2}}\right)^2 (r_f + R_L)$

$\eta = \frac{P_{\text{dc}}}{P_{\text{ac}}} = \frac{\left(\frac{2 I_m}{\pi}\right)^2 R_L}{\left(\frac{I_m}{\sqrt{2}}\right)^2 (r_f + R_L)} = \frac{8}{\pi^2} \frac{R_L}{r_f + R_L} = \frac{0.812}{1 + \frac{r_f}{R_L}}$

$\eta_{\max} = 81.2\%$

• Output Frequency: $f_{\text{out}} = 2 f_{\text{in}}$

Ripple Factor Derivations

Ripple factor is defined as the ratio of the r.m.s. value of the A.C. component to the D.C. component in the rectifier output:

$\text{Ripple Factor} = \frac{I_{\text{ac}}}{I_{\text{dc}}} = \frac{V_{\text{ac}}}{V_{\text{dc}}}$

Given $I_{\text{rms}} = \sqrt{I_{\text{dc}}^2 + I_{\text{ac}}^2}$:

$I_{\text{ac}} = \sqrt{I_{\text{rms}}^2 - I_{\text{dc}}^2}$

$\text{Ripple Factor} = \sqrt{\left(\frac{I_{\text{rms}}}{I_{\text{dc}}}\right)^2 - 1}$

Half-Wave Ripple Factor Calculation

Substituting $I_{\text{rms}} = \frac{I_m}{2}$ and $I_{\text{dc}} = \frac{I_m}{\pi}$:

$\text{Ripple Factor} = \sqrt{\left(\frac{\frac{I_m}{2}}{\frac{I_m}{\pi}}\right)^2 - 1} = \sqrt{\left(\frac{\pi}{2}\right)^2 - 1} = \sqrt{2.4674 - 1} = 1.21$

Full-Wave Ripple Factor Calculation

Substituting $I_{\text{rms}} = \frac{I_m}{\sqrt{2}}$ and $I_{\text{dc}} = \frac{2 I_m}{\pi}$:

$\text{Ripple Factor} = \sqrt{\left(\frac{\frac{I_m}{\sqrt{2}}}{\frac{2 I_m}{\pi}}\right)^2 - 1} = \sqrt{\left(\frac{\pi}{2\sqrt{2}}\right)^2 - 1} = \sqrt{1.2337 - 1} = 0.48$

Comparison Table of Rectifiers

Rectifier Fault Analysis

Transformer Faults

• Shorted Winding: Causes primary fuse to blow; if fuse fails to blow, output voltage drops near zero and transformer overheats.

• Open Winding: D.C. output drops to zero; primary fuse does not blow. Resistance check shows near-infinite resistance.

• Winding Short to Frame: Causes primary fuse to blow. Resistance measurement between lead and chassis casing indicates near-zero resistance.

Diode Faults

• Shorted Diode (Centre-Tap): Primary fuse blows because the shorted diode acts as a wire, short-circuiting the secondary winding when the alternate diode conducts.

• Open Diode (Centre-Tap or Bridge): Rectifier degrades from full-wave operation to half-wave operation.

Filter Circuits

Filter circuits are installed between the rectifier output and the load to eliminate A.C. ripple components while passing D.C.

• Inductor Reactance: $X_L = 2\pi f L$. For D.C. ($f = 0$), $X_L = 0,\Omega$ (passes D.C. readily). Blocks A.C. components.

• Capacitor Reactance: $X_C = \frac{1}{2\pi f C}$. For D.C. ($f = 0$), $X_C = \infty,\Omega$ (blocks D.C.). Passes A.C. components readily.

Filter Circuit Configurations

Capacitor Filter

Consists of a capacitor $C$ placed in parallel across the load $R_L$. Charges to peak voltage $V_m$ during the first quarter-cycle, then slowly discharges through $R_L$ between peaks. Suitable for small load currents ($\le 50,\text{mA}$).

Choke Input Filter

Consists of an inductor $L$ in series with the rectifier output and a capacitor $C$ in parallel across the load. Inductor attenuates A.C. current fluctuations, and capacitor bypasses remaining A.C. ripple.

Capacitor Input Filter ($\pi$-Filter)

Consists of an input capacitor $C_1$ across the rectifier output, a series choke $L$, and an output capacitor $C_2$ across the load.

• Filtering Sequence:

  1. $C_1$ bypasses a large portion of A.C. ripple to ground while passing D.C.

  2. Choke $L$ presents high opposition to residual A.C. ripple while offering minimal opposition to D.C.

  3. $C_2$ bypasses any remaining A.C. ripple passing through the choke.

Output D.C. voltage formula for full-wave capacitor input filter:

$V_{\text{dc}} = V_{p(\text{in})} \left( 1 - \frac{1}{2 f L C R} \right)$

Where $V_{p(\text{in})}$ is the peak full-wave input voltage and $f$ is the output ripple frequency.

Voltage Multipliers

Voltage multipliers utilize diodes and capacitors to deliver a D.C. output that is an integral multiple of the peak A.C. input voltage ($n V_{S(\text{pk})}$).

• Conservation of Energy: No power amplification occurs. When output voltage increases by factor $n$, current output capability decreases by approximately factor $n$.

Half-Wave Voltage Doubler Circuit Operation

• Negative Half-Cycle: Diode $D_1$ is forward-biased (ON) and $D_2$ is reverse-biased (OFF). Capacitor $C_1$ charges to peak source voltage $V_{S(\text{pk})}$.

• Positive Half-Cycle: Diode $D_1$ is reverse-biased (OFF) and $D_2$ is forward-biased (ON). $C_1$ acts in series-aiding with the A.C. source, charging $C_2$ to $2 V_{S(\text{pk})}$.

• Output Voltage: $V_{\text{dc}} = 2 V_{S(\text{pk})}$.

• Limitation: Poor voltage regulation; output voltage drops significantly under heavy load current.

Zener Diodes and Voltage Regulation

A Zener diode is a heavily doped $pn$ junction diode designed to operate safely in the reverse breakdown region.

• Doping Effect: Heavy doping yields a thin depletion region, resulting in a low, sharp breakdown voltage ($V_Z$). Light doping yields a wider depletion region and a higher breakdown voltage.

• Operational Rule: Zener diodes are always connected in reverse bias configuration.

Zener Diode Equivalent Circuit Models

• "ON" State ($V \ge V_Z$): Modeled as a constant voltage battery source $V_Z$.

• "OFF" State ($0 < V < V_Z$): Modeled as an open circuit.

Zener Diode Voltage Regulator Circuit Analysis

In a standard regulator, Zener diode $V_Z$ is placed in parallel across load $R_L$, with series resistor $R$ connected to input supply $E_i$

$V_R = E_i - E_0$

$I = I_Z + I_L$

$R = \frac{E_i - E_0}{I_Z + I_L}$

• Response to Input Voltage Increase: Excess input voltage drops across series resistor $R$. Zener current $I_Z$ increases while load current $I_L$ and output voltage $E_0 = V_Z$ remain constant.

• Response to Load Resistance Decrease: Load current $I_L$ increases. Total current $I$ remains constant because drop across $R$ is fixed; Zener current $I_Z$ decreases by the exact amount $I_L$ increases, maintaining $E_0 = V_Z$

Analysis Methods for Fixed and Variable Conditions

Fixed Input Voltage $E_i$ and Fixed Load $R_L$

1. Remove Zener diode to find open-circuit voltage $V$:

$V = \frac{R_L E_i}{R + R_L}$

2. If $V \ge V_Z$, Zener is ON:

$E_0 = V_Z$

$I_L = \frac{V_Z}{R_L}$

$I = \frac{E_i - V_Z}{R}$

$I_Z = I - I_L$

$P_Z = V_Z I_Z$

3. If $V < V_Z$, Zener is OFF:

$E_0 = V$

$I_Z = 0$

Fixed Input Voltage $E_i$ and Variable Load $R_L$

$R_{L\min} = \frac{R V_Z}{E_i - V_Z}$

$I_{L\max} = \frac{V_Z}{R_{L\min}}$

$I_{L\min} = I - I_{ZM}$

$R_{L\max} = \frac{V_Z}{I_{L\min}}$

Where $I_{ZM}$ is the maximum rated Zener current.

Fixed Load $R_L$ and Variable Input Voltage $E_i$

$E_{i(\min)} = \frac{(R + R_L) V_Z}{R_L}$

$I_{\max} = I_{ZM} + I_L$

$E_{i(\max)} = I_{\max} R + V_Z$

Regulator Fault Analysis

• Shorted Filter Capacitor: Shorts the bridge rectifier output, causing primary fuse to blow.

• Open Filter Capacitor: Drastically increases output ripple and causes a substantial drop in D.C. voltage.