Unit 4: Analogue Electronics

Semiconductor Diodes

P and N Type Semiconductors

Semiconductors are materials whose electrical conductivity lies between that of conductors (such as copper) and insulators (such as glass). At absolute zero temperature, pure (intrinsic) semiconductors behave as perfect insulators. To increase their conductivity, controlled amounts of impurities are added in a process known as doping. Extrinsic semiconductors created by doping are classified into two main types: P-type and N-type.

1. P-type Semiconductors

P-type (Positive-type) semiconductors are produced by doping an intrinsic semiconductor (such as Silicon or Germanium) with a trivalent impurity (atoms containing three valence electrons, such as Boron, Aluminium, Indium, or Gallium).

• Since the impurity atom has only three valence electrons, it can form covalent bonds with only three neighboring silicon atoms. This leaves one bond incomplete, creating a vacancy known as a hole.
• The impurity atom accepts an electron from the neighboring crystal lattice to complete the bond, thereby becoming a negative immobile ion. Hence, trivalent impurities are called acceptor impurities.
• In P-type semiconductors, holes are the majority charge carriers, and electrons are the minority charge carriers.

2. N-type Semiconductors

N-type (Negative-type) semiconductors are produced by doping an intrinsic semiconductor with a pentavalent impurity (atoms containing five valence electrons, such as Phosphorus, Arsenic, or Antimony).

• Four of the five valence electrons form covalent bonds with four neighboring silicon atoms. The fifth electron is weakly bound to the parent atom and is easily detached by thermal energy to become a free conduction electron.
• The impurity atom loses an electron and becomes a positive immobile ion. Hence, pentavalent impurities are called donor impurities.
• In N-type semiconductors, electrons are the majority charge carriers, and holes are the minority charge carriers.

Conductivity, Mobility, and Drift Velocity

In a semiconductor, both free electrons and holes participate in electrical conduction. The total current is the sum of the electronic current and the hole current.

1. Drift Velocity (v_d)

When an external electric field is applied across a semiconductor, the charge carriers experience an electrostatic force. This causes them to accelerate and frequently collide with the lattice ions, resulting in a net average velocity along the direction of (or opposite to) the electric field. This net average velocity is called the drift velocity.

Drift Velocity (v_d) is the average velocity acquired by charge carriers in a semiconductor under the influence of an externally applied electric field.
v_d = μE

Where:
v_d = Drift velocity (m/s)
μ = Mobility of the charge carrier (m²V^-1s^-1)
E = Applied electric field (V/m)

2. Mobility (μ)

Mobility measures how quickly a charge carrier can move through a semiconductor crystal lattice when subjected to an electric field.

Mobility (μ) is defined as the magnitude of the drift velocity acquired by a charge carrier per unit applied electric field.
μ = v_d / E

• Since electrons move in the conduction band where there are fewer obstructions, electronic mobility (μ_e) is always significantly higher than hole mobility (μ_h) in the valence band.
• The unit of mobility is m²V^-1s^-1.

3. Conductivity (σ)

Electrical conductivity is a measure of a material's ability to conduct electric current. The total electrical conductivity of a semiconductor is the sum of the conductivities due to both electrons and holes.

Total Electrical Conductivity (σ):
σ = e(n_eμ_e + n_hμ_h)

Where:
e = Charge of an electron (1.6 × 10^-19 C)
n_e = Density of free electrons (m^-3)
n_h = Density of holes (m^-3)
μ_e = Mobility of electrons (m²V^-1s^-1)
μ_h = Mobility of holes (m²V^-1s^-1)

Exam-Oriented Observation: Since the resistivity (ρ) is the reciprocal of conductivity, the resistivity of a semiconductor is given by:
ρ = 1 / [e(n_eμ_e + n_hμ_h)]

Characteristics of PN Junction Diode

A PN junction is formed by intimately joining a P-type semiconductor to an N-type semiconductor, creating a single continuous crystal structure. This interface is the fundamental building block of many semiconductor devices.

1. Formation of the Depletion Region and Barrier Potential

• Diffusion: Due to the high concentration of holes in the P-region and electrons in the N-region, holes diffuse across the junction from the P-side to the N-side, while electrons diffuse from the N-side to the P-side.
• Recombination: Near the junction, diffusing electrons and holes recombine and neutralize each other.
• Depletion Region: This recombination leaves behind uncompensated positive immobile donor ions on the N-side of the junction and uncompensated negative immobile acceptor ions on the P-side. This region, depleted of free charge carriers, is called the depletion region.
• Barrier Potential (V_o): The accumulation of immobile ions creates an internal electric field directed from the N-side to the P-side. This field opposes further diffusion of majority carriers and establishes a potential difference across the junction known as the barrier potential (approximately 0.7 V for Silicon and 0.3 V for Germanium at room temperature).

2. Biasing of a PN Junction

Biasing refers to the application of an external voltage to the PN junction terminals to modify its behavior.

• Forward Biasing: The positive terminal of the external battery is connected to the P-side, and the negative terminal is connected to the N-side. This external field opposes the internal barrier field, thereby reducing the width of the depletion region and lowering the potential barrier. As a result, a large current flows across the junction even at low voltages.
• Reverse Biasing: The positive terminal of the external battery is connected to the N-side, and the negative terminal is connected to the P-side. The external field reinforces the internal barrier field, widening the depletion region and increasing the potential barrier. Consequently, no majority carrier current flows. Only an extremely small, temperature-dependent reverse saturation current flows due to the drift of minority carriers.

3. V-I Characteristics

The Volt-Ampere (V-I) characteristics represent the graphical relationship between the voltage applied across the diode and the resulting current flowing through it.

• Forward Characteristics: In the forward bias region, current remains very small until the applied voltage overcomes the barrier potential. This threshold voltage is called the Cut-in voltage or Knee voltage (0.7 V for Si, 0.3 V for Ge). Beyond the knee voltage, the current increases exponentially with even minor increases in voltage.
• Reverse Characteristics: In the reverse bias region, a very small reverse saturation current (I_o) flows, which remains nearly constant despite increases in reverse voltage. If the reverse voltage is increased beyond a certain critical value, known as the Breakdown Voltage, the covalent bonds rupture rapidly, causing a sharp, destructive increase in reverse current.

Static and Dynamic Resistance

Unlike a standard linear resistor, a PN junction diode does not obey Ohm's law linearly; its resistance depends on the operating point on the V-I curve.

1. Static Resistance (R_dc)

Static resistance is the resistance offered by a diode when a steady direct current (DC) voltage is applied to it.

Static Resistance (R_dc) is the ratio of the DC voltage across the diode to the DC current flowing through it at a given operating point.
R_dc = V / I

It is determined simply by taking a single coordinate point (V, I) on the V-I curve and calculating their ratio.

2. Dynamic Resistance (r_ac)

Dynamic resistance is the resistance offered by a diode to an alternating current (AC) signal superimposed on a DC bias.

Dynamic Resistance (r_ac) is defined as the ratio of a small change in voltage across the diode to the resulting small change in current through it.
r_ac = ΔV / ΔI

Dynamic resistance is the reciprocal of the slope of the V-I characteristics at a given operating point (Q-point). In the forward bias region above the knee voltage, the slope is very steep, meaning r_ac is extremely small (typically a few ohms). In the reverse bias region, the slope is almost flat, so r_ac is extremely large (megaohms).

Application as a Rectifier: Half-wave and Full-wave Rectifiers

Rectification is the process of converting alternating current (AC), which periodically reverses direction, into unidirectional direct current (DC). Diodes are ideal for this application because they conduct current in only one direction (when forward-biased).

1. Half-wave Rectifier (HWR)

Circuit Diagram Description

A Half-wave Rectifier circuit consists of:

1. An AC mains power source connected to the primary winding of a step-down transformer.
2. A single PN junction diode connected in series with the secondary winding of the transformer.
3. A load resistor (R_L) connected in series with the diode across which the output voltage is taken.

Working

• Positive Half-Cycle: During the positive half-cycle of the input AC voltage, the top terminal of the transformer secondary winding becomes positive with respect to the bottom terminal. This forward-biases the diode, allowing current to flow through the load resistor R_L. The output voltage across R_L mimics the shape of the positive input half-cycle.
• Negative Half-Cycle: During the negative half-cycle, the top terminal of the secondary winding becomes negative relative to the bottom terminal. This reverse-biases the diode, causing it to act as an open circuit. No current flows through R_L, and the output voltage drops to zero.
• Output: The output consists of a series of positive pulses. Half of the input energy is blocked.

Key Mathematical Parameters

• Maximum Efficiency (η_max): 40.6%
• Ripple Factor (γ): 1.21
• Peak Inverse Voltage (PIV): V_m (where V_m is the peak value of secondary AC voltage)

2. Full-wave Rectifier (FWR)

A Full-wave Rectifier utilizes both half-cycles of the input AC wave to produce continuous, unidirectional output current.

Circuit Diagram Description (Center-Tapped Configuration)

A center-tapped full-wave rectifier consists of:

1. An AC mains source connected to the primary winding of a step-down transformer with a center-tapped secondary winding. The center-tap is grounded and serves as the reference common terminal.
2. Two diodes (D₁ and D₂). The anode of D₁ is connected to the upper terminal of the secondary winding, and the anode of D₂ is connected to the lower terminal.
3. A load resistor (R_L) connected between the common cathode junction of the two diodes and the center-tap of the transformer.

Working

• Positive Half-Cycle: The upper secondary terminal becomes positive, and the lower secondary terminal becomes negative relative to the center-tap. Diode D₁ is forward-biased (conducts), while diode D₂ is reverse-biased (does not conduct). Current flows from the upper terminal through D₁, down through the load resistor R_L to the center-tap.
• Negative Half-Cycle: The polarity reverses: the upper secondary terminal becomes negative, and the lower terminal becomes positive relative to the center-tap. Diode D₁ is reverse-biased, while diode D₂ is forward-biased. Current flows from the lower terminal through D₂, down through the load resistor R_L to the center-tap.
• Output: Because current flows through R_L in the exact same direction during both half-cycles, a continuous pulsating DC output voltage is developed across R_L.

Key Mathematical Parameters

• Maximum Efficiency (η_max): 81.2%
• Ripple Factor (γ): 0.482
• Peak Inverse Voltage (PIV): 2V_m

Zener Diode and Voltage Regulation

Standard diodes are damaged by reverse breakdown. A Zener diode, however, is a heavily doped silicon semiconductor diode specifically designed to operate safely in the reverse breakdown region without sustaining damage.

1. Working Principle

• Due to heavy doping, the depletion region of a Zener diode is extremely thin (on the order of 10^-7 m).
• When a reverse bias is applied, a very high electric field is established across this thin depletion region even at low voltages.
• At a specific reverse voltage known as the Zener Voltage (V_Z), the strong electric field pulls valence electrons directly out of their covalent bonds. This rapid generation of free charge carriers causes a sharp increase in reverse current, a phenomenon known as Zener Breakdown.
• Once in the breakdown region, the voltage across the Zener diode remains virtually constant (equal to V_Z) over a very wide range of reverse currents.

2. Zener Diode as a Voltage Regulator

A voltage regulator maintains a constant output voltage across a load resistor despite fluctuations in the input unregulated voltage or changes in the load current.

Circuit Configuration

• The unregulated DC input voltage (V_in) is connected in series with a current-limiting resistor (R_S).
• The Zener diode is connected in parallel (shunt) with the load resistor (R_L), and it is connected in a reverse-biased configuration (cathode to positive line, anode to negative/ground).
• The output voltage (V_o) is measured across the load resistor R_L.

Step-by-Step Working Mechanism

1. Regulation against Input Voltage Fluctuations (Variable V_in, Constant R_L): If the input voltage V_in increases, the total current flowing through the circuit increases. Because the Zener diode is in its breakdown region, it maintains its constant breakdown voltage V_Z across itself and the parallel load resistor R_L. The excess voltage is dropped entirely across the series current-limiting resistor R_S. Conversely, if V_in decreases, the current through the Zener diode drops, but the voltage across the load remains stable at V_Z as long as the diode remains in the breakdown state.
2. Regulation against Load Variations (Constant V_in, Variable R_L): If the load resistance R_L decreases, the current required by the load (I_L) increases. To accommodate this, the current through the Zener diode (I_Z) decreases by the exact same amount, keeping the total current through the series resistor R_S constant. As a result, the voltage drop across R_S remains constant, and the output voltage across R_L remains stabilized at V_Z.

Key Design Formula:
I_S = I_Z + I_L
V_in - V_Z = I_S × R_S

Transistors

NPN and PNP Transistors

A Bipolar Junction Transistor (BJT) is a three-terminal, two-junction semiconductor device used to amplify or switch electrical signals. It consists of three doped regions: Emitter (E), Base (B), and Collector (C).

• Emitter (E): Moderately sized, heavily doped. Its primary role is to inject majority charge carriers into the base.
• Base (B): Extremely thin and lightly doped. It passes the majority of injected carriers through to the collector.
• Collector (C): Physically larger than the emitter, moderately doped. It collects the charge carriers from the base and dissipates heat.

1. NPN Transistors

• Structure: A thin layer of P-type semiconductor sandwiched between two layers of N-type semiconductors.
• Working: To operate the transistor as an amplifier, the Emitter-Base (EB) junction is forward-biased and the Collector-Base (CB) junction is reverse-biased. Electrons (majority carriers in the emitter) are injected from the N-type emitter into the P-type base. Because the base is extremely thin and lightly doped, only a small fraction (about 1% to 5%) of these electrons recombine with holes in the base, producing a small base current (I_B). The remaining 95% to 99% of the electrons drift across the depletion region into the collector under the influence of the strong reverse-bias collector voltage, creating the collector current (I_C).

2. PNP Transistors

• Structure: A thin layer of N-type semiconductor sandwiched between two layers of P-type semiconductors.
• Working: Similar to the NPN transistor, the EB junction is forward-biased and the CB junction is reverse-biased. The majority carriers in the emitter are holes. Holes are injected from the P-type emitter into the N-type base. Most of these holes cross into the P-type collector, while a tiny fraction recombines in the base, giving rise to base current.

Fundamental Transistor Current Equation:
I_E = I_B + I_C

Common Mistake to Avoid: Remember that although electrons and holes move in opposite directions, the conventional current directions are defined by the flow of positive charge. In an NPN transistor, current flows out of the Emitter terminal, whereas in a PNP transistor, conventional current flows into the Emitter terminal.

Characteristics of CB, CE, and CC Configurations

Since a transistor has three terminals, one terminal must be made common to both the input and output circuits to operate as a two-port network. This results in three distinct configurations.

1. Common Base (CB) Configuration

• Description: The Base terminal is common to both the input and output circuits. The input is applied between the Emitter and Base, while the output is taken between the Collector and Base.
• Input Characteristics: Plot of Emitter Current (I_E) vs. Emitter-Base Voltage (V_EB) at constant Collector-Base Voltage (V_CB). It resembles the forward characteristics of a simple diode. At higher values of V_CB, the curve shifts to the left because the depletion region widens, effectively narrowing the base (known as Base Width Modulation or the Early Effect).
• Output Characteristics: Plot of Collector Current (I_C) vs. Collector-Base Voltage (V_CB) at constant Emitter Current (I_E). In the active region, the collector current is almost equal to the emitter current and is independent of V_CB.

2. Common Emitter (CE) Configuration

• Description: The Emitter terminal is common. The input is applied between the Base and Emitter, and the output is taken between the Collector and Emitter.
• Input Characteristics: Plot of Base Current (I_B) vs. Base-Emitter Voltage (V_BE) at constant Collector-Emitter Voltage (V_CE). An increase in V_CE shifts the curves to the right because the effective base width decreases, reducing base recombination and current.
• Output Characteristics: Plot of Collector Current (I_C) vs. Collector-Emitter Voltage (V_CE) at constant Base Current (I_B). It has three distinct operating regions:
  • Active Region: The EB junction is forward-biased, and the CB junction is reverse-biased. I_C is controlled by I_B (I_C = βI_B).
  • Saturation Region: Both junctions are forward-biased. I_C increases rapidly with small changes in V_CE.
  • Cut-off Region: Both junctions are reverse-biased. I_B is zero, and only a tiny reverse leakage current flows.

3. Common Collector (CC) Configuration

• Description: The Collector terminal is common. The input is applied between the Base and Collector, and the output is taken between the Emitter and Collector. It is also referred to as an Emitter Follower because the output voltage closely tracks the input voltage.

Current Gains α and β, and Their Relations

Current gains represent the amplification capabilities of a transistor in its different configurations under DC conditions.

1. Common Base Current Gain (α)

Alpha (α) is defined as the ratio of the collector current to the emitter current.
α = I_C / I_E

Since I_C is always slightly less than I_E due to base recombination, the value of α is always less than unity (typically between 0.95 and 0.99).

2. Common Emitter Current Gain (β)

Beta (β) is defined as the ratio of the collector current to the base current.
β = I_C / I_B

Since the base current is extremely small compared to the collector current, the value of β is much larger than unity (typically ranging from 20 to 500).

3. Derivation of the Relationship Between α and β

We begin with the fundamental transistor current relation:

I_E = I_B + I_C

To establish the relationship, divide the entire equation by the collector current I_C:

I_E / I_C = (I_B / I_C) + (I_C / I_C)

I_E / I_C = (I_B / I_C) + 1

Using our definitions of α and β:
Since α = I_C / I_E, we have: I_E / I_C = 1 / α
Since β = I_C / I_B, we have: I_B / I_C = 1 / β

Substitute these values back into the equation:

1 / α = (1 / β) + 1

Solve for 1 / α by finding a common denominator on the right side:

1 / α = (1 + β) / β

Inverting both sides gives the formula for α in terms of β:

α = β / (1 + β)

Similarly, we can solve for β in terms of α starting from 1 / β = (1 / α) - 1:

1 / β = (1 - α) / α

Inverting both sides gives the formula for β in terms of α:

β = α / (1 - α)

Key Academic Observation: As α approaches 1, the denominator (1 - α) approaches 0, causing β to increase exponentially. This explains why small manufacturing variations in α result in highly variable β values among transistors of the same model.