Bipolar Junction Transistors (BJTs) are three-layer, two-junction semiconductor devices — either NPN or PNP — that can amplify or switch electrical signals.
They build directly on the physics of PN junctions, using two back-to-back junctions to control current flow.

Regions of a BJT:

• Emitter (E): Heavily doped to inject carriers efficiently.
• Base (B): Very thin and lightly doped to allow carriers to pass through with minimal recombination.
• Collector (C): Moderately doped and larger in area to collect carriers and dissipate heat.

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3.2 Modes of Operation

The operation of a BJT depends on the biasing of its two PN junctions: the Emitter-Base (E-B) junction and the Collector-Base (C-B) junction.

Mode	E-B Junction	C-B Junction	Characteristics
Cut-off	Reverse Biased	Reverse Biased	$I_C \approx 0$, transistor is OFF
Active (Forward)	Forward Biased	Reverse Biased	$I_C = \beta I_B$, used for amplification
Saturation	Forward Biased	Forward Biased	$I_C$ limited by circuit, transistor acts as closed switch
Reverse Active	Reverse Biased	Forward Biased	$I_E = \beta_R I_B$, reverse gain $\beta_R \ll \beta$

Detailed Operating Conditions:

1. Cut-off Mode:

   • Both junctions are reverse-biased: $V_{BE} < V_{BE(on)}$ (typically < 0.7 V for Si) and $V_{BC} < 0$
   • Voltage condition: The base voltage ($V_{base}$) is lower than both the collector voltage ($V_{collector}$) and emitter voltage ($V_{emitter}$)
   • Collector current: $I_C \approx I_{CBO}$ (leakage current, typically nA range)
   • Transistor acts as an open switch

2. Active Mode (Normal Operation):

   • E-B junction forward-biased: $V_{BE} \approx 0.7$ V (Si), $V_{BE} \approx 0.3$ V (Ge)
   • C-B junction reverse-biased: $V_{CB} > 0$ or equivalently $V_{CE} > V_{CE(sat)}$
   • Voltage condition: The base voltage ($V_{base}$) is higher than the emitter voltage ($V_{emitter}$) but lower than the collector voltage ($V_{collector}$)
   • Collector current: $I_C = \beta I_B$ where $\beta$ typically ranges from 50 to 300
   • This mode is used for linear amplification

3. Saturation Mode:

   • Both junctions forward-biased: $V_{BE} > V_{BE(on)}$ and $V_{BC} > 0$
   • Voltage condition: The base voltage ($V_{base}$) is higher than both the emitter voltage ($V_{emitter}$) and collector voltage ($V_{collector}$)
   • Collector-emitter voltage: $V_{CE(sat)} \approx 0.2$ V (Si)
   • Current relationship $I_C = \beta I_B$ no longer holds; $I_C < \beta I_B$
   • Transistor acts as a closed switch in digital circuits

4. Reverse Active Mode:

   • E-B junction reverse-biased and C-B junction forward-biased
   • Rarely used; has poor characteristics with $\beta_R \approx 0.1$ to $5$ (much less than $\beta$)
   • Emitter current: $I_E = \beta_R I_B$

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3.3 Carrier Transport in Active Mode

In active mode, the forward bias on the E-B junction injects majority carriers from the emitter into the base.
These carriers diffuse across the thin base and are swept into the collector by the reverse-biased C-B junction’s electric field.

Transport Mechanisms:

1. Injection: High emitter doping → large carrier injection into the base.
2. Diffusion: Carriers move through the base via diffusion due to a concentration gradient.
3. Collection: Carriers reaching the base-collector junction are quickly swept into the collector.

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3.4 Current Relationships

The fundamental relationship between the three terminal currents in a BJT is governed by Kirchhoff's Current Law:

$$I_E = I_C + I_B$$

where:

• $I_E$ = Emitter current (largest)
• $I_C$ = Collector current
• $I_B$ = Base current (smallest, typically 1-2% of $I_E$)

3.4.1 Common-Base Current Gain ($\alpha$)

The common-base current gain $\alpha$ (also called the base transport factor) is defined as:

$$\alpha = \frac{I_C}{I_E}$$

For well-designed BJTs, $0.95 \leq \alpha \leq 0.999$, typically $\alpha \approx 0.98$ to $0.99$.

The parameter $\alpha$ can be further decomposed as:

$$\alpha = \gamma \cdot \alpha_T \cdot (1 - \delta)$$

where:

• $\gamma$ = Emitter injection efficiency (fraction of emitter current due to majority carriers)
• $\alpha_T$ = Base transport factor (fraction of injected carriers reaching collector)
• $(1 - \delta)$ = Recombination factor accounting for carrier loss in base

3.4.2 Common-Emitter Current Gain ($\beta$)

The common-emitter current gain $\beta$ (also called $h_{FE}$ in hybrid parameters) is:

$$\beta = \frac{I_C}{I_B}$$

The relationship between $\alpha$ and $\beta$ is derived from $I_E = I_C + I_B$:

$$\alpha = \frac{I_C}{I_E} = \frac{I_C}{I_C + I_B} = \frac{1}{1 + \frac{1}{\beta}}$$

Solving for $\beta$:

$$\beta = \frac{\alpha}{1 - \alpha}$$

Conversely:

$$\alpha = \frac{\beta}{\beta + 1}$$

Typical values: $\beta$ ranges from 50 to 300 (sometimes up to 500 for high-gain transistors).

3.4.3 Active Mode Current Equations

In active mode, the collector current can be expressed as:

$$I_C = \beta I_B = \alpha I_E$$

The emitter current:

$$I_E = \left(1 + \frac{1}{\beta}\right) I_C = (\beta + 1) I_B$$

Example: If $\alpha = 0.98$, then:

$$\beta = \frac{0.98}{1 - 0.98} = \frac{0.98}{0.02} = 49$$

If $I_B = 20 \, \mu\text{A}$, then:

• $I_C = \beta I_B = 49 \times 20 \, \mu\text{A} = 0.98 \, \text{mA}$
• $I_E = I_C + I_B = 0.98 + 0.02 = 1.0 \, \text{mA}$

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3.5 Ebers–Moll Model

The Ebers–Moll model describes a BJT in all regions of operation, using two coupled diodes.

Collector Current Equation:

$$I_C = I_S \left( e^{V_{BE}/V_T} - 1 \right) - \alpha_R I_S \left( e^{V_{BC}/V_T} - 1 \right)$$

Emitter Current Equation:

$$I_E = \alpha_F I_S \left( e^{V_{BE}/V_T} - 1 \right) - I_S \left( e^{V_{BC}/V_T} - 1 \right)$$

Where:

• $I_S$ = Saturation current of the transistor
• $V_T$ = Thermal voltage (~25.85 mV at room temperature)
• $\alpha_F$ = Forward common-base gain
• $\alpha_R$ = Reverse common-base gain

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3.6 Configurations of BJT

A BJT can be connected in three basic configurations, each with different characteristics and applications. The choice of configuration depends on the desired gain characteristics, impedance levels, and application requirements.

3.6.1 Common-Emitter (CE) Configuration

In the Common-Emitter configuration, the input is applied between base-emitter and the output is taken between collector-emitter; the emitter is the common (grounded) terminal. This is the most widely used amplifier configuration because it provides both high current gain and high power gain.

Circuit Characteristics:

• Input: Base-Emitter ($V_{BE}$, $I_B$)
• Output: Collector-Emitter ($V_{CE}$, $I_C$)
• Common terminal: Emitter (grounded)

Current Relationships:

The emitter current relation is:

$$I_E = I_C + I_B$$

The current gain in CE configuration is:

$$\beta = \frac{I_C}{I_B}$$

where $\beta$ typically ranges from 20 to 200 for general-purpose BJTs.

Another important current ratio is:

$$\alpha = \frac{I_C}{I_E}$$

with $\alpha < 1$, usually around 0.95-0.99.

The relationship between $\alpha$ and $\beta$ is:

$$\beta = \frac{\alpha}{1 - \alpha}, \quad \alpha = \frac{\beta}{\beta + 1}$$

Operating Principle:

A small change in base current $I_B$ causes a much larger change in collector current $I_C$ → this is the key amplification mechanism.

Key Parameters:

• Current gain: $A_i = \beta$ (high, typically 50-300)
• Voltage gain: $A_v = -\frac{\beta R_C}{r_e}$ (high, typically 100-500)
• Input impedance: $Z_{in} = \beta r_e$ (low to moderate, ~1-5 kΩ)
• Output impedance: $Z_{out} \approx R_C$ (moderate, ~1-10 kΩ)
• Phase shift: 180° (inverting amplifier - input and output are out of phase)

Applications: Voltage amplifiers, power amplifiers, switching circuits

3.6.2 Common-Base (CB) Configuration

In the Common-Base configuration, the base terminal is common to both input and output; the input is applied between emitter and base, and output is taken between collector and base.

Circuit Characteristics:

• Input: Emitter-Base ($V_{EB}$, $I_E$)
• Output: Collector-Base ($V_{CB}$, $I_C$)
• Common terminal: Base (grounded)

Current Relationships:

The input current at the emitter is large, since:

$$I_E = I_B + I_C$$

Therefore, the collector current is slightly less than emitter current, giving:

$$\alpha = \frac{I_C}{I_E} < 1$$

This means the current gain is less than unity (unity or less).

Operating Principle:

Because the current gain is less than one, the CB amplifier attenuates current, not amplifies it. However, it is a non-inverting voltage amplifier → input and output voltages are in phase.

It has high voltage gain due to:

• Low input resistance (forward-biased emitter-base diode)
• High output resistance (reverse-biased collector-base junction)

Voltage Gain:

The Common-Base voltage gain is:

$$A_v = \frac{\alpha R_L}{r_e}$$

where $R_L$ is the load resistance and $r_e$ is the emitter resistance.

Resistance Gain:

The CB configuration exhibits significant resistance gain:

$$A_R = \frac{R_L}{R_{in}}$$

where $R_{in}$ is very low and $R_L$ is typically much larger.

Key Parameters:

• Current gain: $A_i = \alpha \approx 0.98$ (less than unity but close to 1)
• Voltage gain: $A_v = \frac{\alpha R_C}{r_e}$ (high, similar to CE)
• Input resistance: $R_{in} = r_e$ (very low, ~20-50 Ω)
• Output resistance: $R_{out}$ (very high, ~100 kΩ - 1 MΩ)
• Phase shift: 0° (non-inverting - input and output are in phase)

Applications: Not commonly used in general amplifiers, but useful in high-frequency applications due to low input capacitance and stable behavior; RF circuits, current buffers

3.6.3 Common-Collector (CC) Configuration (Emitter Follower)

In the Common-Collector configuration, the collector terminal is common to both input and output (usually tied to supply/ground through a resistor). The input signal is applied to the base, and the output is taken from the emitter, making it an Emitter Follower (output follows input).

Circuit Characteristics:

• Input: Base-Collector ($V_{BC}$, $I_B$)
• Output: Emitter-Collector ($V_{EC}$, $I_E$)
• Common terminal: Collector (usually connected to $V_{CC}$)

Current Relationships:

The load resistor is connected in series with the emitter, so the load current equals the emitter current.

Since:

$$I_E = I_C + I_B$$

the emitter current is larger than collector current, which leads to a high current gain.

Current Gain:

The current gain of the CC configuration is approximately:

$$A_i \approx 1 + \beta$$

which is much larger than $\beta$ of a CE amplifier.

Operating Principle:

It provides very high input impedance (hundreds of kΩ) and low output impedance, making it excellent for impedance matching.

The voltage gain is slightly less than 1, but the circuit provides no phase inversion, unlike CE configuration.

Key Parameters:

• Current gain: $A_i = \beta + 1$ (high)
• Voltage gain: $A_v = \frac{\beta R_E}{r_e + (\beta + 1)R_E} \approx 1$ (unity, ~0.95-0.99)
• Input impedance: $Z_{in} = (\beta + 1)R_E$ (very high, ~10-500 kΩ)
• Output impedance: $Z_{out} = \frac{R_S}{\beta + 1} || R_E$ (very low, ~10-100 Ω)
• Phase shift: 0° (non-inverting, output follows input)

Applications: Impedance matching circuits, buffer stages, voltage regulators, driver stages

Comparison Table:

Parameter	Common-Emitter	Common-Base	Common-Collector
Current Gain	High ($\beta$)	Low ($\alpha < 1$)	High ($\beta + 1$)
Voltage Gain	High (~100-500)	High (~100-500)	Unity (~1)
Power Gain	Very High	High	Medium
Input Impedance	Medium	Very Low	Very High
Output Impedance	Medium	Very High	Very Low
Phase Shift	180°	0°	0°

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3.7 Output Characteristics

In the common-emitter configuration, the $I_C$ vs. $V_{CE}$ characteristics exhibit distinct operating regions:

Characteristic Regions:

1. Cut-off Region: $I_B = 0$, $I_C \approx I_{CEO}$ (leakage)
2. Active Region: $I_C = \beta I_B$ (nearly flat curves)
3. Saturation Region: $V_{CE} < V_{CE(sat)} \approx 0.2$ V, transistor fully ON

In the active region, the curves are nearly horizontal, indicating that:

• $I_C$ is primarily controlled by $I_B$
• $I_C$ shows slight dependence on $V_{CE}$

The small positive slope in the active region is due to the Early Effect (also called base-width modulation).

3.7.1 Early Effect (Base-Width Modulation)

Definition: The Early Effect is the variation of the effective base width with changes in the collector-base reverse bias voltage. As $V_{CB}$ increases, the depletion region at the collector-base junction widens, effectively reducing the neutral base width $W_B$.

Physical Mechanism:

• Increasing $V_{CE}$ (or $V_{CB}$) increases the reverse bias on the C-B junction
• This widens the C-B depletion region, encroaching into the base
• The effective base width decreases: $W_B \rightarrow W_B - \Delta W$
• Carrier gradient in base increases: $\frac{dn}{dx}$ increases
• Collector current increases slightly

Mathematical Description:

The collector current including the Early Effect is:

$$I_C = I_S e^{V_{BE}/V_T} \left(1 + \frac{V_{CE}}{V_A}\right)$$

where:

• $V_A$ = Early voltage (typically 50-200 V)
• $I_S$ = Saturation current
• $V_T$ = Thermal voltage ($\approx 26$ mV at 300 K)

The output resistance in active mode is:

$$r_o = \frac{\partial V_{CE}}{\partial I_C}\bigg|_{I_B=\text{const}} = \frac{V_A + V_{CE}}{I_C} \approx \frac{V_A}{I_C}$$

Practical Implications:

• Limits the maximum voltage gain of amplifiers
• Introduces finite output impedance ($r_o$)
• Causes slight variation in $\beta$ with $V_{CE}$
• More pronounced in transistors with narrow base widths

Graphical Interpretation: When the $I_C$-$V_{CE}$ curves in the active region are extrapolated backward, they meet at a common point on the negative $V_{CE}$ axis at voltage $-V_A$ (the Early voltage).

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3.8 Summary of Key Equations

Current Relationships:

Kirchhoff's Current Law: $$I_E = I_C + I_B$$

Common-Base Current Gain: $$\alpha = \frac{I_C}{I_E}, \quad 0.95 \leq \alpha \leq 0.999$$

Common-Emitter Current Gain: $$\beta = \frac{I_C}{I_B} = \frac{\alpha}{1 - \alpha}, \quad \text{typically } 50 \leq \beta \leq 300$$

Active-Mode Relationships: $$I_C = \beta I_B = \alpha I_E$$

$$I_E = (\beta + 1) I_B = \frac{I_C}{\alpha}$$

Ebers-Moll Model:

Collector Current: $$I_C = I_S \left( e^{V_{BE}/V_T} - 1 \right) - \alpha_R I_S \left( e^{V_{BC}/V_T} - 1 \right)$$

Emitter Current: $$I_E = \alpha_F I_S \left( e^{V_{BE}/V_T} - 1 \right) - I_S \left( e^{V_{BC}/V_T} - 1 \right)$$

Active-Mode Approximations:

For $V_{BE} \gg V_T$ and $V_{BC} < 0$ (reverse-biased C-B junction):

$$I_C \approx I_S e^{V_{BE}/V_T}$$

$$I_E \approx \alpha_F I_S e^{V_{BE}/V_T}$$

Including Early Effect:

$$I_C = I_S e^{V_{BE}/V_T} \left(1 + \frac{V_{CE}}{V_A}\right)$$

where $V_A$ is the Early voltage.

Output Resistance: $$r_o = \frac{V_A + V_{CE}}{I_C} \approx \frac{V_A}{I_C}$$

Small-Signal Parameters:

Transconductance: $$g_m = \frac{\partial I_C}{\partial V_{BE}}\bigg|_{V_{CE}=\text{const}} = \frac{I_C}{V_T} \approx \frac{I_C}{26 \text{ mV}}$$

Input Resistance (Emitter Resistance): $$r_e = \frac{V_T}{I_E} \approx \frac{26 \text{ mV}}{I_E}$$

Base-Emitter Input Resistance: $$r_{\pi} = \frac{\partial V_{BE}}{\partial I_B}\bigg|_{V_{CE}=\text{const}} = \frac{\beta}{g_m} = \beta r_e$$

These equations are essential for designing amplifiers and switching circuits using BJTs.

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References

For detailed derivations and additional information, refer to:

1. S. M. Sze & K. Ng, Physics of Semiconductor Devices, Chapter 5: Bipolar Transistors
2. B. G. Streetman & S. Banerjee, Solid State Electronic Devices, Chapter 6: BJTs
3. R. F. Pierret, Advanced Semiconductor Fundamentals, Chapters 3-5
4. Donald A. Neamen, Semiconductor Physics and Devices, Chapter 10: The Bipolar Transistor

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