8.1 BJT Operation, Biasing & Small-Signal Amplifiers

8.1 BJT Operation, Biasing & Small-Signal Amplifiers

1. BJT Semiconductor Physics & Operational Modes

A Bipolar Junction Transistor (BJT) is a three-terminal, current-controlled semiconductor device consisting of two back-to-back $pn$ junctions. The three regions are the Emitter (E), Base (B), and Collector (C).

Construction & Doping Profiles

• Emitter (E): Heavily doped ($N_E \gg N_B > N_C$) to inject maximum majority carriers into the base.
• Base (B): Very lightly doped and physically ultra-thin (order of micrometers) to minimize carrier recombination.
• Collector (C): Moderately doped and physically largest to collect majority carriers and dissipate heat.

Fundamental Terminal Current & Gain Relations

Summing currents according to Kirchhoff's Current Law (KCL):

$$I_E = I_B + I_C$$

Where $I_C$ comprises the injected collector current and reverse saturation leakage current ($I_{CBO}$):

$$I_C = \alpha I_E + I_{CBO} \approx \alpha I_E$$

• Common-Base Current Gain (Alpha, $\alpha$): Ratio of collector current to emitter current in active mode ($0.95 \le \alpha \le 0.999$).

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

• Common-Emitter Current Gain (Beta, $\beta$ or $h_{fe}$): Ratio of collector current to base current ($20 \le \beta \le 500$).

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

• Exact Conversion Formulas:

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

• Thermal Voltage ($V_T$): Key temperature-dependent parameter defined as $V_T = \frac{k T}{q}$. At room temperature ($T = 300\text{ K}$ or $27^\circ\text{C}$):

$$V_T \approx 25.85\text{ mV} \approx 26\text{ mV}$$

BJT Operational Modes

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2. DC Biasing Networks & Q-Point Stabilization

Biasing establishes a stable DC Operating Point (Q-point) defined by $(V_{CEQ}, I_{CQ})$ on the transistor characteristic curves. The goal of proper bias design is to maintain the Q-point near the center of the active region AC load line, preventing distortion (clipping) under large signal swings.

Comparison of BJT Biasing Networks

A. Fixed-Bias Circuit

• Base Current: $I_B = \frac{V_{CC} - V_{BE}}{R_B}$
• Collector Current: $I_C = \beta I_B$
• Collector-Emitter Voltage: $V_{CE} = V_{CC} - I_C R_C$
• Evaluation: Highly unstable Q-point. $I_C$ depends directly on $\beta$, which varies widely with temperature and manufacturing tolerances.

B. Emitter-Stabilized Bias Circuit

• Base Current: $I_B = \frac{V_{CC} - V_{BE}}{R_B + (1+\beta)R_E}$
• Collector Current: $I_C = \beta I_B$
• Collector-Emitter Voltage: $V_{CE} = V_{CC} - I_C(R_C + R_E)$
• Evaluation: Emitter resistor $R_E$ provides negative feedback, stabilizing $I_C$ against $\beta$ variations.

C. Voltage-Divider Bias (Voltage-Self Bias)

• Thevenin Equivalent Parameters:

$$V_{TH} = V_{CC} \cdot \frac{R_2}{R_1 + R_2}, \quad R_{TH} = R_1 \parallel R_2 = \frac{R_1 R_2}{R_1 + R_2}$$

• Exact Analysis:

$$I_B = \frac{V_{TH} - V_{BE}}{R_{TH} + (1+\beta)R_E}, \quad I_C = \beta I_B, \quad V_{CE} = V_{CC} - I_C(R_C + R_E)$$

• Approximate Analysis Criterion: Valid when $\beta R_E \ge 10 R_2$:

$$V_B \approx V_{CC} \cdot \frac{R_2}{R_1 + R_2}, \quad V_E = V_B - V_{BE}, \quad I_C \approx I_E = \frac{V_E}{R_E}$$ $$\text{Notice that } I_C \text{ is completely independent of } \beta!$$

D. Collector-Feedback Bias

• Base Current: $I_B = \frac{V_{CC} - V_{BE}}{R_B + \beta(R_C + R_E)}$
• Collector-Emitter Voltage: $V_{CE} = V_{CC} - (I_C + I_B)R_C \approx V_{CC} - I_C R_C$

Stability Factors

Stability factor $S(I_{CO})$ measures the sensitivity of $I_C$ to changes in reverse saturation current $I_{CO}$:

$$S(I_{CO}) = \frac{\partial I_C}{\partial I_{CO}} = (1+\beta) \frac{1 + R_B/R_E}{1 + \beta + R_B/R_E}$$

• Ideal stability occurs when $S(I_{CO}) \to 1$, achieved when $R_B / R_E \ll 1$.
• For Fixed Bias: $S(I_{CO}) = 1 + \beta$ (worst stability).
• For Voltage Divider Bias with $R_{TH} \ll \beta R_E$: $S(I_{CO}) \approx 1$ (best stability).

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3. Small-Signal AC Modeling & Transistor Equivalent Circuits

Under small-signal AC operation, the transistor is replaced by a linear equivalent circuit model.

AC Dynamic Emitter Resistance ($r_e$)

Derived from the ideal diode equation at the forward-biased BE junction:

$$r_e = \frac{V_T}{I_{EQ}} \approx \frac{26\text{ mV}}{I_{EQ}}$$

Hybrid-$\pi$ Model Parameters

• Input Resistance ($r_{\pi}$): Resistance looking into the base with emitter grounded.

$$r_{\pi} = (1+\beta)r_e = \frac{\beta V_T}{I_{CQ}}$$

• Transconductance ($g_m$): Ratio of AC collector current to AC base-emitter voltage.

$$g_m = \frac{I_{CQ}}{V_T} = \frac{1}{r_e}$$

• Output Resistance ($r_o$): Represents Early Effect (base-width modulation with Early voltage $V_A$).

$$r_o = \frac{V_A + V_{CEQ}}{I_{CQ}} \approx \frac{V_A}{I_{CQ}}$$

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4. Single-Stage BJT Amplifier Configurations

A. Common Emitter (CE) Amplifier (Bypassed $R_E$)

• Input Impedance: $Z_{in} = R_1 \parallel R_2 \parallel \beta r_e$
• Output Impedance: $Z_{out} = R_C \parallel r_o \approx R_C$
• No-Load Voltage Gain: $A_v = -\frac{R_C}{r_e} = -g_m R_C$
• Loaded Voltage Gain: $A_{vL} = -\frac{R_C \parallel R_L}{r_e}$
• Phase Relationship: $180^\circ$ phase inversion between input and output.

B. Common Emitter with Unbypassed Emitter Resistor ($R_E$)

• Input Impedance: $Z_{in(base)} = \beta(r_e + R_E) \approx \beta R_E$
• Voltage Gain: $A_v \approx -\frac{R_C \parallel R_L}{R_E}$
• Benefit: Gain is independent of $r_e$ and temperature, though magnitude is reduced.

C. Common Collector (CC) Amplifier / Emitter Follower

• Input Impedance: $Z_{in} = R_1 \parallel R_2 \parallel [\beta (r_e + R_E \parallel R_L)] \approx R_1 \parallel R_2 \parallel \beta (R_E \parallel R_L)$
• Output Impedance: $Z_{out} = R_E \parallel \left(r_e + \frac{R_{TH}}{\beta}\right)$
• Voltage Gain: $A_v = \frac{R_E \parallel R_L}{r_e + (R_E \parallel R_L)} \approx +1$
• Phase Relationship: $0^\circ$ phase shift (in-phase follower).
• Primary Application: Impedance matching buffer (High $Z_{in}$, Low $Z_{out}$).

D. Common Base (CB) Amplifier

• Input Impedance: $Z_{in} = R_E \parallel r_e \approx r_e$ (Very low, $10-50\ \Omega$)
• Output Impedance: $Z_{out} \approx R_C$
• Voltage Gain: $A_v = +\frac{R_C \parallel R_L}{r_e}$
• Current Gain: $A_i \approx \alpha \approx 1$
• Phase Relationship: $0^\circ$ phase shift.
• Primary Application: High-frequency RF amplifiers (no Miller effect on $C_{bc}$).