Unit 4 - Notes

CSE212 11 min read

Unit 4: Transistor Biasing and Thermal Stabilization

The Operating Point and Bias Stability

The Operating Point (Q-Point)

For a transistor to function properly as an amplifier, it must operate in the active region (Base-Emitter junction forward-biased, Base-Collector junction reverse-biased). The DC values of collector current ( $I_C$ ) and collector-emitter voltage ( $V_{CE}$ ) when no AC signal is applied establish the Quiescent Point (Q-point) or Operating Point.

Optimal Placement: For maximum undistorted output swing, the Q-point is ideally placed near the center of the active region on the DC load line.
Shifting Q-Point: If the Q-point shifts towards the saturation region, the peak of the output signal may be clipped. If it shifts towards the cutoff region, the trough of the signal may be clipped.

Need for Bias Stability

The operating point is highly sensitive to variations in temperature and transistor parameters. The primary factors causing Q-point instability are:

Reverse Saturation Current ( $I_{CBO}$ ): Doubles for every $10^\circ C$ rise in temperature. Since $I_C = \beta I_B + (1+\beta)I_{CBO}$ , an increase in $I_{CBO}$ significantly increases $I_C$ .
Base-Emitter Voltage ( $V_{BE}$ ): Decreases at a rate of approximately $-2.5 \text{ mV}/^\circ C$ as temperature rises. A lower $V_{BE}$ increases base current $I_B$ , which in turn increases $I_C$ .
Current Gain ( $\beta$ or $h_{FE}$ ): Varies significantly between transistors of the exact same type due to manufacturing tolerances, and also increases slightly with temperature.

Stability Factors

To quantify how well a biasing circuit stabilizes $I_C$ against these variations, three stability factors are defined:

$S$ (Stability against $I_{CBO}$ ): $S = \frac{\partial I_C}{\partial I_{CBO}}$ (Assuming $V_{BE}$ and $\beta$ constant). Ideal value is 1.
$S'$ (Stability against $V_{BE}$ ): $S' = \frac{\partial I_C}{\partial V_{BE}}$ (Assuming $I_{CBO}$ and $\beta$ constant). Ideal value is 0.
$S''$ (Stability against $\beta$ ): $S'' = \frac{\partial I_C}{\partial \beta}$ (Assuming $I_{CBO}$ and $V_{BE}$ constant). Ideal value is 0.

General formula for S:
$S = \frac{1 + \beta}{1 - \beta \left( \frac{\partial I_B}{\partial I_C} \right)}$

Biasing Circuits

Collector to Base Bias (Collector Feedback Bias)

In this circuit, a base resistor $R_B$ is connected between the collector and the base of the transistor, rather than directly to the supply $V_{CC}$ .

Operation: If temperature increases, $I_C$ increases. This causes an increased voltage drop across the collector resistor $R_C$ . Consequently, the voltage at the collector ( $V_C$ ) decreases. Since $I_B = \frac{V_C - V_{BE}}{R_B}$ , a drop in $V_C$ reduces $I_B$ . The reduction in $I_B$ decreases $I_C$ , counteracting the initial rise.
Stability Factor ( $S$ ): $S \approx \frac{1 + \beta}{1 + \beta(R_C / (R_C + R_B))}$ . This is smaller than $(1+\beta)$ , meaning it provides better stability than a fixed bias circuit.
Disadvantage: Provides AC negative feedback, which reduces the voltage gain of the amplifier.

Emitter Feedback Bias

An emitter resistor $R_E$ is added to the standard fixed-bias circuit.

Operation: If $I_C$ increases, the emitter current $I_E$ ( $\approx I_C$ ) increases, causing a larger voltage drop across $R_E$ . This raises the emitter voltage $V_E$ . Since $V_{BE} = V_B - V_E$ , a higher $V_E$ reduces $V_{BE}$ (if $V_B$ is fixed), which drops $I_B$ and subsequently lowers $I_C$ .
Feedback Mechanism: This is a form of series current negative feedback.

Collector Emitter Feedback Bias

This circuit combines both Collector-to-Base feedback and Emitter feedback by utilizing both $R_B$ connected to the collector and an emitter resistor $R_E$ .

Operation: It offers superior stabilization compared to using either method alone. Any increase in $I_C$ drops the collector voltage and raises the emitter voltage, drastically reducing $V_{CE}$ and strongly pinching off the base current $I_B$ to restore the Q-point.

Self Bias, Emitter Bias, or Voltage Divider Bias

This is the most widely used biasing circuit because it makes the Q-point almost independent of $\beta$ . It uses a voltage divider ( $R_1$ and $R_2$ ) across $V_{CC}$ to bias the base, and an emitter resistor $R_E$ .

Thevenin Equivalent:
- $V_{TH} = V_{CC} \frac{R_2}{R_1 + R_2}$
- $R_{TH} = R_1 || R_2 = \frac{R_1 R_2}{R_1 + R_2}$
KVL at Input Loop: $V_{TH} - I_B R_{TH} - V_{BE} - I_E R_E = 0$
Since $I_E \approx I_C = \beta I_B$ , we can solve for $I_C$ :
$I_C \approx \frac{V_{TH} - V_{BE}}{R_E + (R_{TH}/\beta)}$
Stability Factor ( $S$ ):
$S = (1+\beta) \frac{1 + \frac{R_{TH}}{R_E}}{1 + \beta + \frac{R_{TH}}{R_E}}$
If $R_E \gg R_{TH} / \beta$ , $I_C$ becomes completely independent of $\beta$ . Ideally, $S \approx 1$ .

Stabilization against Variations in $V_{BE}$ and $\beta$ for the Self Bias Circuit

Variations in $\beta$

Manufacturing tolerances mean $\beta$ can vary by a factor of 3 or more. In the self-bias circuit, looking at the equation $I_C \approx \frac{V_{TH} - V_{BE}}{R_E + R_{TH}/\beta}$ , if we design the circuit such that $R_E \gg \frac{R_{TH}}{\beta}$ , the term containing $\beta$ becomes negligible. Thus, $I_C \approx \frac{V_{TH} - V_{BE}}{R_E}$ , meaning $I_C$ is highly stabilized against $\beta$ variations.

Variations in $V_{BE}$

$V_{BE}$ drops by $2.5 \text{ mV}/^\circ C$ . Looking at $I_C \approx \frac{V_{TH} - V_{BE}}{R_E}$ , to stabilize $I_C$ against $V_{BE}$ changes, $V_{TH}$ must be designed to be much larger than $V_{BE}$ (typically $V_{TH} \ge 10 V_{BE}$ ). If $V_{TH} \gg V_{BE}$ , small fluctuations in $V_{BE}$ will have a negligible percentage effect on the numerator, keeping $I_C$ stable.

General Remarks on Collector Current Stability

Negative Feedback is Key: All effective biasing schemes rely on DC negative feedback. When $I_C$ tries to increase, the circuit automatically reduces $I_B$ .
Trade-off with AC Gain: Components that provide DC stability (like $R_E$ or $R_B$ ) also provide AC negative feedback, reducing the amplifier's signal gain.
Bypass Capacitors: To preserve AC gain while maintaining DC stability, bypass capacitors ( $C_E$ ) are used. $C_E$ is placed in parallel with $R_E$ to provide a low-impedance path to ground for AC signals, effectively shorting $R_E$ for AC while keeping it active for DC stabilization.

Thermal Runaway and Thermal Stability

Thermal Runaway

Thermal runaway is a destructive positive feedback loop:

Current flows through the transistor, causing power dissipation at the collector junction ( $P_D \approx V_{CE} I_C$ ).
This power dissipation raises the junction temperature ( $T_j$ ).
An increase in $T_j$ increases the reverse saturation current $I_{CBO}$ .
Since $I_C = \beta I_B + (1+\beta)I_{CBO}$ , $I_C$ increases.
An increase in $I_C$ causes higher power dissipation ( $P_D$ ), which further raises the temperature.
This cycle continues until the junction melts and the transistor is permanently destroyed.

Thermal Stability Condition

To prevent thermal runaway, the rate at which heat is removed from the junction must exceed the rate at which internal heat generation increases with temperature.

Condition: $\frac{\partial P_C}{\partial T_j} < \frac{1}{\theta}$
Where $P_C$ is the power dissipated at the collector and $\theta$ is the thermal resistance (in $^\circ C/W$ ) from junction to ambient.
Prevention: Use of Heat Sinks (lowers $\theta$ ), ensuring $V_{CE} < \frac{V_{CC}}{2}$ , and using bias circuits with low Stability Factors ( $S$ ).

Bias Compensation

When biasing circuits alone cannot provide sufficient stability without unacceptably reducing amplifier gain or requiring too large a power supply, Bias Compensation techniques are used. These use temperature-sensitive devices to counteract transistor parameter variations.

Thermistor and Sensistor Compensation

Thermistor (Negative Temperature Coefficient - NTC): Its resistance decreases as temperature increases.
- Usage: A thermistor $R_T$ can be placed in parallel with $R_2$ in a voltage divider bias circuit. As temperature rises, $I_C$ tries to rise. However, $R_T$ decreases, which lowers the equivalent resistance of the lower half of the voltage divider. This reduces $V_{TH}$ and base voltage, thereby reducing $I_B$ and returning $I_C$ to its original value.
Sensistor (Positive Temperature Coefficient - PTC): Its resistance increases as temperature increases.
- Usage: A sensistor can be placed in place of $R_1$ (the upper resistor in a voltage divider) or in series with the emitter resistor $R_E$ . If placed as $R_1$ , a temperature rise increases its resistance, dropping the voltage at the base, thus reducing $I_B$ and compensating for the rise in $I_C$ .

Diode Compensation

For $V_{BE}$ : A forward-biased diode of the same material as the transistor is placed in the bias circuit. As temperature rises, the diode's voltage drop decreases at the same rate as the transistor's $V_{BE}$ , keeping the bias conditions constant.
For $I_{CBO}$ : A reverse-biased diode is connected to the base. As temperature rises, the leakage current of the diode increases and draws current away from the base, exactly matching the increase in $I_{CBO}$ inside the transistor.

Understanding the Datasheet of Transistors (BC547, BC548, BC557, BC558, BC107)

When reading a transistor datasheet, key parameters include:

$V_{CEO}$ (Collector-Emitter Voltage): Maximum safe voltage between collector and emitter with the base open.
$I_{C(max)}$ : Maximum continuous collector current.
$P_{D(max)}$ : Maximum total power dissipation.
$h_{FE}$ (DC Current Gain or $\beta$ ): Ratio of $I_C$ to $I_B$ . Often grouped by suffix (e.g., A, B, C for different gain bands).
$f_T$ (Transition Frequency): Frequency at which current gain drops to unity (1).

Specific Transistors Overview

BC547 & BC548:
- Type: NPN Epitaxial Silicon Transistor.
- Application: General-purpose switching and amplification (Audio frequency).
- Specs: $I_{C(max)} = 100 \text{ mA}$ . $V_{CEO} = 45\text{V}$ (BC547), $30\text{V}$ (BC548). $P_{D} = 500 \text{ mW}$ .
- Note: Highly common in hobbyist and academic lab circuits.
BC557 & BC558:
- Type: PNP Epitaxial Silicon Transistor.
- Application: Complementary pairs to the BC547 and BC548 respectively (used in Class B/AB push-pull amplifiers).
- Specs: Mirror equivalents of the 547/548 in terms of voltage, current, and power limits, but with opposite polarity.
BC107:
- Type: NPN Transistor in a Metal Can Package (TO-18).
- Application: General-purpose, low-noise amplification, signal processing.
- Specs: $I_{C(max)} = 100 \text{ mA}$ , $V_{CEO} = 45\text{V}$ , $P_{D} = 300 \text{ mW}$ .
- Note: The metal can provides better shielding against electromagnetic interference and slightly different thermal characteristics compared to plastic TO-92 packages.

Introduction to PSpice

What is PSpice?

PSpice (Personal Simulation Program with Integrated Circuit Emphasis) is a widely used analog circuit and digital logic simulation software. It allows engineers and students to design circuits, analyze their behavior under various conditions, and optimize parameters before physical prototyping.

Basic Workflow in PSpice

Schematic Capture: Drawing the circuit using a graphical interface (like OrCAD Capture). Components are placed and wired together.
Netlist Generation: The software translates the graphical schematic into a text-based "netlist" that the SPICE engine can read. The netlist defines nodes and the components connected between them.
Simulation Setup: Defining the type of analysis to be performed.
Running the Simulation: The engine solves differential and algebraic equations to predict circuit behavior.
Waveform Viewing (Probe): Analyzing the output graphs (voltages, currents, power over time or frequency).

Types of Analysis in PSpice

Bias Point (DC Operating Point): Calculates the DC voltages and currents at all nodes and branches. This is crucial for verifying the Q-point in transistor biasing.
DC Sweep: Sweeps a DC source (voltage or current), temperature, or component parameter over a specified range to see the steady-state transfer characteristics (e.g., producing transistor output characteristic curves).
AC Sweep (Frequency Response): Performs a small-signal linear analysis to find the frequency response (Gain and Phase vs. Frequency) of an amplifier. Uses Bode plots.
Transient Analysis: Analyzes the circuit's behavior in the time domain when subjected to time-varying signals (sine waves, pulses).

Example of a Simple SPICE Netlist for a Biased Transistor

SPICE

* Voltage Divider Biased BJT
VCC 1 0 12V
R1 1 2 47k
R2 2 0 10k
RC 1 3 2.2k
RE 4 0 1k
Q1 3 2 4 Q2N2222
.model Q2N2222 NPN(Is=1e-14 Bf=100)
.OP  ; Request DC Operating Point analysis
.END

Node 0 is always ground.
Q1 defines the transistor connections (Collector, Base, Emitter) and its model.
.OP commands the engine to calculate the bias stability and operating point.

Unit 3

Unit 5