Stability Factor Analysis Fixed Bias And Voltage Divider Bias Circuits
Introduction
In the realm of electronics, particularly in amplifier design, stability is a paramount concern. A stable amplifier ensures that its operating point, or quiescent point (Q-point), remains consistent despite variations in temperature, transistor parameters (such as β), and supply voltage. These variations can significantly affect the amplifier's performance, leading to distortion, clipping, or even thermal runaway. To quantify this stability, we use a metric called the stability factor (S). The stability factor indicates the sensitivity of the collector current (I_C) to changes in various parameters. This article delves into the derivation of stability factors for the fixed bias circuit and analyzes a voltage divider bias circuit, providing a comprehensive understanding of these crucial concepts.
3. (a) Stability Factor Derivation for Fixed Bias Circuit
The fixed bias circuit, while simple in its configuration, is notoriously susceptible to variations in transistor parameters. This is primarily due to its high dependence on the transistor's current gain (β) and the reverse saturation current (I_CO). The stability factor, in this context, helps us quantify how much the collector current (I_C) changes with respect to changes in I_CO, V_BE, and β. Understanding these stability factors is crucial for designing reliable and predictable amplifier circuits. Let's delve into the derivation of the stability factor for a fixed bias circuit with respect to I_CO, V_BE, and β.
Stability Factor with Respect to I_CO (S(I_CO))
The stability factor with respect to I_CO, denoted as S(I_CO), quantifies the change in collector current (I_C) for a given change in the reverse saturation current (I_CO). Mathematically, it is expressed as:
S(I_CO) = dI_C / dI_CO
To derive this, we start with the collector current equation for a BJT:
I_C = βI_B + (1 + β)I_CO
In a fixed bias circuit, the base current (I_B) is determined by the base resistor (R_B) and the supply voltage (V_CC) and is relatively independent of I_C. Therefore, we can differentiate the above equation with respect to I_CO, keeping I_B constant:
dI_C / dI_CO = (1 + β)
Thus, the stability factor with respect to I_CO for a fixed bias circuit is:
S(I_CO) = 1 + β
This result highlights a significant drawback of the fixed bias configuration: the stability factor is directly proportional to β. For transistors with high β values, even small changes in I_CO can lead to substantial variations in I_C, making the circuit highly unstable. This high sensitivity to I_CO makes the fixed bias circuit less desirable for applications requiring stable operation over a wide temperature range.
Stability Factor with Respect to V_BE (S(V_BE))
The base-emitter voltage (V_BE) also influences the collector current, especially with temperature variations. The stability factor with respect to V_BE, S(V_BE), indicates the change in I_C for a change in V_BE and is defined as:
S(V_BE) = dI_C / dV_BE
To derive this, we need to consider the relationship between I_C and V_BE. The collector current can also be expressed using the Shockley diode equation (approximated for a BJT):
I_C ≈ I_S * exp(V_BE / V_T)
Where I_S is the saturation current and V_T is the thermal voltage (approximately 26 mV at room temperature). However, for the fixed bias configuration, it's more practical to relate V_BE to the base current. The base current is given by:
I_B = (V_CC - V_BE) / R_B
Substituting this into the collector current equation (I_C = βI_B + (1 + β)I_CO) and differentiating with respect to V_BE, we get:
dI_C / dV_BE = -β / R_B
Therefore, the stability factor with respect to V_BE is:
S(V_BE) = -β / R_B
The negative sign indicates that an increase in V_BE leads to a decrease in I_C. The magnitude of S(V_BE) depends on β and R_B. A higher β and a lower R_B result in a larger change in I_C for a given change in V_BE, making the circuit more sensitive to V_BE variations. This further emphasizes the instability issues associated with the fixed bias circuit.
Stability Factor with Respect to β (S(β))
The current gain (β) is a crucial parameter that varies significantly between transistors and is also temperature-dependent. The stability factor with respect to β, S(β), measures the change in collector current (I_C) for a given change in β:
S(β) = dI_C / dβ
Starting with the collector current equation:
I_C = βI_B + (1 + β)I_CO
Differentiating with respect to β, we get:
dI_C / dβ = I_B + I_CO
Thus, the stability factor with respect to β for a fixed bias circuit is:
S(β) = I_B + I_CO
Since I_CO is usually much smaller than I_B, we can approximate this as:
S(β) ≈ I_B
This indicates that the change in collector current with respect to β is approximately equal to the base current. Higher base current implies a greater sensitivity to variations in β. This high dependence on β is a major limitation of the fixed bias circuit, as even small changes in β can cause significant shifts in the Q-point, leading to unpredictable circuit behavior. In summary, the fixed bias circuit's stability factors reveal its inherent sensitivity to variations in I_CO, V_BE, and β. The high values of S(I_CO) and S(β), and the dependence of S(V_BE) on β and R_B, make the fixed bias configuration unsuitable for applications requiring stable and predictable performance.
3. (b) Analysis of a Voltage Divider Biased Circuit
The voltage divider bias circuit is a widely used biasing technique known for its improved stability compared to the fixed bias configuration. This enhanced stability is achieved by making the bias point less sensitive to variations in transistor parameters, particularly β. The voltage divider bias circuit employs a voltage divider network (comprising resistors R1 and R2) to establish a stable base voltage, which in turn stabilizes the collector current. This section analyzes a specific voltage divider biased circuit to determine its operating point and stability characteristics.
Circuit Parameters and Given Values
Consider a voltage divider biased circuit with the following parameters:
- R1 = 39 kΩ
- R2 = 82 kΩ
- RC = 3.3 kΩ
- RE = 1 kΩ
- VCC = 18 V
- β = 120 (for the silicon transistor)
Our goal is to find the key operating point parameters, including the base voltage (V_B), emitter voltage (V_E), collector current (I_C), collector-emitter voltage (V_CE), and subsequently assess the circuit's stability.
Calculation of Base Voltage (V_B)
The base voltage (V_B) is determined by the voltage divider network formed by R1 and R2. Using the voltage divider formula:
V_B = V_CC * (R2 / (R1 + R2))
Substituting the given values:
V_B = 18 V * (82 kΩ / (39 kΩ + 82 kΩ)) V_B = 18 V * (82 / 121) V_B ≈ 12.18 V
The base voltage is approximately 12.18 V. This voltage serves as the reference for establishing the bias point of the transistor.
Calculation of Emitter Voltage (V_E)
The emitter voltage (V_E) is related to the base voltage by the base-emitter voltage drop (V_BE). For a silicon transistor, V_BE is typically around 0.7 V. Therefore:
V_E = V_B - V_BE V_E = 12.18 V - 0.7 V V_E ≈ 11.48 V
The emitter voltage is approximately 11.48 V. This voltage, along with the emitter resistance (RE), determines the emitter current (I_E).
Calculation of Emitter Current (I_E)
The emitter current (I_E) is calculated using Ohm's law applied to the emitter resistor (RE):
I_E = V_E / RE I_E = 11.48 V / 1 kΩ I_E ≈ 11.48 mA
The emitter current is approximately 11.48 mA. In a transistor, the collector current (I_C) is approximately equal to the emitter current (I_E), especially for high β values. Therefore, we can approximate I_C ≈ I_E.
Approximation of Collector Current (I_C)
As mentioned, the collector current (I_C) is approximately equal to the emitter current (I_E):
I_C ≈ I_E ≈ 11.48 mA
For a more precise calculation, we can use the relationship:
I_C = αI_E
Where α is the common-base current gain, related to β by:
α = β / (β + 1) α = 120 / (120 + 1) α ≈ 0.992
So,
I_C = 0.992 * 11.48 mA I_C ≈ 11.39 mA
The collector current is approximately 11.39 mA. This value is crucial for determining the Q-point and the voltage drops across the collector resistor (RC).
Calculation of Collector-Emitter Voltage (V_CE)
The collector-emitter voltage (V_CE) is the voltage drop across the transistor from the collector to the emitter. It can be calculated using Kirchhoff's Voltage Law (KVL) around the collector-emitter loop:
V_CC = I_C * RC + V_CE + I_E * RE
Since I_C ≈ I_E, we can approximate:
V_CC ≈ I_C * RC + V_CE + I_C * RE V_CE = V_CC - I_C * (RC + RE)
Substituting the values:
V_CE = 18 V - 11.39 mA * (3.3 kΩ + 1 kΩ) V_CE = 18 V - 11.39 mA * 4.3 kΩ V_CE = 18 V - 48.98 V V_CE ≈ -30.98 V
There seems to be an error in the calculation, as V_CE cannot be negative. Let's re-evaluate the calculation:
V_CE = 18 V - (0.01139 A * (3300 Ω + 1000 Ω)) V_CE = 18 V - (0.01139 A * 4300 Ω) V_CE = 18 V - 48.977 V V_CE ≈ -30.977 V
The error likely stems from an overestimation of I_C. Let's recalculate I_C more precisely using the initial I_E value and the current gain: Given I_E = 11.48 mA, we use the relation I_C = αI_E where α = β/(β+1) = 120/121 ≈ 0.9917.
Thus, I_C ≈ 0.9917 * 11.48 mA ≈ 11.38 mA. Now, calculate V_CE again: V_CE = V_CC - I_C * RC - I_E * RE V_CE = 18 - (11.38 * 10^-3 * 3300) - (11.48 * 10^-3 * 1000) V_CE = 18 - 37.554 - 11.48 V_CE ≈ -31.034 V The error persists, indicating a potential issue in the problem statement or the component values. The calculated V_CE value is negative, which is physically impossible in a properly biased BJT amplifier. This situation typically arises when the assumed operating point drives the transistor into saturation or cutoff regions incorrectly, but in a real-world scenario, it often indicates an issue like incorrect resistance values relative to V_CC, which are causing a miscalculation. However, given the values, the correct approach to calculating V_CE is as follows:
V_CE = V_CC - I_C * RC - I_E * RE V_CE = 18 - (0.01138 * 3300) - (0.01148 * 1000) V_CE ≈ 18 - 37.554 - 11.48 V_CE ≈ -31.034 V
Given the constraints and the typical behavior of such a circuit, it’s crucial to re-evaluate the component values or the supply voltage to ensure that the transistor operates in the active region.
Discussion on the Results and Circuit Stability
Based on the calculations, there's an anomaly indicating that the transistor is not operating correctly within the active region, primarily because the calculated V_CE is negative, an impossible physical state for a BJT in normal amplifier operation. This suggests that there may be a problem with the choice of resistor values relative to the supply voltage, potentially causing the transistor to be in saturation or cutoff. Voltage divider bias, in general, offers better stability than fixed bias because changes in transistor parameters (like β) have less impact on the Q-point. The resistors R1 and R2 provide a stable base voltage, and the emitter resistor RE introduces negative feedback, which helps to stabilize the collector current against changes in temperature and transistor characteristics. However, the component values must be chosen carefully to ensure proper biasing.
Improvements and Considerations
To ensure the circuit operates correctly, the resistor values need to be chosen such that the transistor operates in the active region. Typically, this involves selecting values such that V_CE is a positive value and lies approximately between 1/3 to 2/3 of V_CC. In practical design, it is important to:
- Re-evaluate Resistor Values: Adjust RC, RE, R1, and R2 to achieve a stable and appropriate Q-point.
- Consider Transistor Specifications: Ensure the chosen transistor can handle the calculated currents and voltages.
- Thermal Stability: Analyze and ensure thermal stability to avoid thermal runaway, especially at higher temperatures.
The voltage divider bias configuration offers a significant improvement in stability over fixed bias, but the selection of component values is critical for ensuring proper operation and optimal performance. The negative V_CE suggests that a design review is necessary to adjust the biasing conditions.
Conclusion
Understanding the stability factors of different biasing configurations is crucial for designing robust and reliable amplifier circuits. While the fixed bias circuit offers simplicity, its high sensitivity to variations in transistor parameters makes it less suitable for most practical applications. The voltage divider bias, on the other hand, provides significantly improved stability by reducing the circuit's dependence on β and temperature variations. However, careful selection of component values is essential to ensure the transistor operates in the active region and achieves the desired performance characteristics. The analysis of the voltage divider biased circuit highlights the importance of accurate calculations and the need for design reviews to ensure the circuit operates as intended. By understanding these principles, engineers can design amplifiers that deliver stable and predictable performance across a wide range of operating conditions.
