Unit 5 - Notes

CSE212 8 min read

Unit 5: Transistor Hybrid Models and Multistage Amplifiers

1. Two Port Devices and the Hybrid Model

A transistor can be treated as a two-port network. A two-port network has an input port (two terminals) and an output port (two terminals). For small-signal, low-frequency operation, transistors are often modeled using hybrid parameters (h-parameters) because they offer a highly accurate representation of the transistor's behavior and are easily measurable.

Variables in a Two-Port Network

For any two-port network, there are four key variables:

$V_1$ : Input Voltage
$I_1$ : Input Current
$V_2$ : Output Voltage
$I_2$ : Output Current

In the hybrid model, $I_1$ and $V_2$ are considered independent variables, while $V_1$ and $I_2$ are dependent variables.

The Hybrid Equations

The relationship between these variables is given by the following linear equations:

$V_1 = h_{11}I_1 + h_{12}V_2$
$I_2 = h_{21}I_1 + h_{22}V_2$

Where the h-parameters are defined as follows:

$h_{11}$ ( $h_i$ ): Input impedance with output short-circuited (). Units: Ohms ().
- $h_{11} = \frac{V_1}{I_1} \Big|_{V_2 = 0}$
$h_{12}$ ( $h_r$ ): Reverse voltage transfer ratio with input open-circuited (). Units: Dimensionless.
- $h_{12} = \frac{V_1}{V_2} \Big|_{I_1 = 0}$
$h_{21}$ ( $h_f$ ): Forward current gain with output short-circuited (). Units: Dimensionless.
- $h_{21} = \frac{I_2}{I_1} \Big|_{V_2 = 0}$
$h_{22}$ ( $h_o$ ): Output admittance with input open-circuited (). Units: Siemens (S) or Mhos ().
- $h_{22} = \frac{I_2}{V_2} \Big|_{I_1 = 0}$

Note: The model is called "hybrid" because the parameters have mixed dimensions (impedance, admittance, and dimensionless ratios).

2. Determination of the h-parameters from the Characteristics

The h-parameters can be determined graphically from the static characteristic curves of a transistor. We will use the Common Emitter (CE) configuration as an example, where $V_1 = V_{BE}$ , $I_1 = I_B$ , $V_2 = V_{CE}$ , and $I_2 = I_C$ .

A. From Input Characteristics ( $V_{BE}$ vs. $I_B$ at constant $V_{CE}$ )

The input characteristic curve plots Input Current ( $I_B$ ) on the y-axis against Input Voltage ( $V_{BE}$ ) on the x-axis for various fixed values of Output Voltage ( $V_{CE}$ ).

Finding $h_{ie}$ (Input Impedance):
- Keep $V_{CE}$ constant.
- Measure the change in $V_{BE}$ ( $\Delta V_{BE}$ ) for a small change in $I_B$ ( $\Delta I_B$ ) around the operating point (Q-point).
- $h_{ie} = \frac{\Delta V_{BE}}{\Delta I_B} \Big|_{V_{CE} = \text{constant}}$
Finding $h_{re}$ (Reverse Voltage Ratio):
- Keep $I_B$ constant.
- Measure the change in $V_{BE}$ ( $\Delta V_{BE}$ ) for a small change in $V_{CE}$ ( $\Delta V_{CE}$ ).
- $h_{re} = \frac{\Delta V_{BE}}{\Delta V_{CE}} \Big|_{I_B = \text{constant}}$

B. From Output Characteristics ( $I_C$ vs. $V_{CE}$ at constant $I_B$ )

The output characteristic curve plots Output Current ( $I_C$ ) on the y-axis against Output Voltage ( $V_{CE}$ ) on the x-axis for various fixed values of Input Current ( $I_B$ ).

Finding $h_{fe}$ (Forward Current Gain):
- Keep $V_{CE}$ constant.
- Measure the change in $I_C$ ( $\Delta I_C$ ) for a small change in $I_B$ ( $\Delta I_B$ ) around the Q-point.
- $h_{fe} = \frac{\Delta I_C}{\Delta I_B} \Big|_{V_{CE} = \text{constant}}$
Finding $h_{oe}$ (Output Admittance):
- Keep $I_B$ constant.
- Measure the change in $I_C$ ( $\Delta I_C$ ) for a small change in $V_{CE}$ ( $\Delta V_{CE}$ ). This is the slope of the output characteristic curve.
- $h_{oe} = \frac{\Delta I_C}{\Delta V_{CE}} \Big|_{I_B = \text{constant}}$

3. Analysis of a Transistor Amplifier Circuit using h-parameters

To analyze an amplifier, the transistor is replaced by its h-parameter equivalent circuit. The external circuit components (source voltage $V_S$ , source resistance $R_S$ , load resistance $R_L$ ) are then connected to this model.

Standard Assumptions & Variables:

$Z_L$ : Load Impedance
$Z_S$ : Source Impedance
$V_S$ : Source Voltage

1. Current Gain ( $A_i$ )

Current gain is the ratio of output current to input current.
$A_i = \frac{I_2}{I_1}$
Using the output equation: $I_2 = h_f I_1 + h_o V_2$ and knowing that $V_2 = -I_2 Z_L$ (the negative sign indicates current leaving the port):
$I_2 = h_f I_1 - h_o I_2 Z_L$
$I_2 (1 + h_o Z_L) = h_f I_1$
$A_i = \frac{-h_f}{1 + h_o Z_L}$

2. Input Impedance ( $Z_i$ )

Input impedance is the ratio of input voltage to input current.
$Z_i = \frac{V_1}{I_1}$
Using the input equation: $V_1 = h_i I_1 + h_r V_2$ . Substitute $V_2 = -I_2 Z_L = A_i I_1 Z_L$ :
$V_1 = h_i I_1 + h_r (A_i I_1 Z_L)$
$Z_i = h_i + h_r A_i Z_L$

3. Voltage Gain ( $A_v$ )

Voltage gain is the ratio of output voltage to input voltage.
$A_v = \frac{V_2}{V_1}$
Since $V_2 = -I_2 Z_L = A_i I_1 Z_L$ and $V_1 = I_1 Z_i$ :
$A_v = \frac{A_i I_1 Z_L}{I_1 Z_i}$
$A_v = \frac{A_i Z_L}{Z_i}$

4. Output Impedance ( $Z_o$ )

Output impedance is measured looking back into the output terminals with the source voltage $V_S$ set to zero (shorted) and the load disconnected.
$Z_o = \frac{V_2}{I_2} \Big|_{V_S = 0}$
By solving the input loop with $V_S = 0$ : $I_1(R_S + h_i) + h_r V_2 = 0 \implies I_1 = \frac{-h_r V_2}{R_S + h_i}$
Substitute $I_1$ into the output equation $I_2 = h_f I_1 + h_o V_2$ :
$Y_o = \frac{1}{Z_o} = h_o - \frac{h_f h_r}{h_i + R_S}$

4. Comparison of Transistor Amplifier Configurations

Transistors can be configured in three ways depending on which terminal is common to both input and output. The configurations dictate the amplifier's characteristics.

Parameter	Common Base (CB)	Common Emitter (CE)	Common Collector (CC) / Emitter Follower
Input Impedance ( $Z_i$ )	Very Low ( $\approx 20\Omega - 50\Omega$ )	Medium ( $\approx 1k\Omega - 2k\Omega$ )	High ( $\approx 50k\Omega - 500k\Omega$ )
Output Impedance ( $Z_o$ )	Very High ( $\approx 1M\Omega - 2M\Omega$ )	Medium ( $\approx 40k\Omega - 50k\Omega$ )	Low ( $\approx 50\Omega - 100\Omega$ )
Current Gain ( $A_i$ )	Less than unity ( $\alpha \approx 0.99$ )	High ( $\beta \approx 50 - 300$ )	High ( $\approx 1 + \beta$ )
Voltage Gain ( $A_v$ )	High ( $\approx 100 - 500$ )	High ( $\approx 100 - 500$ )	Less than unity ( $\approx 0.99$ )
Phase Shift	$0^\circ$ (In-phase)	$180^\circ$ (Out of phase)	$0^\circ$ (In-phase)
Primary Application	High-frequency applications, Current buffer	General purpose audio frequency (AF) amplifiers	Impedance matching, Voltage buffer

5. Cascading Transistor Amplifiers

A single transistor amplifier rarely provides sufficient gain (voltage or current) or the exact input/output impedance required for practical applications (e.g., public address systems, radio receivers). To achieve the desired specifications, multiple amplifier stages are connected in series. This is called cascading.

Coupling Methods

When cascading, the output of one stage must be connected to the input of the next without disturbing the DC biasing conditions (Q-point) of either stage.

RC (Resistor-Capacitor) Coupling:
- Uses a coupling capacitor ( $C_C$ ) to block DC and pass AC.
- Pros: Cheap, compact, excellent audio frequency response.
- Cons: Poor low-frequency response due to capacitor reactance; poor impedance matching.
Transformer Coupling:
- Uses a transformer to link stages.
- Pros: Excellent impedance matching (allows maximum power transfer), high efficiency.
- Cons: Bulky, expensive, poor frequency response (only good for specific RF/IF frequencies).
Direct Coupling:
- The output of one stage is directly connected to the input of the next (no blocking capacitors).
- Pros: Can amplify extremely low frequencies (including DC).
- Cons: High temperature instability (thermal runaway is easily propagated), DC drift.

Effects of Cascading

Loading Effect: The input impedance of the second stage acts as a parallel load to the output impedance of the first stage, effectively reducing the voltage gain of the first stage.
Increased Gain: Overall gain is vastly increased.
Reduced Bandwidth: As stages are added, the overall bandwidth of the amplifier decreases.

6. n-Stage Cascaded Amplifier

In an n-stage cascaded amplifier, the signal passes through $n$ sequential amplification stages.

Overall Gain

Let $A_{v1}, A_{v2}, \dots, A_{vn}$ be the voltage gains of the individual stages (taking into account the loading effect of the subsequent stages).

Overall Voltage Gain ( $A_v$ ):
$A_v = A_{v1} \times A_{v2} \times \dots \times A_{vn}$
Overall Gain in Decibels (dB):
Because gains multiply, when expressed in dB, the total gain is the sum of the individual gains.
$A_{v(dB)} = A_{v1(dB)} + A_{v2(dB)} + \dots + A_{vn(dB)}$

The same multiplicative property applies to the Overall Current Gain ( $A_i$ ).

Bandwidth Shrinkage

When identical amplifier stages are cascaded, the overall lower cutoff frequency ( $f_L^*$ ) increases and the upper cutoff frequency ( $f_H^*$ ) decreases, resulting in a reduced overall bandwidth.

For $n$ identical stages, the cutoff frequencies are modified by a shrinkage factor:
$\text{Shrinkage Factor} = \sqrt{2^{1/n} - 1}$

New Upper Cutoff Frequency ( $f_H^*$ ):
$f_H^* = f_H \sqrt{2^{1/n} - 1}$
(High frequency response degrades / drops earlier)
New Lower Cutoff Frequency ( $f_L^*$ ):
$f_L^* = \frac{f_L}{\sqrt{2^{1/n} - 1}}$
(Low frequency response degrades / starts later)

Therefore, the overall bandwidth $BW^* \approx f_H^*$ (since $f_L^*$ is usually much smaller than $f_H^*$ ), meaning cascading amplifiers significantly reduces the system's bandwidth while exponentially increasing its gain.

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