Unit 5 - Notes

ECE206 12 min read

Unit 5: Transistor Hybrid Models and Multistage Amplifiers

1. Two Port Devices and the Hybrid Model

A two-port network is an electrical circuit with two pairs of terminals: an input port and an output port. A transistor, when used in an amplifier circuit, can be modeled as a two-port network to analyze its small-signal AC behavior.

1.1 The Two-Port Network Concept

A general two-port network (or "black box") has four variables:

Input Voltage ( $v_1$ )
Input Current ( $i_1$ )
Output Voltage ( $v_2$ )
Output Current ( $i_2$ )

Out of these four variables, two can be chosen as independent variables, and the other two become dependent variables. This leads to different sets of parameters (z, y, h, g parameters) to describe the network's behavior.

1.2 The Hybrid (h) Parameters

The Hybrid (h) parameters are particularly suitable for modeling transistors because they are easy to measure from the static characteristic curves. In this model, the input current ( $i_1$ ) and output voltage ( $v_2$ ) are chosen as the independent variables. The input voltage ( $v_1$ ) and output current ( $i_2$ ) are the dependent variables.

The relationship is defined by the following two linear equations:

$v_1 = h_{11}i_1 + h_{12}v_2$
$i_2 = h_{21}i_1 + h_{22}v_2$

These equations describe the relationship between the AC voltages and currents for the two-port network.

1.3 Defining the h-Parameters

The four h-parameters are defined by setting one of the independent variables to zero (which corresponds to an AC short circuit for current or an AC open circuit for voltage).

$h_{11}$ - Input Impedance (unit: Ohms, Ω)
- Defined as the ratio of input voltage to input current when the output is AC short-circuited ( $v_2 = 0$ ).
- $h_{11} = \frac{v_1}{i_1} \bigg|_{v_2=0}$
- Also denoted as $h_i$ (input impedance).
$h_{12}$ - Reverse Voltage Gain (unitless)
- Defined as the ratio of input voltage to output voltage when the input is AC open-circuited ( $i_1 = 0$ ).
- $h_{12} = \frac{v_1}{v_2} \bigg|_{i_1=0}$
- Also denoted as $h_r$ (reverse transfer voltage ratio). It represents the feedback from the output to the input.
$h_{21}$ - Forward Current Gain (unitless)
- Defined as the ratio of output current to input current when the output is AC short-circuited ( $v_2 = 0$ ).
- $h_{21} = \frac{i_2}{i_1} \bigg|_{v_2=0}$
- Also denoted as $h_f$ (forward transfer current ratio). This is the current gain of the transistor.
$h_{22}$ - Output Admittance (unit: Siemens, S or mhos, ℧)
- Defined as the ratio of output current to output voltage when the input is AC open-circuited ( $i_1 = 0$ ).
- $h_{22} = \frac{i_2}{v_2} \bigg|_{i_1=0}$
- Also denoted as $h_o$ (output admittance). Its reciprocal ( $1/h_o$ ) is the output impedance.

The name "hybrid" comes from the mixed units of the parameters (Ohms, Siemens, and unitless ratios).

1.4 The h-Parameter Equivalent Circuit

The two defining equations can be represented by an equivalent circuit model.

The first equation ( $v_1 = h_i i_1 + h_r v_2$ ) corresponds to a Kirchhoff's Voltage Law (KVL) equation at the input loop. It represents a resistance ( $h_i$ ) in series with a voltage source ( $h_r v_2$ ) controlled by the output voltage.
The second equation ( $i_2 = h_f i_1 + h_o v_2$ ) corresponds to a Kirchhoff's Current Law (KCL) equation at the output node. It represents a current source ( $h_f i_1$ ) controlled by the input current, in parallel with an admittance ( $h_o$ ).

This leads to the general h-parameter model:

2. Determination of the h parameters from the characteristics

The h-parameters for a specific transistor configuration (e.g., Common Emitter) can be determined graphically from its static characteristic curves. The parameters are defined as partial derivatives, which are approximated by the slope of the curve at the operating point (Q-point).

The second subscript in the h-parameter notation indicates the configuration: e for Common Emitter, b for Common Base, and c for Common Collector.

For Common Emitter (CE) Configuration:

Input port: Base-Emitter ( $v_1 = v_{be}, i_1 = i_b$ )
Output port: Collector-Emitter ( $v_2 = v_{ce}, i_2 = i_c$ )

The defining equations become:
$v_{be} = h_{ie}i_b + h_{re}v_{ce}$
$i_c = h_{fe}i_b + h_{oe}v_{ce}$

1. Determining $h_{ie}$ and $h_{re}$ from Input Characteristics ( $V_{BE}$ vs $I_B$ )

$h_{ie}$ (Input Impedance):
$h_{ie} = \frac{\Delta V_{BE}}{\Delta I_B} \bigg|_{V_{CE} = \text{constant}}$
This is the reciprocal of the slope of the input characteristic curve at the Q-point.
$h_{re}$ (Reverse Voltage Gain):
$h_{re} = \frac{\Delta V_{BE}}{\Delta V_{CE}} \bigg|_{I_B = \text{constant}}$
This is the change in input voltage required to keep the input current constant when the output voltage changes. It's found by measuring the shift in the input curve for a change in $V_{CE}$ . This value is typically very small.

2. Determining $h_{fe}$ and $h_{oe}$ from Output Characteristics ( $I_C$ vs $V_{CE}$ )

$h_{fe}$ (Forward Current Gain):
$h_{fe} = \frac{\Delta I_C}{\Delta I_B} \bigg|_{V_{CE} = \text{constant}}$
This is the change in collector current for a given change in base current at a constant collector-emitter voltage. It is found by measuring the vertical distance between two adjacent curves at the Q-point's $V_{CE}$ .
$h_{oe}$ (Output Admittance):
$h_{oe} = \frac{\Delta I_C}{\Delta V_{CE}} \bigg|_{I_B = \text{constant}}$
This is the slope of the output characteristic curve at the Q-point. The curves are nearly flat, so $h_{oe}$ is very small (meaning output impedance is very high).

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

The h-parameter model is used to analyze the small-signal AC performance of an amplifier. We will analyze a standard Common Emitter (CE) amplifier with voltage divider bias.

Steps for AC Analysis:

Draw the DC equivalent circuit to find the Q-point ( $I_{CQ}, V_{CEQ}$ ).
Draw the AC equivalent circuit:
- Replace DC voltage sources ( $V_{CC}$ ) with ground (AC short).
- Replace capacitors ( $C_1, C_2, C_E$ ) with short circuits (assuming their reactance is negligible at the operating frequency).
- Replace the transistor with its h-parameter model.
Derive expressions for gain, input impedance, and output impedance.

AC Equivalent Circuit for a CE Amplifier

Where:

$R_B = R_1 || R_2$ (biasing resistors)
$R_L$ is the load resistor.
$R_S$ is the source internal resistance.

Derivations for the CE Amplifier

1. Current Gain ( $A_i$ )

The current gain of the amplifier stage is defined as $A_i = \frac{i_L}{i_i} = \frac{-i_c}{i_b}$ . (The negative sign indicates a 180° phase shift).
From the output loop of the h-parameter model:
$i_c = h_{fe}i_b + h_{oe}v_{ce}$
The output voltage is $v_{ce} = -i_c R_C$ . Substituting this:
$i_c = h_{fe}i_b - h_{oe}(i_c R_C)$
$i_c(1 + h_{oe}R_C) = h_{fe}i_b$
Therefore, the current gain is:
$A_i = \frac{i_c}{i_b} = \frac{h_{fe}}{1 + h_{oe}R_C}$

2. Input Impedance ( $Z_i$ )

$Z_i$ is the impedance looking into the base of the transistor: $Z_i = \frac{v_{in}}{i_b} = \frac{v_{be}}{i_b}$ .
From the input loop of the h-parameter model:
$v_{be} = h_{ie}i_b + h_{re}v_{ce}$
Substitute $v_{ce} = -i_c R_C = -(A_i i_b)R_C$ :
$v_{be} = h_{ie}i_b + h_{re}(-A_i i_b R_C)$
$v_{be} = i_b(h_{ie} - h_{re}A_i R_C)$
Therefore, the input impedance is:
$Z_i = \frac{v_{be}}{i_b} = h_{ie} - h_{re}A_i R_C$
Total Input Impedance of the Circuit ( $Z_{in}$ ): This is what the source "sees". It is the parallel combination of the biasing resistors and $Z_i$ .
$Z_{in} = R_B || Z_i = (R_1 || R_2) || Z_i$

3. Voltage Gain ( $A_v$ )

Voltage gain is defined as $A_v = \frac{v_o}{v_{in}} = \frac{v_{ce}}{v_{be}}$ .
We know $v_{ce} = -i_c R_C = -(A_i i_b) R_C$ .
We also know $v_{be} = i_b Z_i$ .
Dividing the two equations:
$A_v = \frac{-A_i i_b R_C}{i_b Z_i} = \frac{-A_i R_C}{Z_i}$
Substituting the expressions for $A_i$ and $Z_i$ gives the full formula, but this form is often more useful.

4. Output Impedance ( $Z_o$ )

$Z_o$ is the impedance looking back into the collector terminal (with the load $R_L$ removed). To find it, we set the independent source ( $v_s$ ) to zero and apply a test voltage ( $v_x$ ) at the output, then find the resulting current ( $i_x$ ). Then $Z_o = v_x / i_x$ .
When $v_s = 0$ , $i_b = 0$ . This simplifies the h-parameter model.
The current source $h_{fe}i_b$ becomes zero (an open circuit).
The circuit simplifies to the admittance $h_{oe}$ in parallel with the collector resistor $R_C$ .
Therefore, the output impedance is:
$Z_o = \frac{1}{h_{oe}} || R_C$
Note: This is an approximation. A more rigorous derivation includes the effect of $h_{re}$ and $R_S$ , but for most practical cases, this is accurate enough. The impedance of the transistor itself is $1/h_{oe}$ .

4. Comparison of Transistor Amplifier Configurations

Transistors can be connected in three basic configurations: Common Emitter (CE), Common Base (CB), and Common Collector (CC). Each has distinct characteristics that make it suitable for different applications.

Parameter	Common Emitter (CE)	Common Base (CB)	Common Collector (CC)
Current Gain ( $A_i$ )	High (e.g., > 50)	Low (slightly less than 1, ~ $h_{fb}$ )	High (e.g., > 50)
Voltage Gain ( $A_v$ )	High (e.g., > 100)	High (e.g., > 100)	Low (slightly less than 1)
Power Gain	Very High (Highest)	Moderate	Low
Input Impedance ( $Z_i$ )	Moderate (e.g., 1-2 kΩ)	Very Low (e.g., 20-100 Ω)	Very High (e.g., > 100 kΩ)
Output Impedance ( $Z_o$ )	High (e.g., 40-50 kΩ)	Very High (e.g., > 1 MΩ)	Very Low (e.g., < 100 Ω)
Phase Shift	180° (output inverted)	0° (in phase)	0° (in phase)
Primary Application	General purpose voltage/current amp	High-frequency amp, Impedance matching	Buffer, Impedance matching
Summary	Provides both voltage and current gain. Most widely used configuration.	Provides voltage gain but no current gain. Good for matching low to high impedance.	Provides current gain but no voltage gain. Good for matching high to low impedance. Also called an "Emitter Follower".

5. Cascading Transistor Amplifiers

A single amplifier stage often does not provide sufficient gain or the desired input/output impedance characteristics. To overcome this, multiple amplifier stages are connected in series, or cascaded. The output of one stage becomes the input of the next.

Need for Cascading

Increased Gain: The primary reason is to achieve a much larger overall voltage or current gain.
Impedance Matching: To connect a source with high internal impedance to a low impedance load, or vice-versa. For example, a CE stage (high gain) might be followed by a CC stage (low output impedance) to drive a low-impedance load effectively.
Frequency Response: To achieve a desired bandwidth or frequency response characteristic.

Coupling Methods

The way stages are connected is called coupling. The coupling network must pass the AC signal from one stage to the next while blocking the DC bias of one stage from affecting the next.

RC (Resistor-Capacitor) Coupling:
- The most common method.
- A coupling capacitor ( $C_c$ ) connects the collector of one stage to the base of the next.
- Advantages: Simple, inexpensive, provides good frequency response for audio frequencies.
- Disadvantages: Poor low-frequency response (capacitor blocks low frequencies/DC), cannot be used for amplifying DC signals.
Transformer Coupling:
- A transformer is used to couple the output of one stage to the input of the next.
- Advantages: Excellent impedance matching capabilities (by choosing the turns ratio), can lead to higher gain.
- Disadvantages: Bulky, expensive, limited frequency response (poor at very low and very high frequencies). Often used in power amplifiers and radio-frequency (RF) circuits.
Direct Coupling:
- The output of one stage is directly connected to the input of the next stage without any coupling components.
- Advantages: Excellent low-frequency and DC amplification, simple circuit layout.
- Disadvantages: Prone to thermal drift and instability. Any change in the Q-point of one stage affects all subsequent stages. Used in operational amplifiers (op-amps).

6. n-Stage Cascaded Amplifier

When n amplifier stages are cascaded, the overall gain is the product of the individual stage gains.

Overall Voltage Gain ( $A_v$ )

Consider an n-stage amplifier where $A_{v1}, A_{v2}, ..., A_{vn}$ are the voltage gains of the individual stages.

$A_{v(\text{total})} = A_{v1} \times A_{v2} \times \dots \times A_{vn}$

Important Note: The gain of each stage ( $A_{vk}$ ) depends on the input impedance of the next stage ( $Z_{in(k+1)}$ ), which acts as the load for the current stage.

$A_{v1}$ is calculated with $Z_{in2}$ as its load.
$A_{v2}$ is calculated with $Z_{in3}$ as its load.
...and so on.
$A_{vn}$ is calculated with the final load resistor $R_L$ as its load.

So, $A_{vk} = \frac{-A_{ik} \times R_{Lk}}{Z_{ik}}$ , where the load for stage k, $R_{Lk}$ , is the input impedance of stage k+1, $Z_{in(k+1)}$ .

Gain in Decibels (dB)

Gains are often expressed in decibels (dB), which is a logarithmic scale. This is convenient because it turns multiplication of gains into addition.

Voltage Gain in dB:
$A_{v(\text{dB})} = 20 \log_{10} |A_v|$
Current Gain in dB:
$A_{i(\text{dB})} = 20 \log_{10} |A_i|$
Power Gain in dB:
$A_{p(\text{dB})} = 10 \log_{10} |A_p|$

For a cascaded amplifier, the total gain in dB is the sum of the individual stage gains in dB.

$A_{v(\text{total, dB})} = A_{v1(\text{dB})} + A_{v2(\text{dB})} + \dots + A_{vn(\text{dB})}$

Example: If a two-stage amplifier has stage gains of $A_{v1} = -50$ and $A_{v2} = -80$ :

Total Voltage Gain:
$A_{v(\text{total})} = (-50) \times (-80) = 4000$ . The overall phase shift is 360° (or 0°).
Total Voltage Gain in dB:
$A_{v(\text{total, dB})} = 20 \log_{10}(4000) \approx 20 \times 3.6 = 72 \text{ dB}$ .
Alternatively, using individual dB gains:
$A_{v1(\text{dB})} = 20 \log_{10}(50) \approx 34 \text{ dB}$
$A_{v2(\text{dB})} = 20 \log_{10}(80) \approx 38 \text{ dB}$
$A_{v(\text{total, dB})} = 34 \text{ dB} + 38 \text{ dB} = 72 \text{ dB}$ .

Unit 4

Unit 6