Hybrid h-Parameter Model

One popular technique for analyzing BJT amplifier circuits is the hybrid h-parameter model. In this method, the transistor is treated as a two-port network, where the base–emitter terminals form the input port and the collector–emitter terminals form the output port.

• The behavior of the transistor is described using hybrid parameters (h-parameters), which relate the input voltage and output current to the input current and output voltage.
• These parameters are called hybrid because they combine different electrical quantities such as resistance, gain, and admittance.
• This model simplifies the analysis of transistor amplifiers and can be applied to common emitter (CE), common base (CB), and common collector (CC) configurations.

Two-Port Network Representation

A transistor amplifier can be represented as a two-port network with the following variables:

• V_1: Input voltage
• I_1: Input current
• V_2: Output voltage
• I_2: Output current

If input current I_1 and output voltage V_2 are taken as independent variables, the hybrid parameter equations are:

V_1 = h_{11} I_1 + h_{12} V_2

I_2 = h_{21} I_1 + h_{22} V_2

These equations describe the electrical relationship between the input and output of the transistor.

Definition of Hybrid Parameters

The four h-parameters are defined as follows.

Input Impedance Parameter (h_{11})

h_{11} = \frac{V_1}{I_1}

It represents the input resistance of the transistor when the output is short-circuited.

Reverse Voltage Gain (h_{12})

h_{12} = \frac{V_1}{V_2}

It represents the reverse voltage gain, showing how much the output voltage affects the input voltage.

Forward Current Gain (h_{21})

h_{21} = \frac{I_2}{I_1}

It represents the forward current gain of the transistor.

Output Admittance (h_{22})

h_{22} = \frac{I_2}{V_2}

It represents the output conductance of the transistor when the input port is open.

Analysis of BJT Amplifier Using the Hybrid Model

When a BJT amplifier is analyzed using the hybrid model, the circuit includes:

• Source resistance R_s
• Load impedance Z_L
• Input impedance Z_{in}

Using the h-parameter model, several amplifier characteristics can be determined.

Current Gain

The ratio of output current to input current.

A_I = \frac{I_2}{I_1}

For the hybrid model,

A_I = \frac{-h_f}{1 + h_o Z_L}

where:

• A_I: Current gain
• h_f: Forward current gain parameter
• h_o: Output admittance parameter
• Z_L: Load impedance.

Input Impedance

Ratio of the input voltage to the input current of the amplifier.

Z_{in} = \frac{V_1}{I_1}

It depends on the input parameter h_i and the effect of load impedance.

Voltage Gain

Ratio of output voltage to input voltage.

A_v = \frac{V_2}{V_1}

For the hybrid model, the voltage gain depends on the forward current gain, input resistance, and load resistance.

Voltage Gain Considering Source Resistance

When the source resistance R_s is included, the voltage gain becomes

A_{VS} = A_v \frac{Z_{in}}{Z_{in} + R_s}

Current Gain Considering Source Resistance

Using Norton’s equivalent circuit, the effective current gain becomes

A_{IS} = A_I \frac{R_s}{Z_{in} + R_s}

Output Impedance

Ratio of output voltage to output current when the input source is deactivated.

Z_o = \frac{V}{I}

Analysis of a CE Transistor Amplifier Using h-Parameters

Consider a common emitter amplifier with self-biasing.

During AC analysis:

• There is a short circuit in the DC supply V_{CC}.
• Bypass and coupling capacitors function as short circuits.
• The h-parameter equivalent model of the transistor is used in its stead.

Using the simplified hybrid model, the amplifier parameters are obtained.

Input Impedance

Z_i = R_B \parallel h_{ie}

where

R_B = R_1 \parallel R_2

and,

• R_B: Equivalent base resistance formed by R_1 and R_2 in parallel
• R_1, R_2: Biasing resistors used to set the base voltage
• h_{ie}: Input resistance of the transistor in common emitter configuration

Output Impedance

Z_o = R_C

where,

• R_C: Collector resistor connected in the collector circuit
• Z_o : Output impedance of the amplifier

Voltage Gain

A_v = - \frac{h_{fe}(R_C \parallel R_L)}{h_{ie}}

where

• h_{fe}: Forward current gain of the transistor in CE configuration
• R_C: Collector resistor connected in the collector circuit
• R_L: Load resistance connected at the output of the amplifier
• h_{ie}: Input resistance of the transistor in common emitter configuration

Current Gain

A_I = -h_{fe}

Analysis of CE Transistor with Unbypassed Emitter Resistance R_E

If the emitter bypass capacitor C_E is removed, the emitter resistance appears in the AC circuit. This changes several amplifier characteristics.

Effects of Unbypassed R_E

• Input resistance increases
• Voltage gain decreases
• Output impedance remains approximately the same

The input impedance becomes

Z_{in} = R_B \parallel \left[ h_{ie} + (1 + h_{fe})R_E \right]

The voltage gain becomes

A_v = -\frac{h_{fe} R_C}{h_{ie} + (1 + h_{fe})R_E}

Hybrid Parameters for Different Configurations

Hybrid parameters vary depending on the transistor configuration.

Common Emitter (CE)

• High current gain
• Moderate input resistance
• Moderate output resistance

Common Collector (CC)

• High input resistance
• Low output resistance
• Voltage gain approximately equal to unity

Common Base (CB)

• Low input resistance
• High output resistance
• High voltage gain but current gain less than unity

Advantages

• Simple Analysis: Offers a straightforward approach to transistor amplifier circuit analysis.
• Applicable to Various Configurations: CE, CB, and CC amplifier configurations can all make use of the same model.
• Experimentally Measurable Parameters: Direct experimental measurements can yield h-parameters.
• Appropriate for Low-Frequency Analysis: Excellent for examining transistor amplifiers operating at low frequencies.

Disadvantages

• Unsuitable for High Frequencies: When analyzing high-frequency amplifiers, accuracy declines.
• Parameter Variation: Temperature and biasing are two examples of operational circumstances that affect h-parameter values.
• Complex Circuit Equations: Complicated circuit equations might result from in-depth investigation.