h-parameter model of BJT

BJT is a non-linear device. To analyze a non-linear device that is complex, hence non-linear characteristics of BJT are approximated into linear characteristics and this approximation is possible only in small signals. The small signal model is also called the h-parameter or hybrid parameter model.

The BJT h-parameter model is based on the definition of hybrid parameters. The mathematical model for two-port networks known as the h-parameter model (Hybrid Parameter model) can be developed.

The transistor is a two-port network. The transistor is also a current-operated device.

V₁ = h₁₁I₁ + h₁₂I₂→1

I₂ = h₂₁I₁ + h₂₂V₂→2

Consider I₁, V₂ as independent variables and V₁, and I₂ as dependent variables.

h₁₁= V₁/I₁ | V₂=0 — short circuit input impedance = h_i = h_ie

h₁₂ = V₁/V₂ |I₁=0 — open circuit reverse voltage gain = h_r = h _re

h₂₁= I₂/I₁ | V₂=0 –short circuit forward gain = h_f = h_fe

h₂₂ = I₂/V₂ | I₁=0 — open circuit output admittance= h₀=h_0e

Advantages of h-parameter model:

1. h-parameters are real numbers at audio frequencies.

2. The h-parameter model is easy to measure.

3. The h-parameter model can be calculated from the transistor static characteristic current.

4. The h-parameter model can be used in circuit analysis and design.

5. The h-parameter model can be easily convertible from one configuration to another.

6. Transistors manufacturers specify the values of h-parameters.

Typical Values of h-parameters:

Parameter	Common emitter(CE)	Common collector(CC)	Common Base(CB)
h_i	1100 ohm	1100 ohm	22 ohm
h_r	2.5 x 10^-4	1	3 x 10^-4
h_f	50	51	-0.98
h₀	25 µA/v	25 µA/v	-0.49 µA/v

Input voltage = V_i

Input current = i_i

Output voltage = V₀

Output current = i₀

V_i= h_iI_i + hrV₀→3

I₀= h_fI_i + h₀V₀ →4

Design a circuit with 3 and 4 we get the h-parameter model.

Draw an h-parameter model to common emitter configuration (Analysis):

A transistor amplifier can be constructed by connecting an external load and signal source as indicated in the below figure and biasing the transistor properly.

The h-parameter also known as the hybrid parameter model is shown below figure:

h-parameter model of above configuration

1. Current gain A_I: The current gain of a transistor amplifier is defined as the ratio of the output current to the input current, indicating the amplification factor.

A_I = I_L/I₁= – I₂/I₁

I₂= h_feI₁ + h_0eV₂

V₂= I_LR_L = – I₂R_L

I₂ = h_feI₁ + h_0e( – I₂R_L)

A_I = -h_fe/1+h_oeR_L

2. Input Resistance or Input Impedance:

R_i=V₁/I₁

V₁=h_ieI₁+h_reV₂

V₁/I₁= h_ie+V₂/I₁

V₂= I_LR_L

R_i = h_ie + h_re A_IR_L

3. Voltage gain(Av): The ratio of output voltage to input voltage gives the voltage gain of the transistor.

A_v=V_o/V_i

=V₂/V₁

Av = A_IR_L/R_i

4. Output admittance (y_o) :

To determine the current gain AI, we disregard RL (output load resistance) and set Vs (input voltage source) to zero, effectively making Rs (input source resistance) zero as well.

Io = I2/V2 | R2 =0, Vs=0

I₂ = h_fe I₁ + h_oe V₂

Yo=hoe – h_fehre/h_ie.

R_o = 1/Y₀

5. Output admittance (Y_os) :

0 = (R_s + h_ie)I₁ + h_reV₂

I₁/V₂ = -h_re / R_s + h_ie

Y_os = h_oe – h_feh_re/R_s + h_ie

6. Input resistance (r_is or z_is):

R_is = R_s + R_i

7. Voltage gain including the source (Avs) :

A_vs = V₂/V_s

= A_v V₁/V_s

V₁ = V_s x R_i/R_s + R_i

A_vs = A_vR_i/R_s + R_i

8. Current gain including source (A_IS):

A_IS = I_L/I_s

A_IS = A_I (R_s/R_s + R_I)

9. Power gain (A_P):

A_PC = A_V X A_I

A_V = A_I X R_L/R_i

A_P = A_I² R_L / R_i

Problem:

A common emitter amplifier is drawn by a voltage source of internal resistance R_s = 800 ohm, load impedance R_L= 1000 ohm the h-parameters are h_ie = 1 KΩ, h_re = 2 X 10^-4 h_fe = 50, h_oe = 25 µA/V. Determine A_I, R_I, A_V, R_O, Z_is or R_isand R_os or Z_os.

Solution: A_I = -h_fe / 1 + h_oeR_L= -50/1+25 X 10^-6 X 1000 = -48.7805.

R_i = h_ie + h_re A_I R_L

= 1 x 10³ + 2 X 10^-4 X (-48.7805) X 1000

= 990.2439.

A_V = A_IR_L/ R_i

= (-48.7805 X 1000) / 990.2439

= -49.2611

R₀ = 1/Y₀

= -13.33 KΩ

R_is = R_s + R_i

= 800 + 990.2439

= 1790.2439

R_os = 1/ Y_os

= -32727.2727.

Applications of h-parameter model:

Transistor Amplifier Design: Transistor amplifier design, particularly for BJTs and FETs, heavily relies on the h-parameter model for comprehensive analysis and effective design. It simplifies the complex behavior of transistors, making it easier to determine gain, input/output impedance, and stability.
Filter Design: H-parameters aid filter design (low-pass, high-pass, band-pass, band-stop) by predicting frequency response, attenuation, and phase shift.
Small-Signal Analysis: The h-parameter model is ideal for analyzing the small-signal behavior of circuits. This is essential for understanding how circuits respond to small variations in input signals, which is crucial for linear circuit design.
Feedback System Analysis: Engineers use H-parameters to assess and develop feedback systems. They help determine the loop gain, stability margins, and performance of feedback amplifiers and control systems.
Matching Networks: The h-parameter model assists in designing matching networks that maximize power transfer between different stages of a circuit. This is particularly important in RF and microwave circuits where impedance matching is critical.

FAQs of the h-parameter model:

1. What is the h-parameter model?

The h-parameter model (or hybrid parameter model) is a two-port network representation used to characterize linear electronic circuits.
It describes the relationships between input and output voltages and currents using four h-parameters: h11, h12, h21, and h22.

2. Why is the h-parameter model used in electronics?

The h-parameter model is popular for several reasons:
- Simplifies analysis: It simplifies the analysis of complex circuits, especially those involving transistors.
- Practical measurements: H-parameters are relatively easy to measure in the lab compared to other two-port parameters.
- Widely used: Manufacturers often provide h-parameters in transistor datasheets, making them readily available for design.

3. What do the h-parameters represent?

h11 (Input Impedance): The input impedance seen at the input port when the output port is short-circuited.
Reverse Voltage Gain (h12): This parameter quantifies the voltage at the input port relative to the output voltage when the input is left open.
h21 (Forward Current Gain): The ratio of output current to input current when the output port is short-circuited.
h22 (Output Admittance): The output admittance seen at the output port when the input port is open-circuited.

4. How are h-parameters used in circuit analysis?

H-parameters are used to analyze various circuits, including:
- Transistor amplifiers: Determine gain, input/output impedance, and stability.
- Filters: Design and analyze the frequency response of different filter types.
- Feedback systems: Assess loop gain, stability, and overall performance.

5. What are the limitations of the h-parameter model?

Frequency-dependent: H-parameters can vary with frequency, making them less accurate at higher frequencies.
Small-signal model: They are only valid for small signal analysis, where the input signal does not significantly alter the operating point of the device.

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