H Parameter Analysis

H parameters, or hybrid parameters, are the standard two-port model used to characterize BJT amplifiers in datasheets and GATE problems. They mix impedance and admittance units in one matrix, which suits the BJT perfectly because its input is low impedance and its output is high impedance.

Core Concept

A BJT is a two-port network: port 1 is the base-emitter pair, port 2 is the collector-emitter pair. H parameters describe how voltage and current at one port relate to the other. The equations are V1 = h11*I1 + h12*V2 and I2 = h21*I1 + h22*V2, where h11 has units of ohms, h12 is dimensionless, h21 is dimensionless, and h22 has units of siemens.

For the common emitter configuration, the four parameters become hie (input impedance in ohms), hre (reverse voltage ratio, typically 10^-4), hfe (forward current gain = beta), and hoe (output admittance in siemens). Datasheets for transistors like the BC547 list hfe, hie, and hoe directly for specific Q-points.

H parameters change with configuration. If you have CE parameters (hie, hre, hfe, hoe), you can derive CB or CC parameters using standard conversion formulas. In most practical amplifier analysis, hre is neglected (it is very small) and hoe is neglected when 1/hoe >> RL, simplifying the model to just hie and hfe.

Key Equations

Port equations: V1 = hie*I1 + hre*V2 and I2 = hfe*I1 + hoe*V2.

Voltage gain (CE, neglecting hre and hoe): AV = -hfe * RL / hie where RL is the collector load resistance.

Current gain (CE): AI = -hfe / (1 + hoe * RL). When hoe*RL << 1, AI ≈ -hfe.

Input impedance: Rin = hie - hre * hfe * RL / (1 + hoe * RL). With hre neglected, Rin = hie.

Output admittance: Yout = hoe - hre*hfe / (hie + Rs) where Rs is source resistance. Output impedance Rout = 1/Yout.

Given:
  BC547 at IC = 1 mA, VCE = 5V (from datasheet)
  hie = 2.5 kΩ, hfe = 150, hoe = 25 µS, hre = 2×10⁻⁴
  RL = 4.7 kΩ (collector resistor)

Why this formula:
  Compute voltage gain and input impedance using simplified
  h-parameter model (hre and hoe neglected first, then checked).

Check hoe*RL:
  hoe * RL = 25×10⁻⁶ * 4700 = 0.1175
  Since 0.1175 < 0.1 is NOT satisfied, include hoe correction.

Voltage gain (exact):
  AI = -hfe / (1 + hoe*RL) = -150 / (1 + 0.1175)
     = -150 / 1.1175
     = -134.2

  AV = AI * RL / hie
     = -134.2 * 4700 / 2500
     = -630740 / 2500

Calculation:
  AV = -252.3

Simplified (neglecting hoe):
  AV = -hfe * RL / hie = -150 * 4700 / 2500 = -282

Final Answer:
  AV (exact) = -252.3
  AV (simplified) = -282
  Difference: ~11%, significant when hoe*RL > 0.1

Exam Tip: GATE problems often give all four h parameters and ask for AV or Rin. The simplification hre = 0 and hoe = 0 is valid only when hoe * RL < 0.1. Always check this condition. If hoe * RL is 0.1 or more, use the full formula for AI = -hfe/(1 + hoe*RL). Skipping this check and using AV = -hfe*RL/hie directly can cost 5-10% accuracy and the full marks.

Key Properties

• hie (input impedance): typically 1-5 kΩ for CE at IC = 1 mA. Equals rπ in the hybrid-pi model.
• hfe (forward current gain): equals beta. Printed on datasheets. BC547 lists hfe = 110-800 depending on grade.
• hoe (output admittance): typically 10-50 µS. Reciprocal is rO. At 25 µS, rO = 40 kΩ.
• hre (reverse voltage ratio): typically 10^-4. Small enough to neglect in most hand calculations.
• H parameters depend on the Q-point. Datasheets specify them at particular IC and VCE values.
• Conversion between CB, CE, and CC h parameters is done using standard tables available in Millman & Halkias.

Quick Revision

• V1 = hie*I1 + hre*V2; I2 = hfe*I1 + hoe*V2.
• hie = rπ (input resistance, ohms).
• hfe = beta (forward current gain, dimensionless).
• hoe = 1/rO (output admittance, siemens).
• hre is negligible in most practical calculations.
• AV = -hfe*RL/hie when hoe*RL << 1.
• Always check hoe*RL before simplifying; if > 0.1, use AI = -hfe/(1+hoe*RL).
• Exam trap: Students apply CE h parameter formulas to a CB or CC circuit without converting parameters first, getting the wrong sign and magnitude for AV.