SMALL-SIGNAL HYBRID-Π EQUIVALENT CIRCUIT

Content BJT – Small Signal Amplifier BJT complete Hybrid equivalent circuit BJT approximate Hybrid model Objectives Develop the small-signal models of transistor that are used in analysis of linear amplifier.

Basic knowledge.. Ohm’s Law Kirchoff’s Law Thevenin and Norton’s Theorem All electronic circuit analysis require these for mathematical manipulation.

Small signal hybrid-  equivalent circuit of bipolar transistor Need to develop a small-signal equivalent cct -- use hybrid-  model because is closely related to the physic of transistor. Treat transistor as two-port network.

2-port system AC analysis require simplification of transistors as 2-port system. Simplification leads to new parameters / definitions.

2-port system cont.. ‘Single ended’ 2-port system has 1 input port shorted to 1 output port. Alternative view =>system has a common input/output port. Three terminal device  device which only three connection leads, i.e transistor falls into this category.

Single-ended 2-port network

Differential 2-port network The ‘differential 2-port’ network will be the basis for forthcoming analysis of all types of transistors (BJT and FET).

Port variables 2-port network analysis is all about current and voltage by breaking down voltage direction (-ve to +ve or +ve to –ve) and current direction (to or from). Each current and voltage has 2 possible directions.

2-port variables Below are the equations for BJT’s derived from 2-port network simplification.

Small signal hybrid-π equivalent circuit Based on 2-port network, 1 input port and 1 output port shorted together to form a common port of both input and output. Transistor has input and output ports shorted (emitter) resulting a small-signal 2-port hybrid- π network.

Cont.. Figure shows i B vs. v BE with small-time varying signal superimposed at Q-pt. Since sinusoidal signals are small, the slope at Q-pt treated as a constant, has units of conductance. The inverse of this conductance is small-signal resistance, r π

Cont.. We can relate small-signal input base current to small-signal input voltage by: Finding r π from Q-point slope lead to: r π also known as diffusion resistance and is a function of Q-point parameters.

Cont.. Now, we consider the output terminal characteristic of BJT. Assume o/p collector current is independent of collector-emitter voltage  collector-current is a function of base-emitter voltage, so the equation: From eq 5.2 in Chapter 5 Neaman,

Cont.. After substitution and rearrange the above, we obtain: The term ICQ / VT is a conductance. Since this term relates current in collector to a voltage in B-E circuit, it is called transconductance and is written: Transconductance also a function of Q-pt parameters and directly proportional to dc bias current.

Cont.. Using these new parameters  develop a simplified small-signal hybrid-π equivalent cct for npn BJT. Phasor components given in parentheses. This circuit can be inserted into ac equivalent circuit shown previously.

Small-signal hybrid-  equivalent circuit using transconductance Transconductance parameter g m =I CQ /V T r  =V T /I CQ

Cont.. We can relate small-signal collector current to small-signal base current for o/p of equivalent cct. Where β is called ac common-emitter current gain. Thus:

Current gain parameter Small-signal hybrid-  equivalent circuit using common-emitter current gain

Small-signal circuit parameters

Small-signal voltage gain Combine BJT equivalent cct to ac equivalent cct.

Small-signal voltage gain Voltage gain, A v = ratio of o/p voltage to i/p voltage. Small-signal B-E voltage is called control voltage, V be or V . The dependent current source is g m V  flows through R C  produce –ve C-E voltage at the output.

Cont.. From the input portion of the circuit: The small-signal voltage gain is:

Example 1 Given :  = 100, V CC = 12V V BE = 0.7V, R C = 6k, V T =0.026V, R B = 50k and V BB = 1.2V Calculate the small-signal voltage gain.

Solutions

Example 2 Given V CC =5V, V BB =2V, R B =650kΩ, R C =15kΩ, β=100 and V BE(on) =0.7V. Determine a) Q-points, b) g m and r  c) voltage gain.

Early effect Early Voltage (V A )

Early voltage Figure above show current-voltage characteristic for constant values of B-E voltage. The curves are linear with respect to C-E voltage in forward-active mode. The slope is due to base-width modulation effect  Early Effect. When the curves extrapolated at zero current, they meet a point on –ve voltage axis, v ce = -V A. V A --- Early voltage with typical value in range of 50 < V A < 300V.

Hybrid-π equivalent circuit with Early Effect Early Effect => collector current, i C is dependent to collector-emitter voltage, v CE (refer Chapter 5- Neaman): The output resistance, r O : Substitute and rearrange both equation,

Cont.. Hence, small-signal transistor output resistance, r O become: r O is equivalent to Norton resistance  r O is parallel with dependent current sources.

Modified bipolar equivalent circuits including r O due to Early Effect. Transconductance parameter Current gain parameter r o =V A /I CQ

Self study for pnp transistor From Neaman textbook, Ac equivalent circuit – pg 386 Transconductance and current gain – pg 386 & 387 Small-signal hybrid-π equivalent circuit – pg 387 Do example 6.3

Expanded hybrid-π equivalent circuit Include 2 additional resistance, r b and r μ. r b  series resistance of semiconductor material. Since r b << r μ., r b is neglected (short cct) at low freq. r μ  reverse-biased diffusion resistance of B-C junction. Typically in megaohms and neglected (open cct). Normally, in hybrid-π model, we neglect both r b and r μ.

Other small-signal parameters -h parameter h-parameter -> relate small-signal terminal currents and voltages of 2-port network. The linear r/ship between terminal currents and voltages are: Where: i for input r for reverse f for forward o for output e for common-emitter

h-parameter These equations represent KVL at input and KCL at output applied to h-parameter model for common-emitter BJT.

h-parameter in relation to hybrid-π