LECTURE 2: SMALL-SIGNAL HYBRID-π EQUIVALENT CIRCUIT OF BIPOLAR TRANSISTOR (BJT)

Lecture’s content Objectives BJT – Small Signal Amplifier Develop the small-signal models of transistor that are used in analysis of linear amplifier. BJT – Small Signal Amplifier Small-signal hybrid-π equivalent circuit of BJT Small-signal hybrid-π equivalent circuit using transconductance Small-signal hybrid-π equivalent circuit using common current gain Small-signal voltage gain Hybrid- π equivalent circuit including Early Effect Expanded hybrid- π equivalent circuit Other small-signal parameters and equivalent circuits

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. for transistor -- use hybrid- model because is closely related to the physic of transistor. Treat transistor as two-port network.

Small signal hybrid- equivalent circuit of bipolar transistor…cont. We can treat the BJT as a two port network The input port is between the base and emitter and the output port is between the collector to emitter.

Small signal hybrid- equivalent circuit of bipolar transistor…cont. Figure shows iB vs. vBE with small-time varying signal superimposed at Q-pt. Since sinusoidal signals are small, the slope at Q-pt can be treated as a constant, which has units of conductance. The inverse of this conductance is small-signal resistance, rπ

Small signal hybrid- equivalent circuit of bipolar transistor…cont. Input Base-Emitter Port We can relate small-signal input base current to small-signal input voltage by: Finding rπ from Q-point slope lead to: rπ is called diffusion resistance or base-emitter input resistance and is a function of Q-point parameters. VT is known as thermal voltage.

Small signal hybrid- equivalent circuit of bipolar transistor…cont. Output Collector-Emitter Port 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,

Small signal hybrid- equivalent circuit of bipolar transistor…cont. After substitution and rearrange the above, we obtain: The term Is exp (vbe/VT) is quiescent collector current, ICQ 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 is also a function of Q-pt parameters and directly proportional to dc bias current.

Small signal hybrid- equivalent circuit of bipolar transistor…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 gm=ICQ/VT r=VT/ICQ Transconductance parameter

Small-signal hybrid- equivalent circuit using transconductance 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 Current gain parameter

Small-signal voltage gain cont.. Combine BJT equivalent cct to ac equivalent cct. Small-signal hybrid-π model

Small-signal voltage gain cont.. Voltage gain, Av = ratio of o/p signal voltage to i/p signal voltage. Small-signal B-E voltage is called the control voltage, Vbe or V. The dependent current source is gmV flows through RC produce –ve C-E voltage at the output.

Small-signal voltage gain cont.. From the input portion of the circuit, using voltage divider: The small-signal voltage gain is:

Example 1 Given :  = 100, VCC = 12V VBE = 0.7V, RC = 6k, VT=0.026V, RB = 50k and VBB = 1.2V Calculate the small-signal voltage gain.

Solutions 1. 2. 3. 4. 5. 6.

Example 2 Given VCC=3.3V, VBB=0.850V, RB=180kΩ, RC=15kΩ, β=120 and VBE(on)=0.7V. Determine: Q-points values, ICQ and VCEQ b) gm and r c) voltage gain.

Hybrid-π equivalent circuit including Early effect Early Voltage (VA)

Hybrid-π equivalent circuit including Early effect **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, vce = -VA. VA --- Early voltage Typical values of VA are in the range of 50 < VA < 300V.

Hybrid-π equivalent circuit including Early Effect Early Effect => collector current, iC is dependent to collector-emitter voltage, vCE (refer Chapter 5-Neaman): The output resistance, rO: Substitute and rearrange both equation,

Hybrid-π equivalent circuit including Early effect cont.. Hence, small-signal transistor output resistance, rO become: rO is equivalent to Norton resistance  rO is parallel with dependent current sources.

Modified bipolar equivalent circuits including rO due to Early Effect. Transconductance parameter ro=VA/ICQ Current gain parameter

Example 3 Determine the small signal voltage gain, including the effect of the transistor output resistance. Given :  = 100, VCC = 12V, VA=50 V VBE = 0.7V, RC = 6k, VT=0.026V, RB = 50k and VBB = 1.2V Calculate the small-signal voltage gain.

Solutions 1. 2. 3.

Solutions 4. 5. 6. 7. 8.

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, rb and rμ. rb  series resistance of semiconductor material between the external base B and an idealized internal base region B’. Since rb << rμ., rb is neglected (short cct) at low freq. rμ  reverse-biased diffusion resistance of B-C junction. Typically in megaohms and neglected (open cct). Normally, in hybrid-π model, we neglect both rb and rμ.

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 Equation 1: KVL at input, hie in series with dependent voltage source, hreVce Equation 2: KCL at output, hoe is in parallel with dependent current source, hfeIb. Equation 1 Equation 2

h-parameter model of C-E BJT Common-emitter transistor

h-parameter h-parameter in relation with hybrid-π are shown below: