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

2. 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

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

4. 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.

5. 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.

6. 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

7. 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.

8. 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,

9. 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.

10. 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.

11. Small-signal hybrid- equivalent circuit using transconductance
gm=ICQ/VT
r=VT/ICQ
Transconductance parameter

12. 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:

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

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

15. 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.

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

17. 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.

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

19. 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.

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

21. 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.

22. 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,

23. 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.

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

25. 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.

26. Solutions 1. 2. 3.

27. Solutions 4. 5. 6. 7. 8.

28. 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

29. 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.

30. 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

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

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