Question 19.8: Determine the small-signal voltage gain, input resistance an......
Determine the small-signal voltage gain, input resistance and output resistance of the circuit in Example 19.7.
The mid-band gain of the amplifier is approximately -R_{C}/r_{e} where r_{e} is given by
r_{e} \approx \frac{1}{40I_{E}} \approx \frac{1}{40 \times 1.04 \times 10^{-3}} \approx 24 \OmegaTherefore
small-signal voltage gain \approx – \frac{R_{C}}{r_{e}} = -\frac {5.6 k\Omega}{24 \Omega} \approx -233
The input resistance of the circuit is given by the parallel combination of R_{1}, R_{2} and the resistance seen looking into the base of the transistor. In the circuit in Example 19.6, the resistance seen looking into the base of the transistor is approximately equal to h_{fe}R_{E}, so it is normally so high that it may be ignored. However, the presence of the decoupling capacitor in the circuit in Example 19.7 removes the effect of R_{E} for small signals. The resistance seen looking into the base of the transistor is now h_{ie}, which is likely to be of similar magnitude to the parallel combination of R_{1} and R_{2}, so cannot be ignored. From Equation 19.9, we know that
h_{ie} \approx h_{fe}r_{e}In this case r_{e} is about 24 Ω, so h_{ie} will be a few kilohms (depending on the value of h_{fe}) and the input resistance of the complete amplifier r_{i} will be R_{1}//R_{2}//h_{ie} . For a given type of transistor, we can determine the range of h_{fe} from the data sheet and so deduce the likely range of h_{ie} and hence the input resistance. For example, if we know that h_{fe} is between 100 and 400, then we can compute a range for the input resistance.
If h_{fe} = 100
h_{ie} \approx h_{fe}r_{e} = 100 \times 24 Ω = 2.4 kΩ \\ r_{i} = R_{1}//R_{2}//h_{ie} = 82 kΩ//13 kΩ//2.4 kΩ = 2.0 kΩIf h_{fe} = 400
h_{ie} \approx h_{fe}r_{e} = 400 \times 24 Ω = 9.6 kΩ \\ r_{i} = R_{1}//R_{2}//h_{ie} = 82 kΩ//13 kΩ//9.6 kΩ = 5.2 kΩThus, the small-signal input resistance for such a circuit, using such a transistor, would be in the range 2.0–5.2 kΩ.
Small-signal output resistance
The output resistance of the circuit is similar to that of a simple common-emitter amplifier, which we have previously shown to be approximately equal to R_{C} . Therefore
small-signal output resistance \approx R_{C} = 5.6 kΩ