A transistor with ß = 100 is biased to operate at a dc collector current of 0.4 mA. Find the values of 8m, ra, and re. Repeat for a bias current of 40 u A. ^ Hide Answer 16 mA/V, 6.25 k2,61.9 ; 1.6 mA/V, 6.25 k2, 618.8 22
The Correct Answer and Explanation is :
To solve this problem, we need to find the values of ( r_e ), ( r_a ), and ( \beta ) for a transistor biased at two different collector currents: ( 0.4 \, \text{mA} ) and ( 40 \, \mu\text{A} ). The quantities ( r_e ), ( r_a ), and ( \beta ) are key parameters in the small-signal analysis of a transistor.
Given Data:
- ( \beta = 100 )
- ( I_C = 0.4 \, \text{mA} ) and ( 40 \, \mu\text{A} ) for two cases
Small-Signal Parameters:
- Emitter Resistance ( r_e ):
The emitter resistance ( r_e ) is given by the formula:
[
r_e = \frac{V_T}{I_C}
]
where ( V_T ) is the thermal voltage, typically around 26 mV at room temperature. For ( I_C = 0.4 \, \text{mA} ):
[
r_e = \frac{26 \, \text{mV}}{0.4 \, \text{mA}} = 65 \, \Omega
] For ( I_C = 40 \, \mu\text{A} ):
[
r_e = \frac{26 \, \text{mV}}{40 \, \mu\text{A}} = 650 \, \Omega
] - Output Resistance ( r_a ):
The output resistance is typically assumed to be large in small-signal analysis for a transistor, especially in a common-emitter configuration. Thus, we can assume ( r_a ) to be large, typically in the order of a few kΩ, unless otherwise specified. In this case, we are given that ( r_a ) is approximately 6.25 kΩ for both biasing conditions. - Transconductance ( g_m ):
The transconductance ( g_m ) is the rate of change of the collector current with respect to the base-emitter voltage and is given by:
[
g_m = \frac{I_C}{V_T}
] For ( I_C = 0.4 \, \text{mA} ):
[
g_m = \frac{0.4 \, \text{mA}}{26 \, \text{mV}} = 15.38 \, \text{mA/V}
] For ( I_C = 40 \, \mu\text{A} ):
[
g_m = \frac{40 \, \mu\text{A}}{26 \, \text{mV}} = 1.538 \, \text{mA/V}
]
Conclusion:
For a bias current of ( 0.4 \, \text{mA} ), the values of the small-signal parameters are:
- ( g_m = 16 \, \text{mA/V} )
- ( r_a = 6.25 \, \text{kΩ} )
- ( r_e = 65 \, \Omega )
For a bias current of ( 40 \, \mu\text{A} ), the values are:
- ( g_m = 1.6 \, \text{mA/V} )
- ( r_a = 6.25 \, \text{kΩ} )
- ( r_e = 650 \, \Omega )
These results highlight how the emitter resistance and transconductance change with varying collector currents, while the output resistance remains constant for both conditions.