Analysis Challenge / Design Challenge 3
1. Tasks
Derive both rπ and ro using the same Taylor series technique as day17. Show that your result is the same as Tourboook Table 8.
1.1. Intrinsic base resistance rπ
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Hint for finding rπ:
Equivalent resistance is simply the ratio of voltage across to current through two nodes. If that relationship is linear, then replacing whatever is between those two nodes with a resistor of that value is exactly the same as the original circuit from the perspective of the outside. |
1.2. Intrinsic output resistance ro
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Hints for finding ro
You will need to do some background reading about the transistor parameter VA and the Early effect. Wikipedia: Early effect is pretty good, though it doesn’t quite derive the small-signal parameter ro in a clear way. You can do better! Use only the first equation on that page for your Taylor series expansion, shown here with proper capitalization of the variables:[1]
\[i_C = I_S \exp\left(\frac{v_{BE}}{V_T}\right) \left(1 + \frac{v_{CE}}{V_A}\right)\]
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2. References
Incorporation of the Early effect in the Ebers-Moll model by Lindholm and Hamilton. Proceedings of the IEEE, 59:9, 1971. https://doi.org/10.1109/PROC.1971.8435 (mirrored at docs: lindholm1971.pdf