The Transistor — Question 5

Question 5

The exponential collector law sets transconductance and input resistance

In forward active operation use $I_C=I_Se^{V_{BE}/V_T}$, with $V_T=.0250\,\mathrm V$ and constant $\beta=120$. The bias current is $I_C=1.50\,\mathrm{mA}$.

1. Derive and evaluate $g_m=dI_C/dV_{BE}$ and $r_\pi=dV_{BE}/dI_B$.

2. For a small $v_{be}=1.00\,\mathrm{mV}$ variation, find first-order collector and base current variations, and the ratio of the quadratic correction to the linear collector term.

Question 5 - Solution

The exponential collector law sets transconductance and input resistance

(a) Method and calculation

$$\boxed{g_m=I_C/V_T=.0600\,\mathrm S}.$$$$I_B=I_C/\beta\Rightarrow\boxed{r_\pi=\beta/g_m=2.00\,\mathrm{k\Omega}}.$$

(b) Interpretation and further calculation

$$\boxed{i_c=g_mv_{be}=60.0\,\mu\mathrm A,\quad i_b=v_{be}/r_\pi=.500\,\mu\mathrm A}.$$$$e^x-1=x+x^2/2+\cdots,\quad x=v_{be}/V_T=.0400\Rightarrow\boxed{\frac{\text{quadratic}}{\text{linear}}=x/2=.0200}.$$

Physical check

The exponential curvature sets the small-signal limit. Transconductance depends on bias current, whereas beta also controls the input current required.