Exercise 3.1 Biasing a circuit with an NPN

The collector current of a BJT with some finite $\mathit{\beta fe}$ is set at $I_C$ in the circuit at the right hand side of the figure below.

a)

Give an approximate expression for the value of the base resistor $R_B$ to get a collector current $I_C$. In other words: express $R_B$ in terms of the target $I_C$ and other (circuit and component) parameters. For the approximate expression, you may assume a fair estimation for the resulting $V_{\mathit{BE}}$.

b)

For better accuracy, now derive an exact expression for $R_B$, using the element equations of the BJT. For that assume that $I_{C0}$ is known for the BJT. Note that typically $\frac{I_C}{I_{C0}}>>>1$ and hence the ”+1” can be ignored.

c)

Given is $\mathit{\beta fe}=200$, $I_C=1\mathit{mA}$, $I_{C0}=10^{-13}A$, $\mathit{kT}∕q=25\mathit{mV}$ and a supply voltage $V_{\mathit{CC}}=10V$. Calculate the base resistor $R_B$ using the relations you found at a) and b).

d)

$R_B$ is chosen such that $I_C=1\mathit{mA}$. Due to some unforeseen circumstance, the transistor breaks down and is replaced by one that appears to have a collector current of $2\mathit{mA}$. Assume that for this transistor, the $I_{C0}$ is identical to the previous one and that only $\mathit{\beta fe}$ is different. Give an estimation for $\mathit{\beta fe}$ of this new transistor.

e)

In the circuit at the left hand side of Figure 3.1, the BJT is biased at $I_C=1\mathit{mA}$, while its current gain $\mathit{\beta fe}=200$. We now place an identical transistor parallel to the original one. Will the total collector current be larger than, smaller than or be equal to $2\mathit{mA}$?