Exercise 5.1 An amplifier with 2 outputs

Given is the circuit below. The transistor is biased in “normal mode” (active forward, $V_{\mathit{CE}}>V_{\mathit{CE},\mathit{knee}}$). The behaviour of the transistor is then as a good approximation:

The transistor is biased at an emitter bias current $I_E=1\mathit{mA}$ for $V_{\mathit{BE}}=0.6V$ and has a current gain $\mathit{\beta fe}=99$. Furthermore, $R_{B2}=220k\mathrm{\Omega }$, $R_E=500\mathrm{\Omega }$ and $V_{\mathit{CC}}=5V$.

a)

Derive an expression for the value of $R_{B1}$ required to set the specified $I_E$, as a function of the parameters of the other components (wherever applicable) and determine its numerical value.

b)

Explain the impact of temperature variations on $I_E$ and $I_C$.

c)

Determine the small signal parameters $g_m$ and $\mathit{\beta fe}∕g_m$ for the BJT in this circuit in the bias point. Give you answer numerically (pay attention to the dimensions! ) and draw the small-signal equivalent circuit of the transistor. For the temperature of the circuit, $\frac{\mathit{kT}}{q}=25\mathit{mV}$.

d)

Draw the small-signal equivalent circuit of the circuit in a frequency range where the capacitors $C_{\mathit{in}}$ , $C_1$ and $C_2$ have a negligible impedance.

e)

Derive an equation for $R_C$, to ensure that the amplitude of $v_{\mathit{OUT}1}$ is equal to that of $v_{\mathit{OUT}2}$.

f)

Derive the phase relation between $v_{\mathit{out}1}$ and $v_{\mathit{out}2}$.

g)

Derive an expression for the output resistance of the circuit, as seen on the port where $v_{\mathit{out}2}$ is defined.

h)

Derive an expression for the input resistance of the circuit