High-Q harmonic oscillators

Q 10.1

For the harmonic oscillator circuits shown in the figure below you can assume that the current gain of the transistor is very large, i.e.

αfe \to \infty

pict

(a): Draw a small-signal equivalent circuit.
(b): Derive the condition for oscillation for this circuit.
(c): Derive an expression for the (radian) frequency of oscillation.
(d): Derive an expression for the required value of $g_{m}$ to get stable oscillation.

Q 10.2

The figure below shows the circuit schematic of a Hartley oscillator. Transistor T has

αfe = 99

and is ideal otherwise. Furthermore, set

C = 1 nF

and

L_{1} = 1 mH

. The current through

R_{B 1}

and

R_{B 2}

may be neglected.

pict

(a): What is the function of $C_{0}$ and what can you say about its size?
(b): The transistor is set at $I_{C} = 5 mA$ ; calculate the $g_{m}$ of the transistor.
(c): Draw a small-signal equivalent of this oscillator circuit.
(d): Formulate the oscillation condition and derive an expression for the oscillation frequency.
(e): Derive an expression for the value of $L_{2}$ to obtain stable oscillation at the (radian) frequency derived in the previous question.