Instability criteria

One method of oscillator analysis is to determine the input impedance of an input port neglecting any reactive components. If the impedance yields a negative resistance term, oscillation is possible. This method will be used here to determine conditions of oscillation and the frequency of oscillation.
An ideal model is shown to the right. This configuration models the common collector circuit in the section above. For initial analysis, parasitic elements and device non-linearities will be ignored. These terms can be included later in a more rigorous analysis. Even with these approximations, acceptable comparison with experimental results is possible.
Ignoring the inductor, the input impedance can be written as

Where v₁ is the input voltage and i₁ is the input current. The voltage v₂ is given by

v₂ = i₂Z₂

Where Z₂ is the impedance of C₂. The current flowing into C₂ is i₂, which is the sum of two currents:

i₂ = i₁ + i_s

Where i_s is the current supplied by the transistor. i_s is a dependent current source given by

Where g_m is the transconductance of the transistor. The input current i₁ is given by

Where Z₁ is the impedance of C₁. Solving for v₂ and substituting above yields

Z_in = Z₁ + Z₂ + g_mZ₁Z₂

The input impedance appears as the two capacitors in series with an interesting term, R_in which is proportional to the product of the two impedances:

If Z₁ and Z₂ are complex and of the same sign, R_in will be a negative resistance. If the impedances for Z₁ and Z₂ are substituted, R_in is

If an inductor is connected to the input, the circuit will oscillate if the magnitude of the negative resistance is greater than the resistance of the inductor and any stray elements. The frequency of oscillation is as given in the previous section.
For the example oscillator above, the emitter current is roughly 1 mA. The transconductance is roughly 40 mS. Given all other values, the input resistance is roughly

This value should be sufficient to overcome any positive resistance in the circuit. By inspection, oscillation is more likely for larger values of transconductance and/or smaller values of capacitance. A more complicated analysis of the common-base oscillator reveals that a low frequency amplifier voltage gain must be at least four to achieve oscillation.^[2] The low frequency gain is given by:

If the two capacitors are replaced by inductors and magnetic coupling is ignored, the circuit becomes a Hartley oscillator. In that case, the input impedance is the sum of the two inductors and a negative resistance given by:

R_in = − g_mω²L₁L₂

In the Hartley circuit, oscillation is more likely for larger values of transconductance and/or larger values of inductance.

Oscillation amplitude

The amplitude of oscillation is generally difficult to predict, but it can often be accurately estimated using the describing function method.