# What requirements of a feedback network are needed to produce oscillation?

I'm trying to understand how to analyze simple oscillators, and would like to know if my conclusions about the required feedback network are correct:

The simplest way to build an oscillator is with:

1. A single transistor amplifier
2. A feedback network

where the output of the transistor amplifier is fed back into its input.

Since, according to the Barkhausen criteria, the circuit must have unity loop gain and 0° net phase shift. Since the simplest amplifier (#1) is a common-emitter amplifier, which has a 180° phase shift, we need the feedback network to introduce another 180° phase shift.

Practically speaking, such a feedback network will attenuate. This is okay as long as the amplifier's gain is large enough such that gain * attenuation = 1. So the key requirement is a feedback network with a 180° phase shift.

Consider the feedback network below. The "resistors" aren't really resistors, but rather complex impedances (the symbol for these wasn't available):

We need Vout to equal -k * Vin, for some k > 0 (also, k can't be so small that k*gain < 1). But Vout = Vin * (Z3/[Z2+Z3]). So we need Z3/[Z2+Z3] < 0. Mathematically, this happens if

1. Z2 and Z3 are pure reactances, no resistance
2. If Z2 is a capacitor, Z3 is an inductor; if Z2 is an inductor, Z3 is a capacitor
3. The magnitude of Z3's impedance is greater than Z2

Is the above correct?

• Only R2 is in the FB loop. It might be best to draw the full circuit you are proposing. Commented Jun 26 at 4:01
• If you want to use the box-style impedance symbol, I believe you can switch between US-style (zigzag) and European-style (box) resistors in the circuit editor by double-clicking the resistor and changing a dropdown setting. Commented Jun 26 at 14:01

Practically speaking, such a feedback network will attenuate.

Not at all if you choose the right type of topology (such as the common emitter Colpitts oscillator): -

If you break the loop (dotted line) and plot the circuit gain (base to inductor output) you get this: -

In other words, C1, C2 and L1 (the CLC filter) actually have unity gain and can deliver the phase shift needed to make a successful oscillator.

Images from my basic website.

In short:

In order to satisfy Barkhausen`s oscillation condition we have the following criteria:

• For a non-inverting amplifier the feedback network must be able to produce zero deg phase shift at one single freqency fo only (classical solution: 2nd-order bandpass)

• For an inverting gain stage we need a passive network of at least 3rd-order which is able to produce a 180deg phase shift at one single frequency fo only. Classical solutions are based on a 3rd-order RC-lowpass, CR highpass or allpass topologies. Another well-known solution for the feedback path is based on two integrator stages in series (Double-Integrator-oscillator).

• In both cases, we must ensure that (a) there is negative feedback for DC (negative loop gain, stable operational point) and (b) the loop gains magnitude is (slightly) larger than 0dB at f=fo.