### 4.1 Feedback Oscillators.

Many systems require an input in the form of a periodic, usually sinusoidal, waveform — for example, the LO needed to drive the heterodyne receivers considered in the last few sections. These oscillators can take many forms. Cavity controlled lasers are a form of feedback oscillator. So are the transistor circuit oscillators used in domestic radios and TV. Here we will use the example of a simple electronic oscillator, but remember that similar results can be shown to apply for many other types of coherent signal source. The example we'll use is illustrated in figure 4.1a. This shows a transistor *phase shift oscillator*.

All feedback oscillators require some device or mechanism which provides *gain* combined with a *feedback* arrangement which sends some of the system's output back to be re-amplified after a suitable __time delay__. In 4.1a the gain is provided by the transistor. The time delayed feedback is provided by the capacitors & resistors marked *C* and *R*. Although this system uses a particular type of transistor and feedback *network* we can generalise its behaviour into the arrangement illustrated in figure 4.1b.

This shows an *amplifier* which has a voltage gain, , whose output and input are linked via a feedback network. This returns a fraction, , of the output voltage to the amplifier's input. Note that both the amplifier's gain and the feedback factor are frequency dependent. In general, both the amplifier and the feedback network will alter the magnitude and the phase of the signal. To take this into account it is normal to treat both and as complex values.

The amplifier and feedback network form a *loop*. An initial signal fluctuation

at the amplifier's input will produce an output

which, in turn, produces a new ‘echoed’ input signal

back at the amplifier's input. This new input will, in turn, be amplified and produce a new echo at the input, which... etc.

After *n* ‘trips around the loop’ the amplitude of the newest echo will be

By looking at this expression we can see that if the echoes will fade away. However, if we arrange that then the size of the echoes tends to grow with time (or at least remains constant if we arrange that ).

As a result we find that an initial signal produces a sustained, repeating signal whose amplitude doesn't fade away with time provided we can ensure that

Provided that

each delayed echo or *cycle* of fluctuation will ‘tack itself onto the tail’ of the previous fluctuation with the same sinusoidal phase. As a result, provided that the two expressions 4.5 & 4.6 are satisfied we only have to ‘give the system a kick’ by providing the initial cycle of input.

The expressions 4.5 & 4.6 taken together are called the *Barkhausen Criterion*. Any system which satisfies this criterion is able to oscillate at any frequency for which the expressions are both true. Note that, in practice, expression 4.6 is usually only satisfied for a one or more discrete frequency values, hence the system can only oscillate at these specific frequencies. It should also be noted that the fact that a system satisfies the criterion __doesn't__ guarantee that it __will__ actually oscillate! The process has to be started by an initial small fluctuation of the correct frequency. If this starting kick is absent the system may just sit in a quiescent state. Fortunately, any small, brief, fluctuation which contains some power at the frequency, *f*, will start a sequence of steady or growing *oscillations* at that frequency. In practice this means that we don't usually have to provide a specific input to start the process. The electrical ‘shock’ of switching on the oscillator's amplifier (the gain source) is usually enough to get things going. If not, the random noise present in all real physical systems can often provide the required starting kick.

Anyone familiar with electronics can see that the feedback oscillator is almost identical to a feedback controlled amplifier system. The only difference is that a feedback amp __shouldn't__ satisfy the Barkhausen Criterion whereas an oscillator __should__! In practice this to one of the basic rules of electronics, “*All feedback amplifiers try to oscillate and all oscillators don't!*” There are lots of different types of electronic feedback oscillator. If you have a look through electronics books you can find ‘Hartley Oscillators’, ‘Colpitts Oscillators’, ‘Wien Bridge Oscillators’, etc, etc. Although their details differ, they all use the same technique of combining a gain section with a feedback arrangement which provides the phase/time delay required for the system to oscillate at a specific frequency. We can represent the behaviour of the whole system in terms of an overall *loop gain*,

and the frequency (or frequencies!) which have a loop *phase shift* where oscillation is possible as the value(s), , such that

In an ideal situation we might expect to arrange that there is just one frequency where __exactly__ equals unity. If this done, an oscillation at this frequency will continue forever without its amplitude becoming either bigger or smaller. However we tend to find that oscillations can only be started when . This gets things going, but it means that the oscillation amplitude tends to increase exponentially as time passes. In a real oscillator system this growth will eventually be limited in some way. For example, in the oscillators shown in figure 4.1 the voltage oscillations will be limited by the size of the voltages on the *power rails* which supply power to the oscillator. (Naturally, the oscillation power has to come from __somewhere__ and there's always a limit on the available power!)

In general, therefore, an oscillator tends to start up, its oscillation amplitude then grows (usually rapidly), until limited by some process or feature of the system. The action of this limiting process is to reduce the effective loop gain until it's modulus is unity. The oscillation then continues with a waveform of an essentially constant amplitude. The above example assumes that we're considering an electronic form of a *simple harmonic oscillator*, hence it generates a sinewave. Other forms of oscillator can produce other types of waveform — squarewaves, triangle-waves, even non-periodic chaotic waves! Each system requires a combination of some gain with some feedback.

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