Many systems require an input in the form of a periodic, usually sinusoidal, waveform. The periodic signal is generated from a special circuit which is known as the “oscillator”. Oscillators convert direct current (DC) from a power supply to an alternating current signal. They are widely used in many electronic devices. Common examples of signals generated by oscillators include clock signals, sounds produced by electronic beepers and video games, and radio frequency signals used for broadcasting.

Oscillators can take many forms, and the most common form is considered to be the feedback oscillator. A feedback oscillator can be built from a transistor, a tube, an op-amp, or any other active (amplifying device). Oscillation is brought about by applying a portion of the amplifier’s output signal to its input.

Basic concepts

The block-diagram of a feedback oscillator is shown in figure 1. The oscillator is made from an amplifier and a feedback network. The feedback network sends some of the system's output back to be re-amplified after a suitable time delay.

Let’s consider that the amplifier has a voltage gain equal to A(f), and the feedback network has a voltage gain equal to B(f). A and B, are both frequency dependent, so they are considered to be functions of the frequency (f). 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 A and B as complex functions.

Let’s consider that a pure harmonic signal Vi is applied at amplifier’s input.

Vi(t)=Vo·exp(-2πft)
(1)

Vi, is of course, a function of time (t).

The amplifier will amplify the signal and will produce an output

Vout=A(f)· Vo·exp(-2πft)
(2)

Vout, will in turn, be applied at the input of the feedback network, which will produce a new “echoed” signal V'out, back to the amplifier’s input:

V’out.= B(f)·A(f)· Vo·exp(-2πft)
(3)

This new signal will be amplified again and will produce a new echo at the input, and the same will happen again and again, and again… After n ‘trips around the loop’ the amplitude of the newest echo will be:

|V|=|A(f)·B(f)|n·|Vi|
(4)

By looking at this expression we can see that if |A(f)·B(f)| <1 the echoes will fade away. However, if we arrange that |A(f)·B(f)| >1,  then the amplitude of the echoes will tend to grow with time. And finally, the amplitude of the echoes will remain constant if we arrange that |A(f)·B(f)|=1.

As a conclusion, we can say that an initial signal will produce a sustained - repeating signal whose amplitude will not fade away with time, if we arrange that

|A(f)·B(f)| ≥1
(5)

At this point you may say that we found a necessary condition for oscillation. Well, this is almost the case. Actually, there is another condition:

Each delayed echo or cycle of fluctuation must ‘tack itself onto the tail’ of the previous fluctuation with the same sinusoidal phase. This condition can be stated in a mathematical expression as follow:

arg A(f) + arg B(f)=2πn,  n=0,1,2,3 …..
(6)

Equation (6) simply states that the total phase difference produced by the loop, must be zero (00) or an integer multiple of 2π rad, which is equivalent to 0o. This is equivalent to say that the feedback signal must be applied in phase with the original input signal. Usually, a typical practical amplifier is an inverter that provides 1800 (π rad) of phase shift by itself. So, in most practical feedback oscillators, the feedback network must provide an additional 1800 (π rad) of phase shift.

As a result, we may say that if the two expressions 5 & 6 are satisfied we only have to ‘give the system a kick’ by providing the initial cycle of input. However, there still unanswered questions: How does oscillation begin? If oscillations depend on feeding the output signal back to the input in phase, then how can the process begin when there is no output signal to feed back? Who or what, is able to give the initial “kick”?

To use a mechanical analogy, consider a tuning fork. It needs a mechanical stimulus – a sharp rap- to start oscillating. Is there an electrical equivalent to that rap that will get an oscillator running?

The answer, of course, is yes. First, no circuit is ever perfectly noise free. In fact, transistors, operational amplifiers, and other active devices have an abundance of noise that initially “rings” the circuit. Second, there is also a variable output voltage (transient) that is caused by the varying collector (or drain) current as the device “comes alive”. In many cases that change in current is sufficient to ring the circuit into oscillation. However, if the circuit is very quiet or very slow to turn on, it is possible that the circuit would be unable to start oscillating. But assuming it does start, we usually want oscillation to occur at a specific frequency.

There are many methods of controlling frequency, but the main concept behind all methods is the use of tuned networks. In other words, feedback oscillators are built to satisfy equations (5) and (6) at a specific frequency or at specific frequencies. Most practical oscillators, satisfy equation (5) at a broad frequency spectrum. However, equation (6) is usually satisfied only at a unique frequency or at specific harmonics (f, 2f, 3f…).  The shape of the output waveform of an oscillator is sinusoidal (the oscillator is a harmonic oscillator), if the two expressions 5 & 6 are both satisfied in only one frequency. The output signal may have a different shape if the conditions are satisfied in multiple frequencies. Actually, if the conditions are satisfied in multiple frequencies, the output waveform is expected to be a linear superposition of multiple sinusoidals that may vary in phase. The shape of the output waveform may also be affected by the “clipping” effect, caused by a saturated amplifier.