Oscillators

Oscillators:

An oscillator may be described as a source of alternating voltage. It is different than amplifier.

An amplifier delivers an output signal whose waveform corresponds to the input signal but whose power level is higher. The additional power content in the output signal is supplied by the DC power source used to bias the active device.

The amplifier can therefore be described as an energy converter, it accepts energy from the DC power supply and converts it to energy at the signal frequency. The process of energy conversion is controlled by the input signal, Thus if there is no input signal, no energy conversion takes place and there is no output signal.

The oscillator, on the other hand, requires no external signal to initiate or maintain the energy conversion process. Instead an output signals is produced as long as source of DC power is connected. Fig. 1, shows the block diagram of an amplifier and an oscillator.

Fig. 1

Oscillators may be classified in terms of their output waveform, frequency range, components, or circuit configuration.

If the output waveform is sinusoidal, it is called harmonic oscillator otherwise it is called relaxation oscillator, which include square, triangular and saw tooth waveforms.

Oscillators employ both active and passive components. The active components provide energy conversion mechanism. Typical active devices are transistor, FET etc.

Passive components normally determine the frequency of oscillation. They also influence stability, which is a measure of the change in output frequency (drift) with time, temperature or other factors. Passive devices may include resistors, inductors, capacitors, transformers, and resonant crystals.

Capacitors used in oscillators circuits should be of high quality. Because of low losses and excellent stability, silver mica or ceramic capacitors are generally preferred.

An elementary sinusoidal oscillator is shown in fig. 2. The inductor and capacitors are reactive elements i.e. they are capable of storing energy. The capacitor stores energy in its electric field.Whenever there is voltage across its plates,and the inductor stores energy in its magnetic field whenever current flows through it. Both C and L are assumed to be loss less. Energy can be introduced into the circuit by charging the capacitor with a voltage V as shown in fig. 2. As long as the switch S is open, C cannot discharge and so i=0 and V=0.

Fig. 2

Now S is closed at t = t_o, This means V rises from 0 to V, Just before closing inductor current was zero and inductor current cannot be changed instantaneously. Current increases from zero value sinusoidally and is given by

The capacitor losses its charge and energy is simply transferred from capacitor to inductor magnetic field. The total energy is still same. At t = t₁, all the charge has been removed from the capacitor plates and voltage reduces to zero and at current reaches to its maximum value. The current for t> t₁ charges C in the opposite direction and current decreases. Thus LC oscillation takes places. Both voltage and current are sinusoidal though no sinusoidal input was applied. The frequency of oscillation is 

The circuit discussed is not a practical oscillator because even if loss less components were available, one could not extract energy with out introducing an equivalent resistance. This would result in damped oscillations as shown in fig. 3.

Fig. 3

These oscillations decay to zero as soon as the energy in the tank is consumed. If we remove too much power from the circuit, the energy may be completely consumed before the first cycle of oscillations can take place yielding the over damped response.

It is possible to supply energy to the tank to make up for all losses (coil losses plus energy removed), thereby maintaining oscillations of constant amplitude.
Since energy lost may be related to a positive resistance, it follows that the circuit would gain energy if an equivalent negative resistance were available. The negative resistance, supplies whatever energy the circuit lose due to positive resistance. Certain devices exhibit negative resistance characteristics, an increasing current for a decreasing voltage. The energy supplied by the negative resistance to the circuit, actually comes from DC source that is necessary to bias the device in its negative resistance region.

Another technique for producing oscillation is to use positive feedback considers an amplifier with an input signal v_in and output v_O as shown in fig. 4.

Fig. 4

The amplifier is inverting amplifier and may be transistorized, or FET or OPAMP. The output is 180° out of phase with input signal            v_O= -A v_in.(A is negative)

Now a feedback circuit is added. The output voltage is fed to the feed back circuit. The output of the feedback circuit is again 180° phase shifted and also gets attenuated. Thus the output from the feedback network is in phase with input signal v_in and it can also be made equal to input signal.

If this is so, V_f can be connected directly and externally applied signal can be removed and the circuit will continue to generate an output signal. The amplifier still has an input but the input is derived from the output amplifier. The output essentially feeds on itself and is continuously regenerated. This is positive feedback. The over all amplification from v_in to v_f is 1 and the total phase shift is zero. Thus the loop gain A β is equal to unity.

When this criterion is satisfied then the closed loop gain is infinite. i.e. an output is produced without any external input.

v_O = A v_error
      = A (v in + v f )
       = A (v_in + β v_O)
or    (1-A β )v_O = A v_in
or   
When A β = 1, v_O / v_in= ¥

The criterion A β = 1 is satisfied only at one frequency.This is known as backhausen criterion.

The frequency at which a sinusoidal oscillator will operate is the frequency for which the total phase shift introduced, as the signal proceeds form the input terminals, through the amplifier and feed back network and back again to the input is precisely zero or an integral multiple of 2p. Thus the frequency of oscillation is determined by the condition that the loop phase shift is zero.

Oscillation will not be sustained, if at the oscillator frequency, A β <1 or A β>1. Fig. 5, show the output for two different contions A β < 1 and A β >1.

Fig. 5

If Aβ is less than unity then Aβ v_in is less than v_in, and the output signal will die out, when the externally applied source is removed. If Aβ>1 then A b v_inis greater than v_in and the output voltage builds up gradually. If A β = 1, only then output voltage is sine wave under steady state conditions.

In a practical oscillator, it is not necessary to supply a signal to start the oscillations. Instead, oscillations are self-starting and begin as soon as power is applied. This is possible because of electrical noise present in all passive components.

Therefore, as soon as the power is applied, there is already some energy in the circuit at f_o, the frequency for which the circuit is designed to oscillate. This energy is very small and is mixed with all the other frequency components also present, but it is there. Only at this frequency the loop gain is slightly greater than unity and the loop phase shift is zero. At all other frequency the Barkhausen criterion is not satisfied. The magnitude of the frequency component f_o is made slightly higher each time it goes around the loop. Soon the f_o component is much larger than all other components and ultimately its amplitude is limited by the circuits own non-lineareties (reduction of gain at high current levels, saturation or cut off). Thus the loop gain reduces to unity and steady stage is reached. If it does not, then the clipping may occur.

Practically, Aβ is made slightly greater than unity. So that due to disturbance the output does not change but if Aβ = 1 and due to some reasons if Aβ decreases slightly than the oscillation may die out and oscillator stop functioning. In conclusion, all practical oscillations involve:

• An active device to supply loop gain or negative resistance.
• A frequency selective network to determine the frequency of oscillation.
• Some type of non-linearity to limit amplitude of oscillations.

Example - 1

The gain of certain amplifier as a function of frequency is A (jω) = -16 x 10⁶ / jω. A feedback path connected around it has β(j ω ) = 10³ / (20 x 10³ + jω )². Will the system oscillate? If so, at what frequency ?

Solution:

The loop gain is 

To determine, if the system will oscillate, we will first determine the frequency, if any, at which the phase angle of equals to 0° or a multiple of 360°. Using phasor algebra, we have

This expression will equal -360° if ,

Thus, the phase shift around the loop is -360° at ω = 2000 rad/s. We must now check to see if the gain magnitude |A β| = 1 at ω = 2 x 10³. The gain magnitude is

Substituting ω = 2 x 10³, we find

Thus, the Barkhausen criterion is satisfied at ω = 2 x 10³ rad/s and oscillation occurs at that frequency (2 x 10³ / 2 π= 318 .3 Hz).