User:Msiddalingaiah/Oscillator Analysis

References

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Maxim Crystal Oscillator Analysis

Colpitts Oscillator

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Ignoring the inductor, the input impedance at the base of a common collector circuit can be written as

 

Where   is the input voltage and   is the input current. The voltage   is given by

 

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

 

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

 

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

 

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

 

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

 

If   and   are complex and of the same sign,   will be a negative resistance. If the impedances for   and   are substituted,   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 smaller values of capacitance.

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:

 

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

Pierce Oscillator

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In a common emmitter circuit,   and   are the feedback (collector to base) and output (collector to  ) impedances respectively. Transistor impedance will be ignored initially. Input impedance at the base terminal is:

 

Where   is the input voltage and   is the input current. The voltage   is given by

 

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

 

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

 

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

 

Solving for   and substituting above yields:

 

Capacitor-Inductor-Capacitor Model

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If   is an inductor,   is a capacitor, and frequencies above resonance:

 
  for  
 
 

Substituting   and  :

 

The input impedance appears as a negative resistance in series with an inductor.

Inductor-Capacitor-Inductor Model

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If   is a capacitor,   is an inductor, and frequencies below resonance:

 
  for  
 
 
 

The input impedance appears as a negative resistance in series with a capacitor.

Wien Bridge Oscillator

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Input admittance analysis

The analysis of the circuit will be performed by looking at the circuit from the negative impedance viewpoint. If a voltage source is applied directly to the input of an ideal amplifier with feedback, the input current will be:

 

Where   is the input voltage,   is the output voltage, and   is the feedback impedance. If the voltage gain of the amplifier is defined as:

 

And the input admittance is defined as:

 

Input admittance can be rewritten as:

 

For the Wien bridge, Zf is given by:

 

 

If   is greater than 1, the input admittance can be thought as of a negative resistance in parallel with an inductance. The inductance is:

 

As a capacitor with the same value of C is placed in parallel with the input, the circuit has a natural resonance at:

 

Substituting and solving for inductance yields:

 

If   is chosen to be 3:

 

Substituting this value yields:

 

Or:

 

Similarly, the input resistance at the frequency above is:

 

For   = 3:

 

The resistor placed in parallel with the amplifier input cancels some of the negative resistance. If the net resistance is negative, amplitude will grow until clipping occurs. Similarly, if the net resistance is positive, oscillation amplitude will decay. If a resistance is added in parallel with exactly the value of R, the net resistance will be infinite and the circuit can sustain stable oscillation at any amplitude allowed by the amplifier.

Notice that increasing the gain makes the net resistance more negative, which increases amplitude. If gain is reduced to exactly 3 when a suitable amplitude is reached, stable, low distortion oscillations will result. Amplitude stabilization circuits typically increase gain until a suitable output amplitude is reached. As long as R, C, and the amplifier are linear, distortion will be minimal.