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**Introduction:**

Resistors R_{1}, R_{2} and R_{3} determine dc bias of the circuit.

The parallel - network in emitter circuit is the stabilizing circuit to prevent signal degeneration. As usual, C_{1} is dc blocking capacitor. Mutually-coupled coils L_{1} and L forming primary and secondary coils of an RF transformer provide required feedback between collector and base circuits. Amount of feedback depends on coefficient of coupling between two coils. CE connected transistor itself provides phase shift of 180° between its input and output circuits. Transformer gives another 180^{o} phase shift and therefore producing the total phase shift of 360^{o} which is necessary condition for producing oscillations. Parallel-tuned LC circuit connected between base and emitter is frequency determining network (FDN) i.e. it produces oscillations at its resonant frequency.

**Circuit Action:**

Moment switch S is closed, collector current is set up that tends to rise to quiescent value. This increase in I_{C} is accompanied by:

1. Expanding magnetic field through L_{1} that links with L and

2. Induced emf called feedback voltage in L.

Two immediate reactions of this feedback voltage are:

- Increase in emitter-base voltage (and base current) and
- Further increase in collector current I
_{C}

It is followed by the succession of cycles of:

1. Increase in feedback voltage

2. Increase in emitter-base voltage and

3. Increase in until saturation is reached.

In the meantime c gets charged. As soon as I_{C} stops to increase, magnetic field of L_{1} stops to expand and therefore no longer induces feedback voltage in L. Having been charged to maximum value begins to discharge through L.

During this time, capacitor having lost its original charge, again becomes completely charged though with opposite polarity. Transistor being in cut-off, capacitor will again start to discharge through L. As polarity of capacitor charge is opposite to that when transistor was in saturation, sequence of reactions now will be:

- Increase in emitter-base bias,
- An increase in I
_{C}, - An expanding magnetic field in L1,
- An induced feedback voltage in L,
- A further increase in emitter-base bias and
- So on till I
_{C}increases to its saturation value

This cycle of operation keeps repeating so long as sufficient energy is supplied to meet losses in LC circuit. Output can be taken out by means of third winding magnetically coupled to L_{1}. It has approximately same waveform as collector current. Frequency of oscillation is equal to resonant frequency of LC circuit.

*Tuned Collector Oscillator:*

Frequency Determining Network (FDN): It is composed of variable capacitor C and coil L that forms primary winding of step-down transformer. Combination of Land C forms the oscillatory tank circuit to set frequency of oscillation.

Resistors R_{1}, R_{2} and R_{3} are utilized to dc bias transistor. Capacitors C_{1} and C_{2 }act to bypass R_{3 }and R_{2 }respectively so that they have no effect on ac operation of circuit. Furthermore, C_{2} provides ac ground for transformer secondary L_{1}.

**Positive Feedback:**

Feedback between collector-emitter circuit and base-emitter circuit is provided by transformer secondary winding L_{1} that is mutually- coupled to L. As far as ac signals are concerned, L_{1} is connected to emitter via low-reactance capacitors C_{2} and C_{1}.

As transistor is connected in CE configuration, it gives phase shift of 180° between its input and output circuits. Another phase shift of 180^{o} is provided by transformer therefore producing total phase shift of 360^{o} between output and input voltages resulting in positive feedback between two.

**Working:**

When supply is first switched on, transient current is developed in tuned LC circuit as collector current rises to its quiescent value. This transient current starts natural oscillations in tank circuit. These natural oscillations induce the small emf into L_{1} by mutual induction that causes corresponding variations in base current. These variations in I_{B} are amplified β times and appear in collector circuit. Part of this amplified energy is utilized to meet losses taking place in oscillatory circuit and balance is radiated out in form of electromagnetic waves. Frequency of oscillatory current is almost equal to resonant frequency of tuned circuit.

f_{0} = 1/2π√LC

*Hartley Oscillator:*

It uses single tapped-coil having two parts marked L_{1} and L_{2} instead of two separate coils. So far as ac signals are concerned, one side of L_{2}is joined to base via C_{1} and the other to emitter via ground and C_{3}. Similarly, one end of L_{1} is joined to collector via C_{2} and other to common emitter terminal via C_{3}. In other words, L_{1 }is in output circuit i.e. collector-emitter circuit while L_{1} is in base-emitter circuit i.e. input circuit. These two parts are inductively-coupled and form the auto-transformer or split-tank inductor. Feedback between output and input circuits is accomplished through autotransformer action that also introduces phase reversal of 180°. This phase reversal between two voltages takes place as they are taken from opposite ends of inductor ( L_{1} - L_{2} combination) with respect to tap that is tied to common transistor terminal i.e. emitter that is ac grounded via C_{3}. Positive feedback is obtained from tank circuit and is coupled to base via C_{1}. Feedback factor is given by ratio of turns in L_{2} and L_{1} i.e by N_{2}/N_{1} and its value ranges from 0.1 to 0.5.

Resistors R_{1} and R_{2} form voltage divider for providing base bias and R_{3} is emitter swamping resistor to add stability to circuit. Capacitor C_{3} gives ac ground thereby preventing any signal degeneration while still giving temperature stabilization. Radio- frequency choke (RFC) gives dc load for collector and also keeps ac currents out of dc supply V_{CC}.

When V_{CC} is first switched on through S, initial bias is established by R_{1} - R_{2} and oscillations are produced due to positive feedback from LC tank circuit ( L_{1} and L_{2} constitute L). Frequency of oscillation is given by f_{0 }= 1/2π√LC where L = L_{1} + L_{2} + 2M

Output from tank may be taken out by means of another coil coupled either to L_{1} or L_{2}.

*Colpitts Oscillator:*

This oscillator is fundamentally the same as Hartley oscillator except for one difference. Colpitts oscillator utilizes tapped capacitance while Hartley oscillator employs tapped inductance. Two series capacitors C_{1} and C_{2} form voltage divider utilized for providing feedback voltage (voltage drop across C_{2} comprises feedback voltage). Feedback factor is C_{1}/C_{2}. Minimum value of amplifier gain for maintaining oscillations is

A_{v}(min) = 1/C_{1}/C_{2} = C_{2}/C_{1}

Tank circuit comprises of two ganged capacitors C_{1} and C_{2} and single fixed coil. Frequency of oscillation is given by:

f_{0} = 1/2π√LC where C = C_{1}C_{2}/(C_{1} + C_{2})

Transistor itself produces the phase shift of 180^{o}. Another phase shift of 180^{o} is given by capacitive feedback therefore giving total phase shift of 360^{o} between emitter-base and collector-base circuits.

Resistors R_{1 }and R_{2} form voltage divider across V_{CC} for providing base bias, R_{3} is for emitter stabilization and RFC gives essential dc load resistance RC for amplifier action. It also prevents ac signal from entering supply dc V_{CC}. Capacitor C_{5} is bypass capacitor while C_{4} conveys feedback from collector-to-base circuit.

When S is closed, abrupt surge of collector current shock-excites tank circuit in oscillations that are sustained by feedback and amplifying action of transistor.

Colpitts oscillator is extensively utilized in commercial signal generators up to 1MHz. Frequency of oscillation are varied by gang-tuning two capacitors C_{1} and C_{2}.

*Clapp Oscillator:*

It is a variation of Colpitts oscillator. It varies from Colpitts oscillator in respect of capacitor C_{3} only which is joined in series with tank inductor. Addition of C_{3}

(i) Enhances frequency stability and (ii) eliminates effect of transistor's parameters on operation of circuit. Operation of this circuit is same as that of Colpitts oscillator. Frequency of oscillation is provided by

f_{0} = 1/2π√LC where 1/C = 1/C_{1 }+ 1/C_{2} + 1/C_{3}

*Crystals:*

For exceptionally high degree of frequency stability, use of crystal oscillators is necessary. Crystal usually utilized is finely-ground wafer of translucent quartz (or tourmaline) stone held between two metal plates and housed in package about size of the postal stamp. Crystal wafers are cut from crude quartz in two different ways. Method of cutting determines crystal's natural resonant frequency and its temperature coefficient. When wafer is cut so that its flat surface is perpendicular to electrical axis, it is known as X-cut crystal. But if wafer is so cut that flat surfaces are perpendicular to mechanical axis, it is known as Y- cut crystal.

**Piezoelectric Effect:**

The quartz crystal described above has strange properties. When mechanical stress is applied across two opposite faces, potential difference is developed across them. It is known as piezoelectric effect. On the other hand, when the potential difference is applied across its two opposite faces, it causes crystal to either expand or contract. If the alternating voltage is applied, crystal wafer is set in vibrations. Frequency of vibration is equal to resonant frequency of crystal as determined by structural characteristics. Where frequency of applied ac voltage equals natural resonant frequency of crystal, the amplitude of vibration will be maximum. As the general rule, the thinner the crystal, the higher its frequency of vibration.

**Equivalent Electrical Circuit: **

It comprises of series RLC_{1} circuit in parallel with the capacitor C_{2}. Circuit has two resonant frequencies:

One is lower series resonance frequency f_{1} that takes place when XL = XC_{1}. In that case, Z = R.

f_{1} = 1/2π√LC_{1}

**Q-factor:**

Equivalent inductance of the crystal is very high as compared to either its equivalent capacitance or equivalent resistance. Due to high ratio, L/R Q-factor of the crystal circuit is 20,000 as compared to maximum of about 1000 for high-quality LC circuits. Consequently, greater frequency stability and frequency discrimination are obtained due to extremely high Q (up to 10_{o}) and high L/R ratio of series RLC_{1} circuit.

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