## Hartley oscillator Definition

Fig. 4 shows the **circuit diagram of a Hartley oscillator** using a transistor in C.E. configuration.

In **hartley oscillator**, tank circuit consists of L_{1}, L_{2} and C_{1}. Stabilized 1self-bias to the amplifier is provided by R_{1}, R_{2}, R_{e} and C_{e}. Radio frequency choke is used to provide load for the collector and the reactance of RF choke is greater than that of L_{2}.

Hence the reactance of RF choke may be neglected in the equivalent circuit. The capacitor C_{c} is used to provide path to the A.C. from the collector to the tank circuit. The capacitor in the base circuit C_{b }can be neglected in the equivalent circuit as it gives low reactance at the frequency of oscillations.

The transistor in common emitter configuration introduces a phase shift of 180^{o} between the output and input voltages. Moreover, a phase shift of 180^{o} between output and input voltages is also introduced by L_{1} and L_{2 }(a transformer).

Hence the net phase shift of 360^{o} or 0^{o} is introduced between the input and output voltages leading to positive feedback which is essential for the sustained oscillations.

## Hartley Oscillator Working

When collector current flows and begins to increase, the charging of capacitor C_{1} begins. After that capacitor C_{1} is fully charged, it begin to discharge through L_{1} and L_{2}. Hence oscillations are produced.

The voltage in coil L_{1} is induced by this oscillations. This voltage is applied between the base and emitter and appears in the collector circuit in the amplified form. The mutual inductance M between L_{1} and L_{2} accomplishes the **feedback of energy from the collector circuit to the base circuit**. Thus, losses taking place in the tank circuit are compensated and hence undamped oscillations are obtained at the output.

## Hartley Oscillator Derivation

The equivalent circuit of hartley oscillator is shown in fig. 5.

Load impedance between output terminals (Z_{L})

Here h_{ie} and Z_{1} are in parallel, so their equivalent impedance is given by

Now Z^{‘} and Z_{3} are in series, so their equivalent impedance is given by

Z^{”} and Z_{2} are in parallel, so the load impedance is given by

Output voltage is given by

Using eqn. (2), we get

Voltage feedback to the input is given by

Using eqn. (1), we get

The voltage gain of C.E. amplifier (without feedback) is given by

According to Barkhausen criterion,

Using eqn. (3), we get

Using values of Z_{1}, Z_{2}, and Z_{3} as given in fig. 24, we get

Comparing the imaginary parts of eqn. (9), we get

Eqn. (11) is the expression for the **Hartley oscillator frequency formula**.

Comparing the real parts of eqn. (9), we get

Using eqn. (10), we get

Which is the condition for sustained oscillation.