Oscillatory circuit or tank circuit

A circuit that produces electrical oscillations of any desired frequency is called an oscillatory circuit or tank circuit. The simplest tank circuit is a capacitor $C$ in parallel with an inductance coil $L$, fed through a two-way plug key APB (Figure 6.2.1a).

How a tank circuit oscillates

Suppose the moving pole P of the key is first thrown to point A. The capacitor charges to the battery potential $V$, the upper plate becoming positive and the lower plate negative — the tank now stores electrostatic energy in $C$.

Now P is thrown to point B (Figure 6.2.1b). The inductor $L$ and capacitor $C$ form a closed loop and the capacitor begins to discharge through $L$. Because the current is changing, the coil develops an induced emf that opposes the growth of current (Lenz's law), so the capacitor does not discharge instantly; instead the current rises gradually and builds a magnetic field around the coil. When the current is maximum the potential difference across the capacitor is zero — all the electrostatic energy of $C$ has been converted into magnetic energy stored in $L$. Assuming no resistance, no energy is lost in this interchange.

Once the capacitor is fully discharged the magnetic field starts to collapse and induces a counter-emf that, again by Lenz's law, keeps the current flowing in the same direction. This current recharges the capacitor with the opposite polarity — upper plate now negative, lower plate positive (Figure 6.2.1c). The capacitor then discharges again in the reverse direction, and the whole charge-discharge cycle repeats, producing an alternating (oscillating) current. The frequency of these oscillations is

Losses in the tank circuit

If there were truly no losses this energy interchange between $L$ and $C$ would continue forever. In a real tank circuit, however, there are resistive and radiation losses in the coil and dielectric losses in the capacitor, so a small part of the energy is lost every cycle. The amplitude therefore decreases steadily to zero, giving damped oscillations (Figure 6.2.2). To obtain sustained oscillations the lost energy must be replaced from outside — this is done by connecting an amplifier to the tank circuit, which is the basis of every LC oscillator.

LC oscillators

LC oscillators use inductance $L$ and capacitance $C$ to generate oscillations; each consists of an amplifier together with an LC tank circuit in its feedback network. According to the nature of the feedback there are two important types:

1. Hartley oscillator — uses inductive feedback.
2. Colpitts oscillator — uses capacitive feedback.

LC oscillators are widely used for high frequencies, in RF signal generators, radio and TV receivers and in high-frequency heating.

Hartley oscillator

In the Hartley oscillator (Figure 6.2.3) the resistors R1, R2 and RE set the DC bias of the transistor, CE is a bypass capacitor and CC provides coupling. The RFC (radio-frequency choke) behaves as a short circuit for DC and an open circuit for AC, so it isolates the DC supply from the collector oscillations. Two inductors L1 and L2 (with a common tapping point) together with the capacitor $C$ form the tank circuit; the output is taken across L1.

Working. At switch-on the collector current charges $C$, which then discharges through L1 and L2, setting up oscillations in the tank. The voltages developed across the two inductors are in opposite phase: the voltage across L2 is fed back to the input and the voltage across L1 is the output. The transistor amplifier provides 180 degrees and the tapped inductor another 180 degrees, giving 360 degrees around the loop, as required.

The frequency of oscillation is $f = \dfrac{1}{2\pi\sqrt{LC}}$ with the total tank inductance $L = L_1 + L_2$ (if the mutual inductance $M$ between the coils is neglected; if it is present, $L = L_1 + L_2 \pm 2M$, with $+2M$ for series-aiding and $-2M$ for series-opposition winding). The feedback ratio is

L1 and L2 are chosen so that $A\beta = 1$, which fixes the required amplifier gain at $A = \dfrac{L_1}{L_2}$ (i.e. $A\beta = A \cdot \dfrac{L_2}{L_1} = 1$).

Colpitts oscillator

The Colpitts oscillator (Figure 6.2.4) has the same biasing arrangement — R1, R2 and RE for DC bias, CE as bypass, CC for coupling and the RFC for AC/DC isolation — but its tank circuit is a single inductor $L$ with two capacitors C1 and C2 in series, the junction between them being grounded.

Working. At switch-on the rising collector current charges C1 and C2. Because their common terminal is earthed, the voltages across the two capacitors are in opposite phase; the voltage across C2 is fed back to the amplifier. When the capacitors are fully charged they discharge through $L$, sustaining the oscillation. The tank provides 180 degrees and the amplifier 180 degrees for a total of 360 degrees.

The voltage feedback fraction is

$\beta = \dfrac{V_f}{V_o} = \dfrac{I X_{C2}}{I X_{C1}} = \dfrac{I/\omega C_2}{I/\omega C_1} = \dfrac{C_1}{C_2}$ …