Oscillators Questions and Answers

Answer: A

The two capacitors appear in series across the inductor:

\[ \begin{aligned} C_{s} &= \tfrac{C_{1}C_{2}}{C_{1}+C_{2}} = 0.5 \ \mathrm{nF} \\ f &= \tfrac{1}{2\pi\sqrt{LC_{s}}} \approx 712 \ \mathrm{kHz} \end{aligned} \]

giving about 712 kHz. Using one nanofarad rather than the series combination gives 503 kHz, which is the commonest error. The series rule for capacitors is what makes the Colpitts calculation differ from the Hartley's.

Answer: B

The two inductive sections add, since they are in series:

\[ f = \frac{1}{2\pi\sqrt{LC}} \approx 1.59 \ \mathrm{MHz} \]

giving about 1.59 MHz. Inductances in series add while capacitances in series combine reciprocally, which is exactly why the Hartley uses the total inductance and the Colpitts the series capacitance. Getting that pair of rules the right way round is most of distinguishing the two calculations.

Answer: C

The two capacitors form a divider, so the fraction returned is set by their ratio - a small capacitor across the input and a large one to earth gives a small fraction. The inductance sets the frequency with them, the quality factor governs stability, and the transconductance supplies the gain that must overcome the fraction. Separating which component sets the frequency from which sets the feedback is what makes an oscillator designable rather than merely recognisable.

Answer: D

The Clapp inserts a small capacitor in series with the inductor, and since it is much smaller than the other two it dominates the resonant frequency - which makes the frequency far less sensitive to the transistor's own capacitances. The other options describe different circuits. Arranging for a stable component to dominate a frequency, so that unstable ones matter little, is a general technique and this is its classic application.

Answer: A

The transistor's junction capacitances appear in parallel with the two divider capacitors and drift with temperature and bias, but the small series capacitor dominates the frequency, so those drifts are swamped. The quality factor and the gain requirement are not what changes. Making a frequency depend on the most stable component available, rather than on the sum of several unstable ones, is what the extra capacitor buys.

Answer: B

The Armstrong, or tickler, uses a separate winding coupled to the tank, so the transformer's turns ratio and winding sense set the fraction and the phase. The capacitive divider is the Colpitts and the tapped inductor the Hartley. Its advantage is that the phase can be reversed simply by reversing the winding, and its drawback is that a transformer is a less convenient component than a pair of capacitors - which is why it has largely fallen out of use.

Answer: C

Both reactive components have temperature coefficients, and since the frequency depends on the square root of their product, the drift is half the sum of their fractional changes. Gain and feedback fraction affect amplitude rather than frequency, and the supply is regulated. That the drift is halved by the square root, and that components with opposite coefficients can be chosen to cancel, are the two facts a stable LC design rests on.

Answer: D

Choosing a capacitor whose capacitance rises with temperature to work with an inductor whose inductance falls lets the two drifts cancel in the product, and capacitors are made in a range of specified coefficients for exactly this purpose. A high quality factor improves stability against phase disturbances rather than against component drift, and gain and current affect amplitude. Cancelling two drifts against each other is cheaper than making either component stable, which is why the technique is standard.

Answer: A

Quartz is piezoelectric: mechanical strain produces a voltage across it and an applied voltage strains it, so an electrical circuit can excite and sense a mechanical resonance. The Hall effect is a transverse voltage in a magnetic field, the photoelectric effect concerns light and electrons, and the Seebeck effect produces a voltage from a temperature difference. It is the mechanical resonance, far sharper than any electrical one, that gives the crystal its extraordinary quality factor.

Answer: B

The mechanical resonance appears as a series inductance, capacitance and resistance, with the electrodes' own capacitance in parallel across the whole thing. That structure gives the crystal its two resonances: a series one where the branch is purely resistive, and a parallel one slightly above it where the branch's net inductance resonates with the holder capacitance. The two-resonance behaviour follows entirely from the equivalent circuit, which is why the circuit is worth knowing rather than merely the frequencies.