RF oscillators
A 10 MHz crystal specified for an 18 pF load oscillates 41 ppm higher if the circuit gives it 16 pF. That is why the crystal's datasheet and the board's have to agree.
01The oscillation condition at radio frequency
An oscillator is a feedback amplifier that deliberately satisfies the Barkhausen criterion: loop gain of magnitude 1 and zero phase at a single frequency. At low frequency the network that selects the frequency is RC, like the Wien bridge in stability and compensation. At radio frequency, LC resonant circuits or crystals are used, which have a much higher quality factor: their phase changes very fast near resonance, and that fixes the frequency precisely and reduces noise. To start up, the loop gain has to be somewhat greater than 1; the amplitude is then limited by the nonlinearity of the transistor.
02LC oscillators
| Oscillator | Feedback network | Frequency |
|---|---|---|
| Colpitts | Capacitive divider \( C_1, C_2 \) with an inductor | \( 1/\big(2\pi\sqrt{L\,C_1C_2/(C_1+C_2)}\big) \) |
| Hartley | Tapped inductor, or two inductors, with a capacitor | \( 1/\big(2\pi\sqrt{(L_1+L_2)C}\big) \) |
| Clapp | Colpitts with a small capacitor in series with the inductor | Set almost entirely by the series capacitor, less sensitive to the transistor |
In the Colpitts, the capacitive divider feeds back a fraction \( C_1/C_2 \) of the output; with a bipolar transistor, the start-up condition calls for approximately \( g_m R \ge C_2/C_1 \), where \( R \) is the load resistance of the tank. With \( L \) = 1 μH and two 100 pF capacitors, it oscillates at 22.5 MHz. The basic circuits were presented in oscillators, in the Electronics Technician program.
03Phase noise: Leeson's model
A real oscillator does not generate a perfect line: the transistor's noise modulates its phase, and the spectrum has skirts on either side of the carrier. David Leeson proposed in 1966 a model that shows what it depends on:
That is why reference oscillators use crystals, with Q of tens or hundreds of thousands, compared with a few hundred for an LC. The phase noise of the reference is what a synthesizer multiplies, as seen in synthesized generators.
04The quartz crystal
A quartz crystal is piezoelectric: it vibrates mechanically when a voltage is applied, and its mechanical resonance looks electrically like a series RLC circuit, the motional branch (\( L_m, C_m, R_m \)), in parallel with the capacitance of the electrodes \( C_0 \). It has two very close resonances:
A crystal with a series resonance at 10 MHz, specified for an 18 pF load. The curve is the reactance of the crystal between its two resonances; the yellow line, the reactance that the crystal has to cancel in order to oscillate with the chosen load. Their crossing is the operating frequency.
Besides the load, the frequency changes with temperature: in the AT cut, the most widely used, it follows a cubic with a few ppm between −20 and 70 °C. The temperature-compensated (TCXO) and oven-controlled (OCXO) oscillators of time and frequency tackle that dependence. The voltage-controlled oscillators (VCO) use a varactor, a diode whose capacitance depends on the reverse voltage, to tune an LC tank; in a VCXO, the varactor pulls a crystal by a few tens of ppm.
05In the lab
Build a Colpitts oscillator with a transistor, measure the frequency with a counter and compare it with the calculated one. Change the C1/C2 ratio and observe the start-up.
In a Pierce oscillator with a logic gate, change the load capacitors and measure the shift in ppm.
Compare with a spectrum analyzer the phase noise of an LC oscillator and a crystal oscillator of similar frequency.
06Common mistakes
- Forgetting the stray capacitances in the crystal load. Those of the board and the IC add up to several picofarads.
- Driving the crystal with too much power. It ages faster and changes frequency; the drive level is specified.
- Designing with the loop gain exactly at 1. The oscillator will not start with tolerances; margin is needed.
- Loading the tank with the output. It lowers the Q and worsens the phase noise; a buffer is preferable.
07Self-assessment
At what frequency does a Colpitts with L = 2 μH, C1 = 220 pF and C2 = 220 pF oscillate?
\( C_{eq} \) = 110 pF: \( 1/(2\pi\sqrt{2\cdot10^{-6}\cdot 110\cdot10^{-12}}) \approx 10.7 \) MHz.
According to Leeson, how much does the close-in phase noise improve when the resonator's Q is doubled?
6 dB.
What shift does a crystal with Cm = 20 fF and C0 = 5 pF have between loads of 18 and 16 pF?
About 41 ppm.
What component makes it possible to tune a VCO?
A varactor, a diode whose capacitance depends on the reverse voltage.
08Further reading
- Chris Bowick. RF Circuit Design. 2nd ed., Newnes, 2008. RF oscillators and the practical design of resonant circuits.
- Thomas H. Lee. The Design of CMOS Radio-Frequency Integrated Circuits. 2nd ed., Cambridge University Press, 2004. Oscillators and a clear treatment of phase noise.
- David B. Leeson. A Simple Model of Feedback Oscillator Noise Spectrum. Proceedings of the IEEE, 54(2), 1966. The paper with the phase noise model.