Oscillators: Colpitts, stability and phase noise
Frequency and start-up condition of the Colpitts proved, frequency stability versus Q proved, phase noise and Leeson's model, crystals, TCXO and OCXO.
Every radio has at least one oscillator: it sets the transmit frequency and the conversion frequency in the receiver. It is not enough that it oscillates: it must have the right frequency, stable, and a clean spectrum. In this lesson we look at the oscillation condition, the Colpitts oscillator with the proof of its frequency and start-up condition, the link between quality factor and frequency stability proved, phase noise and its consequences, and crystals.
The oscillation condition
An oscillator is an amplifier with frequency-selective feedback. By the Barkhausen criterion (Applied Electronics lesson on oscillators), a sinusoidal oscillation is sustained at the frequency where the loop gain has magnitude 1 and phase a multiple of 360°. At switch-on the gain must be somewhat greater than 1, so that noise grows into an oscillation; then non-linearity (amplifier compression, lesson on non-linearity) reduces the average gain to 1 and stabilises the amplitude.
The Colpitts oscillator
The Colpitts uses a π network with one inductor and two capacitors, and an inverting amplifier of transconductance 
; the network provides the other 180° of phase shift.
Theorem. The circuit oscillates at the frequency
![\[ \omega_0 = \frac{1}{\sqrt{LC_s}},\qquad C_s = \frac{C_1C_2}{C_1 + C_2} \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-84a806518585ccaace890912b4172726_l3.png "Rendered by QuickLaTeX.com")
and starts if 
.
Proof. Let 
be the input voltage (on 
) and 
the output voltage (on 
); the amplifier injects the current 
into the output node. Current law at node 1, where the amplifier input draws no current: 
, hence 
and 
. At node 2:
![\[ v_2\left(\frac{1}{R} + j\omega C_2\right) + j\omega C_1v_1 + g_mv_1 = 0 \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-2631aec182564692c808332fa6ab5a00_l3.png "Rendered by QuickLaTeX.com")
Substituting and dividing by 
:
![\[ (1 - \omega^2LC_1)\left(\frac{1}{R} + j\omega C_2\right) + j\omega C_1 + g_m = 0 \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-d2bcc2bac7ffb854b03cdaf8e92efb8f_l3.png "Rendered by QuickLaTeX.com")
The imaginary part vanishes if 
, i.e. 
, which gives 
. The real part then gives 
: this is the gain that exactly sustains the oscillation, and start-up needs a larger value. ∎
Example. 
µH, 
pF: 
pF and 
MHz. With 
kΩ we need 
mS; the design uses a margin of 3-5 times to guarantee start-up over tolerances and temperature. The ratio 
sets how much signal returns to the input.
Quality factor and stability
An oscillator’s frequency shifts when something adds phase to the loop: the transistor capacitance changing with temperature, the supply voltage, the load. A resonator with a high quality factor makes the frequency insensitive to these phase shifts.
Theorem. If a spurious phase shift 
appears in the loop, the frequency shifts by about
![\[ \frac{\Delta\omega}{\omega_0} = \frac{\Delta\varphi}{2Q} \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-30738909b1fd270e27a6750ef8cf64fe_l3.png "Rendered by QuickLaTeX.com")
Proof. Near resonance the impedance phase of a parallel resonator is 
, where 
(lesson on resonant circuits). To keep the total loop phase at zero, the resonator must compensate the spurious phase shift, i.e. work at the frequency where 
. ∎
Example. A phase shift of 0.01 rad with an LC resonator of 
shifts the frequency by 
, 100 ppm: at 1 GHz that is 100 kHz, unacceptable for a radio channel. With a crystal of 
the shift drops to 
.
Phase noise
Component noise also makes the oscillator’s phase fluctuate: the spectrum is not a line, but a line with two “skirts” on either side. It is measured as phase noise 
: the power in a 1 Hz bandwidth at an offset 
from the carrier, relative to the carrier, in dBc/Hz. Leeson’s model estimates it as:
![\[ \mathcal{L}(\Delta f) \approx 10\log_{10}\left[\frac{2FkT}{P_s}\left(1 + \left(\frac{f_0}{2Q\,\Delta f}\right)^2\right)\right] \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-c4771c3acb91512de44e44ce42bace65_l3.png "Rendered by QuickLaTeX.com")
where 
is the amplifier’s noise figure and 
the power in the resonator. The resonator converts the amplifier’s phase noise into frequency noise close to the carrier, falling as 
; again, a high 
and a high power reduce it.
The consequences are practical. In the receiver, a strong signal on an adjacent channel mixes with the local oscillator’s skirts and falls into the wanted channel (reciprocal mixing); in dense digital modulations such as 256-QAM phase noise rotates the constellation points and increases errors (lesson on digital modulation performance).
Crystals
A quartz crystal cut into a plate vibrates mechanically, and through the piezoelectric effect behaves like a resonant circuit with from 
to 
, very stable with temperature. It is the reference of almost all oscillators: temperature-compensated crystal oscillators (TCXO) reach a stability of 0.5 ppm, oven-controlled ones (OCXO) 
. High, variable frequencies are obtained by locking a voltage-controlled oscillator to the crystal with a phase-locked loop (Electrical Communications lesson on FM and PLLs), which within its bandwidth inherits its stability and low phase noise.
