Phase-domain equations

VCOs are used in analog applications such as frequency modulation and frequency-shift keying. The functional relationship between the control voltage and the output frequency for a VCO (especially those used at radio frequency) may not be linear, but over small ranges, the relationship is approximately linear, and linear control theory can be used. A voltage-to-frequency converter (VFC) is a special type of VCO designed to be very linear over a wide range of input voltages.

Modeling for VCOs is often not concerned with the amplitude or shape (sinewave, triangle wave, sawtooth) but rather its instantaneous phase. In effect, the focus is not on the time-domain signal A sin(ωt+θ₀) but rather the argument of the sine function (the phase). Consequently, modeling is often done in the phase domain.

The instantaneous frequency of a VCO is often modeled as a linear relationship with its instantaneous control voltage. The output phase of the oscillator is the integral of the instantaneous frequency.

${\displaystyle {\begin{aligned}f(t)&=f_{0}+K_{0}\cdot \ v_{\text{in}}(t)\\\theta (t)&=\int _{-\infty }^{t}f(\tau )\,d\tau \\\end{aligned}}}$

• ${\displaystyle f(t)}$ is the instantaneous frequency of the oscillator at time t (not the waveform amplitude)
• ${\displaystyle f_{0}}$ is the quiescent frequency of the oscillator (not the waveform amplitude)
• ${\displaystyle K_{0}}$ is called the oscillator sensitivity, or gain. Its units are hertz per volt.
• ${\displaystyle f(\tau )}$ is the VCO's frequency
• ${\displaystyle \theta (t)}$ is the VCO's output phase
• ${\displaystyle v_{\text{in}}(t)}$ is the time-domain control input or tuning voltage of the VCO

For analyzing a control system, the Laplace transforms of the above signals are useful.

${\displaystyle {\begin{aligned}F(s)&=K_{0}\cdot \ V_{\text{in}}(s)\\\Theta (s)&={F(s) \over s}\\\end{aligned}}}$

LC oscillators

Commonly used VCO circuits are the Clapp and Colpitts oscillators. The more widely used oscillator of the two is Colpitts and these oscillators are very similar in configuration.

Crystal oscillators

27 MHz VCXO clock generator IC (TLSI T73227), used in a DVB-T set-top box.

A voltage-controlled crystal oscillator (VCXO) is used for fine adjustment of the operating frequency. The frequency of a voltage-controlled crystal oscillator can be varied a few tens of parts per million (ppm) over a control voltage range of typically 0 to 3 volts, because the high Q factor of the crystals allows frequency control over only a small range of frequencies.

A 26 MHz TCVCXO

A temperature-compensated VCXO (TCVCXO) incorporates components that partially correct the dependence on temperature of the resonant frequency of the crystal. A smaller range of voltage control then suffices to stabilize the oscillator frequency in applications where temperature varies, such as heat buildup inside a transmitter.

Placing the oscillator in a crystal oven at a constant but higher-than-ambient temperature is another way to stabilize oscillator frequency. High stability crystal oscillator references often place the crystal in an oven and use a voltage input for fine control.^[7] The temperature is selected to be the turnover temperature: the temperature where small changes do not affect the resonance. The control voltage can be used to occasionally adjust the reference frequency to a NIST source. Sophisticated designs may also adjust the control voltage over time to compensate for crystal aging.

Clock generators

A clock generator is an oscillator that provides a timing signal to synchronize operations in digital circuits. VCXO clock generators are used in many areas such as digital TV, modems, transmitters and computers. Design parameters for a VCXO clock generator are tuning voltage range, center frequency, frequency tuning range and the timing jitter of the output signal. Jitter is a form of phase noise that must be minimised in applications such as radio receivers, transmitters and measuring equipment.

When a wider selection of clock frequencies is needed the VCXO output can be passed through digital divider circuits to obtain lower frequencies or be fed to a phase-locked loop (PLL). ICs containing both a VCXO (for external crystal) and a PLL are available. A typical application is to provide clock frequencies in a range from 12 kHz to 96 kHz to an audio digital-to-analog converter.

Frequency synthesizers

A frequency synthesizer generates precise and adjustable frequencies based on a stable single-frequency clock. A digitally controlled oscillator based on a frequency synthesizer may serve as a digital alternative to analog voltage controlled oscillator circuits.