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# Voltage-Controlled Oscillators

Module by: Christopher Schmitz. E-mail the author

Summary: This module provides an overview of the Voltage-Controlled Oscillator for use in Phase-Locked Loops. It additionally addresses the software-based Numerically-Controlled Oscillator, NCO (aka. digitally-controlled oscillator, DCO).

## Voltage-Controlled Oscillator

The basic Voltage-Controlled Oscillator (VCO) is modeled as an ideal integrator with gain Ko Ko (rad/V-s). In this way, the Laplace-Transform representation of the VCO is Ko s Ko s. When driven by the control voltage of the loop filter of a Phase-locked loop (PLL), the output of the VCO is radian frequency with units of rad/s.

### Characterizing the VCO

A VCO has a DC-coupled input port where a control voltage may be applied and must also receive power to drive the oscillator circuit. It also has one or more AC-coupled ports where a sinusoidal signal is generated. As the control voltage is increased, the frequency of the output sinusoid increases. Typically, the increase in frequency is greater per unit increase in voltage at lower control voltages and less per unit increase in voltage at higher control voltages. The response can be modeled as locally linear around the anticipated mean control voltage of the VCO when placed into a working PLL.

Since, normally, VCOs have a non-linear input/output relationship that causes the above linearized model to depend heavily upon the frequency of which it is expected to operate. A more accurate representation of Ko Ko would be Ko(norm) = Ko fo Ko(norm)= Ko fo . In this model, Ko(norm)Ko(norm) should be approximately constant and Ko Kois the linear, tangental curve-fit to the measured input/output data. In this way, Ko(norm)Ko(norm) could be the average value of Ko(norm) = Ko fo Ko(norm)= Ko fo measured at various values of fo fo and then Ko Ko could later be extracted (albiet approximately) from the recorded value of Ko(norm)Ko(norm).

### Why is the VCO an Integrator?

The output frequency can be written as wo = wc+KoVc (radians/sec). Here, wc is the "free-running frequency" of the VCO...the frequency at which it is biased to run in absense of the PLL's loop feedback. The output of the VCO can be written as cos(wct+thetao(t)). The instantaneous frequency is given by wo = d/dt [wct+thetao(t)] = wc+dthetao/dt. Comparing with the earlier equation verifies that dthetao/dt = KoVc

## Numerically-Controlled Oscillator

The Numerically-Controlled Oscillator (NCO, aka. Digitally-Controlled Oscillator) is the discrete-time implementation of the VCO. As the VCO is basically an ideal integrator, the discrete time equivalent is given by G(z)= z-1 1-z-1 G(z)= z-1 1-z-1

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