PLL Theory Q&A

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Why is the transfer function of the VCO?

This is only true if you treat the VCO output as a phase. It breaks down like this:

Why is the VCO an integrator?

Consider a sinusoid of frequency ω:

Let the instantaneous phase be θ:

Now, phase just keeps on truckin' 2π, 4π, 6π, 8π ... 1000000π ... up and up:

Frequency is the rate of change of phase:

Conversely, phase is the integral of frequency:

When analysing the VCO output, we don't look at instantaneous voltage or current. We look at phase. The phase (in radians) is:

When the loop is in lock:
Φout = Φin

It's not a square wave or a sine wave. It's a ramp. The higher the frequency, the steeper the slope:

The VCO input is a voltage but the output is a phase. Thought of this way, it's an integrator. Now, an integrator is a low pass filter. So, the VCO is also a low pass filter!

What is the transfer function of an integrator?

Imagine an integrator with gain k

The transfer function is

1/s is the Laplace operator for integration. This is the First Integration Theorem.

Why does increasing loop gain widen the bandwidth?

If you have a response which rolls off with frequency (e.g. K/S) you can raise the bandwidth by increasing k:

How is kpd calculated?

Signals are variously represented around the loop by pulse width, voltage and phase.

The phase detector output is a pulse width modulated (PWM) representation of the quantity Φinout sampled at the loop comparison frequency. The pulses can be positive or negative. The duty cycle represents a fraction of 2π radians lagging or leading.

Phase detector gain kPD (milliamps per radian) is equal to the slope of the transfer characteristic (left).

As long as the comparison frequency is well outside the loop bandwidth, the PWM is de-modulated (i.e. averaged) by the loop filter. The mean value is proportional to phase error:

What's the difference between open and closed loop gain?

Being a feedback system, the PLL can oscillate if the total phase shift around the loop is 360°. To check stability, we "break" the loop.

The normal input (Φin) is earthed for the purpose of this thought experiment. The open loop gain is defined as:

The loop oscillates if |G|=1 and G approaches 180° i.e. if G ≈ -1 at any frequency. It only takes 180° because the phase comparator -ve input contributes another 180°.
 

Closed loop gain is defined as the system transfer function:

Try substituting G = -1 into this equation.