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This is only true if you treat the VCO output as a phase. It breaks down like this:
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: Φ_{out} = Φ_{in} |
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!
Imagine an integrator with gain k
The transfer function is
^{1}/_{s} is the Laplace operator for integration. This is the First Integration Theorem.
If you have a response which rolls off with frequency (e.g. ^{K}/_{S}) you can raise the bandwidth by increasing k:
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 Φ_{in}-Φ_{out} 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 k_{PD} (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:
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 (Φ_{in}) is earthed for the purpose of this thought experiment. The open loop gain is defined as:
The loop oscillates if |G|=1 and ÐG 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.
Copyright © Andrew Holme, 2004. |