Phase Locked Loops

Chris Angove is familiar with both the analog and digital type phase locked loops, both in discrete components and in integrated circuit form. He understands the basic PLL architectures and principles as described in the complex frequency plane and has designed and developed several of the basic PLL components including amplifiers, loop filters, couplers and VCOs. He has developed several PCBs to carry PLL circuits which included extra screening and filtering  to help achieve phase noise, spurious and other requirements.  

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CONTENTS

  1. Phase Locked Loops
  2. Second Order Synthesizer Feedback Loop
  3. Synthesizer Parameters
  1. Phase Locked Loops

    There are many excellent references I often use for anything connected with phase locked loops (PLLs) for example Robins and Gardner.. Synthesisers for a multitude of applications from high stability low phase noise through fast hopping and beyond are widespread in everyday equipment. Cellular or mobile communications, especially digital in the last 10 years or so, have promoted this even more. Synthesizer ICs have progressed from dividers only, through programmable dividers, inclusion of phase detectors, loop filters and even VCOs.

    A PLL is a form of negative feedback, so the theory is totally based on feedback theory. In many branches of electrical engineering a feedback circuit is usually designed to feed a portion of voltage from the output of an amplifier to its input and there are of course various ways of doing that. The simple voltage case would be DC or steady state, that is, considering the conditions after a long time has elapsed. A more realistic and practical feedback model would include complex variations of voltage, frequency and phase with respect to time. A PLL is one example.

    A PLL works at a fixed frequency and phase once steady state has been achieved. A simple example is shown below and would typically be used to provide a relatively high level sinusoidal output, but locked to a high quality, stable, relatively low power or 'reference' signal. A VCO provides appreciable signal, but with relatively crude control of frequency and effeectively no control of phase. A reference frequency oscillator, such as a crystal oscillator source, provides a high quality source but of limited output power. The PLL provides the best of both worlds.

    As we know, the parameter 's' is complex frequency equivalent to jw. Mathematically this is a very useful tool, and rather more than just the simple advantage that 's' is easier to write than 'jw'. However I think there is more to it than this, Laplace transforms to start with. Laplace transforms may be used to transform from the time to the complex frequency domain or, in the opposite direction for its inverse form. Well, I hear you say, 'isn't that what Fourier transforms do?. Well they do but I believe that Laplace transforms are more suitable for considering pulses and transients, which are very important in PLLs, and Fourier transforms are better for steady state conditions. In fact a necessary condition for taking the discrete Fourier of a waveform is that it continues to infinity, ie. has been present for ever. Of course there are mathematical ways of getting around this.

    2ndorder3.jpg

Apologies if the figures in this document have not rendered very well. It was because they were imported from TurboCAD into an early version of Microsoft Word, then exported into HTML which never seemed to work very reliably. When I get time I will generate a nice new set of notes in PDF.

The function of the phase detector is to compare the phase of a fed back sample, originating from the VCO, with that of the reference frequency oscillator. The phase detector output is an error signal proportional to the difference in phase between these two signals, normally just a proportional voltage. The error signal is integrated by the loop filter and used to control the VCO frequency in such a direction that it corrects for the phase error, since it is negative feedback. This will reduce to zero when phase lock is achieved. Although the diagram does not show it, the phase detector might not be able to operate at the VCO frequency and there might be a frequency divider of some type in the feedback path.

We always have 3 varying parameters: phase (f), angular frequency (w) and of course time (t). For a fixed frequency then

pll_3.gif

Whenever phase varies, so does the frequency, since the angular frequency is defined as the derivative of phase with respect to time.

pll_4.gif

Another way of putting that is

pll_5.gif

Although there is a physical electrical connection forming a closed loop, the voltage within that loop is not the only parameter used to control it. Starting at the VCO, a voltage input controls a frequency output or perhaps more accurately a frequency deviation output relative to a fixed frequency, the natural frequency at which the VCO oscillates. The phase detector provides a voltage output proportional to the different between the phases of 2 inputs. So we have frequency, phase and voltage. The dimensions of frequency are 1/TIME so we have time as well. All this gets horrendously confusing and I don't pretend to understand it particularly well and I would suggest reading any of the very good text books on PLLs and synthesizers around. I use Robins and Gardner. but they are probably getting a bit dated now.

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  1. The Second Order Synthesizer Transfer Function

    This is an extension of the phase locked loop but incorporating a change in frequency, so it is called a synthesizer in the sense that it can synthesize the desired frequency, but it is simply an extension of the PLL. (If you put 'synthesiser' into Google it comes back with 'did you mean synthesizer?', but that does not work in reverse so the American spelling is of course much more common, something to bear in mind with search engines. Also there are many more references to audio synthesizers, the type used as musical instruments.)

    2ndorder2.jpg

    In this list of variables, each one is a function of 's' the complex frequency parameter, remembering that s = jw.

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  2. Synthesizer Parameters
    • Phase Noise
    • Harmonics
    • Spurious
    • Output Power
    • Stability
    • AM Noise
    • Lock Time
    • Pulling
    • Pushing
    • Aging
    • Power Consumption / Efficiency
    • Lock Time / Settling Time
    • Warm Up Time

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