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.

- 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.

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

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

Another way of putting that is

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.

- 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.)

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

- q
_{i}(s) is the phase noise voltage of the reference frequency oscillator - q
_{n}(s) is the phase noise voltage of the VCO - q
_{0}(s) is the resultant output phase noise voltage - N is the frequency division ratio
- F(s) is the transfer ratio of the loop filter
- K
_{f}is the phase detector transfer function, or phase detector gain, in V/rad - K
_{V}is the VCO transfer function or sensitivity in Hz/VTo start with, ignoring the noise contribution q

_{n}(s) of the VCO, the output of the phase detector is:Therefore the output is

Rearranging these gives:

### Phase Noise Transfer Function

This equation is the transfer function for the whole (closed) loop. It describes how the closed loop affects the input noise voltage from the reference oscillator. As we are using functions of complex frequency, all parameters are expressed in voltage amplitude and phase, ignoring for the moment the noise contribution from the VCO, which we will look at later.

It is usually quite safe to assume that N, K

_{f}and K_{V}are constant with respect to the frequency within the normally locked range and we can simply plug in the values from the data sheet, taking care where necessary to convert to the correct dimensions. Some datasheets are not that reliable and it might be safer to actually measure the VCO frequency against control voltage characteristic. F(s), the transfer function of the loop filter itself, is however a function of frequency and we have to digress a little to have a closer look at them.We can use a passive or active loop filter.

Passive loop filters are of course simple and reliable and do not require any voltage rails. Provided the capacitors are high quality and the resistor values are as small as possible to minimise the Nyquist/Johnson noise, if one of these is adequate, thats great. However there it may well be difficult to provide enough output voltage to drive the control pin of the VCO.

Active filters normally use at least one operational amplifier, and are very much more flexible but they need voltage supplies, albeit usually of small current requirements. Operational amplifier performance is specified by a multitude of parameters and there are literally hundreds different types available which must be studied carefully before selection. Operational Amplifiers are a whole subject in their own right.

### Active Filters

There are 2 basic active filter configurations, one balanced and one unbalanced which make very good starting points for loop filters envisaged in a typical second order synthesizer feedback loop like we have been talking about above. These are of course best suited to phase detector ICs with balanced or unbalanced outputs respectively. Sometimes we can easily make a balanced output drive an unbalanced loop filter but the reverse is not normally so easy. Have a good look through the manufactures' data sheets. There are usually lots of different configurations possible and copious application notes and examples provided from places like Analog Devices, Maxim and National Semiconductor.

Here is the simplest active low pass filter balanced configuration.

This is the equivalent, but for the un balanced configuration.

The voltage transfer function F(s) below is defined at the output voltage divided by the input voltage. This is a function of complex frequency and so is a quantity expressed in both magnitude and phase. Also these sort of transfer functions generally assume a high input impedance and very low output impedance which must be allowed for when calculating any loading effects that might exist when connected to other components at the input and output. It is not like the correctly matched transmission systems that we might be used to with 50 Ohm systems.

Applying potentiometer and operational amplifier theory and treating everything in terms of the complex frequency variable, s gives:

where

and

### The Closed Loop Transfer Function

Substituting the loop filter transfer function F(s) gives, after some manipulation,

w

_{n}is known as the natural frequency of the loop and is given byw

x_{n}must not be confused with the cutoff frequency of the low pass filter. However there are some interesting relationships between these which*Gardner*will tell you about._{n}is the damping factor

- 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

Telephone 020 8249 0957

Copyright ©2015, Chris Angove

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