Chris Angove has significant design and development experience involving synthesizers, both the fractional N and dual modulus types. He has designed a number of synthesizer components including VCOs, filters, mixers, amplifiers, couplers, attenuators and programming interfaces as well as the compartmentalised PCBs to carry the components. Many of the synthesizers he has worked on were required to meet some tight phase noise specifications, usually over temperature, and much of the development time was devoted to improvements to achieve these. He is also familiar with achieving other requirements such as lock time, spurious emissions, output power level, pushing/pulling, power consumption / efficiency and frequency stability. Some of the synthesizers have been multi-loop so these have presented additional challenges in essentially making several synthesizers work together effectively.
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There are many excellent references I often use for anything connected with phase locked loops (PLLs) for example Robins and Gardner.. In the last 30 years or so, the volume of PLL circuits has exploded (and a few PLL circuits have exploded as well). 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 electronics feedback usually comprises a portion of voltage fed 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. More realistic and practical applications of feedback would include complex variations of voltage with respect to time. A PLL is one example.
A PLL works at a fixed frequency once steady state has been achieved. A simple example is shown below and would typically be used to provide an appreciable level of sinusoidal signal output, but locked to a high quality, stable, relatively low power or 'reference' signal. A VCO provides appreciable signal, but with little accurate control of frequency or, more precisely, phase. A reference frequency oscillator, such as a crystal 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, if only that 's' is less 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 frequency domain or, in its inverse form, from the frequency to the time domain, similar to Fourier transforms.
The function of the phase detector is to compare the phase of a fed back sample of the VCO signal 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. 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. This will reduce to zero when phase lock is achieved.
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. The phase detector provides a voltage output from the comparison of 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 in any great depth.
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This is an extension of the phase locked loop but incorporating a change in frequency. There is a practical limit to the upper frequency at which the phase detector will operate and very often the actual required
To start with, ignoring the noise contribution qn(s) of the VCO, the output of the phase detector is:
Therefore the output is
Rearranging these gives:
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, Kf and KV 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 where necessary to convert to the correct dimensions. 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 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, virtually always using an operational amplifier, are very much more flexible. However they need voltage supplies, albeit usually of small current requirements. Operational amplifier performance is specified by a multitude of parameters and these days literally hundreds are available and are constantly being improved. These must all be studied carefully before selection and the 'stuff in a 741' addage ended some years ago. Operational Amplifiers are a whole subject in their own right.
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.
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.
Applying potentiometer and operational amplifier theory and treating everything in terms of the complex frequency variable, s gives:
Substituting the loop filter transfer function F(s) gives, after some manipulation,
wn is known as the natural frequency of the loop and is given by
xn is the damping factor
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