# Documentation

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## Fractional-N Frequency Synthesis

This model shows how to simulate a phase-locked fractional-N frequency synthesizer. The model multiplies the frequency synFr of a reference signal by a constant synN+synM, to produce a synthesized signal of frequency synFr(synN+synM). A feedback loop maintains the frequency of the synthesized signal at this level. In this example, synN is an integer and synM is a fraction between 0 and 1. This approach has several advantages, since it enables you to approximate the frequency of the synthesized signal with relatively small values for synN and synM.

Fractional-N PLL synthesizers attain improved frequency resolution at the expense of increased circuit complexity and increased phase noise (timing jitter) over their non-fractional counterparts. It also enables the use of a larger reference frequency. For more information, see Selected Bibliography. This model implements a fractional-N scheme with analog phase error correction in the phase detector portion of the design. This eliminates the sidebands introduced by switching the loop divider between N and N+1.

There is a simpler example available, PLL-Based Frequency Synthesis Example, which produces a synthesized signal of frequency synFr*synN/synM, where synN and synM are integers.

### Structure of the Example

The model uses these variables in addition to synN and synM:

• synFr - frequency of the reference signal

• synFq - quiescent frequency in the Continuous-Time VCO block

• synSen - Voltage-Controlled Oscillator input sensitivity

The model initially assigns values to these variables as follows:

• synN = 10

• synM = 0.3

• synFr = 10 MHz

• synFq = 90 MHz

• synSen = 10 MHz/V

The frequency of the synthesized signal at the model's steady state is then 103 MHz. After running the simulation with these values, you can later change them by typing new values in the MATLAB® Command Window, if you want to experiment with the model.

Blocks and Subsystems in the Example

Many of the blocks in this model function in the same way as they do in the PLL-Based Frequency Synthesis Example. This section discusses the subsystems that are different.

Accumulator: The Accumulator subsystem repeatedly adds the constant synM to a cumulative sum. While the sum is less than 1, the output labeled Carry is 0. At a time step when the sum becomes greater than or equal to 1, the carry output is 1 and the cumulative sum is reset to its fractional part. The fraction of the time when the carry output is 1 is equal to synM, while the fraction of the time when it is 0 is equal to 1-synM. The accumulator controls the switching between N and N+1 with the carry output. The accumulator output (state) is used to drive the phase compensation circuitry.

Divide Frequency: Divide by N or N+1 is implemented using a "swallow" counter scheme, as it would likely be done in hardware. The Divide Frequency subsystem divides the frequency of the synthesized signal by synN when the output of the Accumulator subsystem is 0, and divides it by synN+1 when the output is 1. As a result, the average value that the frequency is divided by is

(1-synM)*synN + synM*(synN+1) = synN + synM = 10.3

Phase Detector: Phase-frequency detector and error compensator uses a "dual D" flip-flop for phase detection, along with an integrator, Sample and Hold block, and a simple lead/lag loop filter for error compensation.

### Results and Displays

When you run the simulation the following are the Scopes that are displayed:

Control Signals Scope:

1. The control signal, which the VCO block uses to maintain the frequency of the synthesized signal

2. The VCO Control signal: This is seen to settle to a stable value with Phase compensation set to 'on' (default)

Synthesized Signal Scope:

• The square wave generated based on the VCO output

RF Spectrum Analyzer Spectrum Scope:

• Displays the frequency spurs.

### Exploring the Example

The switch labeled 'Phase Comp Enable' is set to 'on' by default. In this mode timing jitter and sideband frequency spurs are completely eliminated with just a single frequency spur of -80 dB remaining.

Run the simulation with the compensation off: switch in lower position. Now the graphs will show significant phase jitter and sidebands due to the periodic variations of the VCO control voltage.

The Simulink model outputs a very high quality synthesized output. Real world component limitations can now be introduced to study their effects on the system performance.

### Selected Bibliography

Egan, William F., "Fractional-N and Relatives", Frequency Synthesis by Phase Lock, (2nd ed., pp. 371-390). N.Y., John Wiley & Sons, 2000.

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