# Oscillator Injection Locking

If the isolation between 2 oscillators is inadequate injection locking can occur if the frequency difference between the 2 sources is sufficiently small. When comparing the oscillator with a high stability reference source, injection locking may cause the oscillator to phase lock to the reference source and thus have much greater stability (relative to the reference source) than it has when injection locking doesn't occur.

The maximum fractional frequency difference for which injection locking can occur depends on the loaded Q of the Oscillator tank and the amplitude of the injected signal.

When the amplitude of the injected signal is small, the maximum injection locking range is given by:

Δf/fo = (1/2Q)(Ainj/A)

Where

Δf is the frequency locking range

fo is the frequency to which the oscillator tank is tuned.

Q is the oscillator tank loaded Q

Ainj is the amplitude of the injected signal at the oscillator tank

A is the amplitude of the unperturbed oscillator tank signal

The injection locking range is reduced by increasing the oscillator tank circuit Q, operating with high tank circuit power and reducing the amplitude of the injected signal. The latter can be achieved by using a high reverse isolation output buffer, using effective shielding, using well shielded cables, and ensuring that the isolation between the input ports of any device to which 2 or more oscillators are connected is high. In particular all unused outputs should be terminated in well screened loads.

If the frequency difference between 2 oscillators is greater than the injection locking range injection locking will not occur.

If the short term frequency instability of the oscillator's tank frequency is sufficiently large injection locking may not occur as the frequency offset between the tank frequency and the injected signal frequency may not remain in the injection locking range for sufficiently long for locking to occur.

## EXAMPLES

### EXAMPLE 1: Fundamental crystal oscillator with low isolation buffer

For an oscillator like the Wenzel low noise crystal oscillator driving a 2 input port load with an isolation of 40dB between the input ports and a +13dBm source driving the other load input port. The crystal oscillator buffer isolation is about 40dB and a typical fundamental crystal may have a Q of around 100,000 or so. If the crystal dissipation is 1mW (0dBm) and the output is +7dBm then the injected power at the crystal is:

+13dBm - 40dB - 40dB = -67dBm

The ratio of the injected power to the crystal power dissipation is

(-67 + 0)dB = -67dB

thus

Then Ainj/A ~ 4.46x10-4

and the fractional injection locking range is:

Δf/f = (1/2.105)(4.46x10-4) = 2.23 x10-9

### EXAMPLE 2: Overtone crystal oscillator with high isolation buffer

For an E1938A crystal oscillator driving a 2 port load with an isolation of 40dB between the ports and a +13dBm source driving the other load input port. The output to crystal isolation is ~ 100dB and the loaded crystal Q is ~ 1,000,000, the crystal dissipation is ~ 50uW (-26dBm) and the oscillator output is +4dBm thus the injected power at the crystal is:

+13dBm - 40dB - 100dB = -127dBm

The ratio of the injected power to the crystal power dissipation is

(-127 + 26)dB = -101dB

thus

Ainj/A ~ 8.9 x 10-6

and the fractional injection locking range is:

Δf/f = (1/2.106)(8.9 x 10-6) = 4.45 x 10-12

### REFERENCES

1) R. Adler, "A Study of Locking Phenomena in Oscillators," in Proc. IRE, 1946, vol. 34, pp 351-357.

later reprinted by IEEE as:

2) R. Adler, “A study of locking phenomena in oscillators proc. IEEE, vol. 61, pp. 1380-1385, 1973.