linrad support: State of the art crystal oscillators
(Dec 18 2001)

#### Background

The Linrad hardware uses VHF crystal oscillators to convert from HF to VHF and then to 2.5 MHz. The local oscillators have to have very low sideband noise to avoid reciprocal mixing. To avoid the need for narrow filters close to the antenna very low sideband noise levels are needed at frequency separations of 25 kHz and above. Below 25 kHz performance is limited by A/D converters and at close range the bandwidth of the offending signals makes high dynamic range useless. Good VHF crystal oscillators for amateur use are typically designed for very low noise close to the carrier in order to produce a narrowband signal when the frequency is multiplied into the microwave region. It is not self evident that the noise close to the carrier and the noise far away is minimised by the same design strategy.

#### Theoretical model of low noise crystal oscillators

The oscillator can be seen as an amplifier which has feed-back through a filter. If the gain is sufficient to overcome the filter attenuation and if the phase shift is correct oscillations will occur. If the amplitude of the oscillation is limited somehow, the amplifier can be made to operate in linear class A mode and then the Leeson model will describe the main characteristics of the sideband noise.

The Leeson model looks like this:

L(d) = 10*log[((F/(2*Q*d)^2 + 1)* (NkT/P)*(c/d + 1))]

d=frequency offset in Hz.
L(d)=noise level at the frequency offset d in dBc/Hz.
F=center frequency in Hz.
N=noise factor.
k=Boltzmann's constant 1.380E-23 J/degree.
T=Room temperature 290 degrees Kelvin.
P=Power delivered to amplifier input in W.
c=Corner frequency for flicker (1/F) noise in Hz.

To see better what it means we can rewrite the Leeson model like this:

L(d) =
NF - PdBm - 174
+ 10*log[F/(2*Q*d)^2 + 1]
+ 10*log[c/d + 1]

NF=Noise figure in dB
PdBm=Power delivered to amplifier input in dBm.

At large frequency separations only the first line becomes non-zero. It says that the flat noise floor in dBc/Hz simply is the difference between the power sent into the amplifier and the noise floor of the amplifier in dBm/Hz. If one assumes a low noise figure like 5 dB and a very high power level into the amplifier like -3dBm one finds that the flat noise floor is expected at 166dBc/Hz.

Real oscillators can be made much better than predicted by the Leeson model because it is not necessary to make the amplifier see the same source impedance at all frequencies. If the crystal is loaded by the transistor only and not by any extra resistor to ground, the transistor will see a very high source impedance at large frequency offsets and if current feedback is used the noise at large frequency offsets will be much less amplified than the signal at the frequency of resonance.

Close to the carrier, the bandwidth of the filter causes the noise that is produced at the amplifier output to be amplified with a positive feedback that depends on the frequency separation. The gain increases by 20dB/decade.

At some frequency separation flicker noise (1/F noise) will cause phase modulation. To some extent the transistor amplifier is a phase modulator and the current variations through the transistor will change the phase shift through the amplifier very slightly. The flicker noise slopes at 10dB/decade and changes the slope from 20dB/decade to 30dB/decade close to the carrier.

The first IF of the Linrad hardware is 70 MHz. To see what the Leeson equation predicts we use the following estimated values:

PdBm = - 6 dBm
NF = 8 dB
Q = 50000
c = 10000 Hz

The corresponding values for L(d) then become:

L(1MHz) = -160 dBc/Hz
L(100kHz) = -160 dBc/Hz
L(10kHz) = -157 dBc/Hz
L(1kHz) = - 148 dBc/Hz
L(100Hz) = - 123 dBc/Hz
L(10Hz) = - 93 dBc/Hz
L(1Hz) = - 63 dBc/Hz

#### State of the art performance

There are several Internet sites giving performance data for low noise 100 MHz crystal oscillators. Table 1 gives some typical data.
```  Type        1Hz   10Hz   100Hz   1kHz  10kHz  100kHz  1MHz
QEX 2/2001     -     98     128    159    175    178     178
XTO-05        62    100     122    138    155     -       -
500-02268B    73    103     133    161    179    181      -
PN9530         -     95     125    155    168    168      -
ULN 100MHz     -      -     133    160    173    174      -
```
Table 1. Performance data (-dBc/Hz) for commercial state-of-the-art 100 MHz crystal oscillators.

For a state of the art 70 MHz crystal oscillator the flat noise floor should be about -180dBc/Hz and at an offset of 1kHz the phase noise should be at about -163 dBc/Hz with a 30 dB/octave slope when the ratio 70 to 100MHz is taken into account. The Leeson equation does obviously not describe these oscillators well, the flat noise floor is about 20dB lower than one would expect from the model, probably for the reason given above. In the 30dB/octave region the Leeson model would fit nicely if the loaded Q is made three times larger, something I can not guess if it is true or not.

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