– 39 –
These figures suggest that if water is used the frequency should not be much above 10 Mc/s, but that we can go considerably higher with mercury.
(v) Noise.
Before leaving the subject of attenuation we should verify how much can be tolerated. The limiting factor is the noise, due to thermal agitation and to shot effect in the first amplifying valve. The effect of these is equivalent to an unwanted signal on the grid of the first valve, whose component in a narrow band of width f cycles has an R.M.S. value of
VN = 4 k T f (R + Re)
where T is the absolute temperature, k is Boltzmann's constant and R is the resistive component of the impedance of the circuit working into the first valve, including the valve capacities. Re is a constant for the valve and describes the shot effect for the valve.
In the case that we use mercury and do not tune the input the value of R will be quite negligible in comparison with Re, which might typically be 1000 ohms. For a pulse frequency of 1 megacycle we must take f = 106 (the theoretical figure is ½106 but this is only attainable with rather peculiar circuits). At normal temperatures 4 k T = 1.6 x 10−20 and therefore VN = 4 μV. In the case that we use water and tune the input, we have R = Q / w(Cx + Cs) at the worst frequency. Assuming w / 2πQ = 2 Mc/s (see Fig. 41) and Cx + Cs = 20 pf and ignoring the fact that the effect will not be so bad at other frequencies, we have VN = 9 μV.
Now suppose that we wish to make sure that the probability of error is less than p, and that the difference in signal voltage between a digit 0 and a digit 1 is V. Then we shall need
2 dx < p.
(This follows from the fact that a random noise voltage is normally distributed in all its coordinates.) If we put p = 10-32 we find V/VN ≥ 24 , V ≥ 0.1 mV.
(vi) Summary of output power results. Summarising the voltage attenuation and noise questions we have:
(a) There is an attenuation factor depending on the material of the crystal and its cut and for quartz typically giving a loss of 48 dB.
(b) There is a factor R depending on the ratio of band width required to carrier frequency, and the matching factor u between crystal and liquid. In practical cases this amounts to gains of 10 dB with water and 2 dB with mercury.
(c) There is a loss factor Cx/Cx + Cs due to stray capacity Cs across the receiving crystal. This might represent a loss of 6 dB.
