Noise in amplifying valves and mixer noise gives my understanding of valve internal noise. What does this mean for a radio receiver system? There are two main sources of noise in a receiver: internal device noise and thermal noise. For the moment let us assume that the noise performance of the whole receiver is set by the first stage. This is an over-simplification, but we need to start somewhere.

This is a new version of the page. I have rewritten parts of it after revising the mathematics for imperfect coupling. The corrected formulas lead to different conclusions. I have left the old page here so you can see the difference.

## Input noise

The input circuit of a radio receiver can be represented as shown at the right. Note that this applies to either valve or FET inputs, in the normal common-cathode or common-source form.

Vs is the input signal, with a source impedance of Rs - this will typically be 50 ohms. There is a step-up transformer, which will probably be part of a tuned circuit. The voltage step-up is 1:n, so impedance will be transformed as 1:n2. The tuned circuit will have finite Q, so at resonance will present a dynamic resistance Rq; note that this is based on the unloaded Q. In parallel with this is the input resistance of the valve, Rin, which includes any grid bias resistor and socket losses. Rin is much higher than Rq so can often be ignored, but we will work out the parallel combination. The combined resistance will generate thermal noise Vth - we will assume for now that the resistors are at the same temperature. Finally the valve internal noise ('shot' and partition) is represented by Veq; this is the thermal noise which would be generated by a resistance Req at room temperature.

We want to calculate the signal-to-noise ratio, so we need to find the signal voltage at the grid and the noise voltage at the grid. Equivalently, we can work with signal and noise power instead of voltage.

### Signal at grid

The signal voltage at the grid comes from a transformer step-up, combined with a potential divider formed by the source resistance and the transformed grid circuit resistance. The result is

### Noise at grid

There are two noise sources, Vth and Veq. The first one is reduced by a potential divider formed by the grid cicuit resistance and the transformed source resistance. The noise is

### Signal to noise

A useful way to represent the expected signal-to-noise ratio is the noise figure (NF). This is the ratio, expressed in dB, between the output noise of the actual receiver versus the output noise of a hypothetical noiseless receiver when both are fed with a signal consisting of thermal noise from the source resistance. The lower the NF the better (other things being equal, which they are not!). As we are assuming for now that the rest of our receiver is noiseless we can just use the ratio at the input grid.

This can be simplified to

Typically, Rs << Req << Rinq so approximations can be found for small and large n. For small n, the argument of the log is [1+Req/(n2Rs)] i.e. NF improves as n increases. For large n, the argument is [1+n2Rs/Rinq] so NF grows as n increases. There is a broad minimum at intermediate values of n, somewhere in the region given by n2=[sqrt(ReqRinq)]/Rs.

### Examples

Consider an RF pentode operating at 14MHz. Assume Req=1500 ohms. The tuned circuit has a 40pF capacitor, and a Q of 80: this gives Rq=22700 ohms. Assume that Rin can be ignored. Rs is 50 ohms. The minimum NF will be somewhere near n=11, so try 8, 12 and 15. This gives calculated NF of 2.46dB, 2.30dB and 2.61dB respectively.

A beam deflection mixer might have Req=15000 ohms. With the same input circuit as the pentode, the best NF will be near n=19. Calculated NF is 6.62, 6.71, 7.23dB at n=15, 20, 25. Improving the input Q to 150 will give NF 5.43, 5.07, 5.24dB instead.

## Reality?

These results seem impossibly good. However, claims of measured performance at this level have been made for a receiver with a 7360 mixer. The answer seems to be to have a high unloaded Q and a high step-up ratio. This leads to two potential problems: a reduction in strong-signal handling, and undue sensitivity to antenna impedance. The latter has been reported for the SS-1R receiver.

What about the pentode - why is this high performance rarely seen? Why are older receivers with 'noisy' pentodes regarded as being deaf on the higher bands? I think there are three potential issues: input circuit Q, coupling coefficient and grid noise.

Older receivers were usually general coverage. This means that the circuit Q and dynamic impedance were non-optimal at some frequencies. In particular, at the higher frequencies the tuning capacitance would be too high so the dynamic impedance is low. This means that a high step-up, needed for noiser pentodes, is not possible.

General coverage also means that there will not be tight coupling between the antenna and the tuned circuit. Looser coupling means that the receiver is not disturbed too much by antenna impedance variations. Langford-Smith recommends k~0.2 for broadcast receivers, and says that having k as high as 0.4 can start to bring tracking problems. However, loose coupling can also mean:

1. voltage step-up is smaller than expected, so the signal is smaller at the grid
2. thermal noise from the tuned circuit is not attenuated so much by loading from the antenna source impedance, so the noise is greater at the grid

These two effects work in the same direction. The result can be an increase in noise figure. Even a fairly tightly coupled coil will not achieve k=1. I wondered why one 7360 receiver uses capacitive antenna coupling instead of a transformer - this automatically gives full coupling. In this receiver the step-up is n=23. Incorporating the coupling factor k in the expression for noise figure gives

The asymptotic approximations change to [1+k2Req/(n2Rs)] for small n, and [1+n2Rs/(k2Rinq)] for large n. The best noise figure is now near n2=k2[sqrt(ReqRinq)]/Rs, which will be lower than for the tight coupling case. However (and somewhat surprisingly) it turns out that the noise figure need not be degraded provided that the correct value of n is used. Note that n is the turns ratio\$, which is different from the step-up when k is not equal to 1.

\$ Strictly speaking, n is not the turns ratio but the square root of the pri:sec inductance ratio. For typical HF receiver input coils this is close enough.

### Example

Let us revisit the pentode mentioned above: Req=1500 ohms, Rq=22700. Assume k=0.5 - perhaps a communications receiver with an antenna trimmer to cope with impedance variations. Then best NF is near n=5. Calculated NFs for n=3, 5, 8 are 2.96dB, 2.00dB, 2.37dB respectively. These are similar to the k=1 case. Now change to a slightly noiser valve: Req=3000 ohms. The NFs become 4.59dB, 2.91dB, 2.75dB. So although valve noise has increased by 3dB, receiver noise only increases by 0.9dB. Optimising n will improve this. The net result may be an increase in dynamic range, as the noisier valve probably has lower gain and better strong-signal handling.

Going back to the original valve (Req=1500 ohms), let us now include the grid input resistance. For an EF85 this is given as 9k at 50MHz. At 14MHz it will be 115k, as it scales with frequency-2. Calculated NFs are now 3.03dB, 2.1dB and 2.72dB respectively. A small increase on the previous figures, but at this point we are still assuming that the grid input resistance is at room temperature.

## Grid noise

Older pentodes may have relatively low input resistance at higher frequencies. This has the same effect as a reduction in Q. Even worse, much of this resistance is due to induced grid noise. This means that, in effect, the input resistance is much hotter than room temperature - perhaps approaching the cathode temperature. The maths gets more messy now, but fear not, the result is not too bad.

The only change is that an extra factor appears in part of the expression: GT is the ratio of the effective grid input resistance temperature to room temperature. Langford-Smith suggests that GT is about 5. The input turns ratio for best NF will be somewhere near n4=k4ReqRqRin/[Rs2(Rin+5Rq)], which will be slightly smaller than before. This is because a smaller ratio causes a greater load to be applied to the hot resistors, thus attenuating their noise.

If we revisit our example valve at 14MHz the calculated NFs for n=3,5,8 are now 3.16dB, 2.59dB and 3.65dB respectively. As expected, it seems that the optimum turns ratio has reduced as the high n noise figure has become much worse. Now go to 28MHz, with all other impedances unchanged. Rin is 29k. NFs are 3.69dB, 3.98dB and 6.03dB respectively. These figures still seem very good. However, it is likely that many receivers will use a higher turns ratio in order to maintain front-end selectivity. Putting n=12 gives NF=8.69dB. Most of the front-end noise is now coming from the valve grid with a smaller contribution from the tuned circuit; very little comes from normal pentode noise.

The figures show that in the lower part of the HF band the dominant noise source is likely to be thermal noise from the dynamic impedance of the input tuned circuit. Noise is not an issue here anyway, unless a very poor antenna is used. At higher frequencies valve input resistance is the main noise source. High transconductance does not help; indeed it tends to be associated with slightly lower input resistance. It is interesting to note that the Racal R210 receiver (part of the UK Larkspur system) used an EF92 for the first stage, yet still boasted a noise factor of 6dB.

Modern valves, with high input impedance and low grid noise, do not need particularly low shot and partition noise in order to produce an effective HF receiver. The real need is high Q and the right values of coupling and step-up. However, there is a trade-off: high turns ratio reduces strong-signal handling and strong coupling can mean poor antenna matching. A receiver designer can juggle these to get the best compromise. These compromises are explored here. The effect of second stage noise must also be considered.

## Beam deflection mixer noise

There has been some debate on discussion boards about the noise produced by a beam deflection mixer such as 7360, with some attempts to back-calculate from measured receiver noise figures to Req. This will be difficult without detailed knowledge of the input circuit, especially Q, coupling coefficient and turns ratio.

updated 23 Sept 2011: correct formulas including k, and update comments accordingly. Update eHam link.