Common-Emitter Output Impedance.

Work In Progress
Currently in a high state of confusion, and possibly entirely wrong.

There are various equivalent circuits for a transistor, and some have a resistor between collector and emitter, but some have a resistor between base and collector. Looking at the internal construction of a transistor however we see that there is no direct path from collector to emitter, any current must also pass through the base region, so one question to answer is why would the equivalent circuit have a collector to emitter resistor?

The collector to base impedance is the common-base output impedance, and is related to the h-parameter hob, (actually this is the conductance which is the inverse of the resistance) but this is rarely listed, so we need to calculate it from the given h-parameters. The value of hob is usually taken to be hoe / (1 + hfe), so for our BC560B hob = 10/331 uS. The resistance is just the inverse of the conductance, so we have a resistance of 33.1M.

Suppose there is a high source impedance connected to the base, then we would expect most of the current through that 33M becomes base current, and the collector current is then hfe (330) times greater, so the collector resistance is about 33M divided by current gain 330, giving 100k, which is just the value of the inverse of hoe, and is the resistance from emitter to collector included in some equivalent circuits, so does this mean there is no need to include hoe, it has already been taken into account in the effect of the 33M?

Where this goes wrong is when the base is connected to a low source impedance, then according to the above interpretation all the current through the 33M will go through that low impedance, having little or no effect on the collector current? We would then expect the output impedance to be as high as for a common base stage, which is certainly not correct. Clearly some different interpretation is needed. Looking for information about the effect of source impedance Rs on output impedance however proved to be almost entirely unhelpful.

I started with a Spice simulation, which showed no effect from Zs. Spice models are not necessarily reliable, so I searched for a formula, and a few sources gave the output impedance as: Zo = 1 / (hoe - (hfe.hre / (Rs + hie))) . Taking the h-parameter typical values from an old BC560B data sheet I got Zo = 100k for a very high Rs, and for zero Rs I got Zo = - 59k. Yes, a negative resistance. Trying again with a more recent data sheet for the 2SC4117 gave more reasonable results 400k and 2.8M with high and low Rs respectively.

To make matters worse I found an IEEE paper Output resistance of the common-emitter amplifier which states that the ratio of Zo values for high and low source resistance is 1.5, and shows measurement results which appear to support this. Worse still, the higher Zo is for higher Rs, the opposite of the formula, and not what I would expect. The article does agree with my own observation that Spice simulations can fail to show any effect from Zs.

I found a more convincing version in Wireless World, Jan 1965, page 42, a reply to a 'Letter to the Editor' from G.P.Hobbs related to his articles in previous issues, saying that his own measurements gave a ratio of output impedances about 2 or 3 for source impedance varied from zero to infinity, although a calculation gave a ratio of 18. He suggested the problem could be that the 'typical' parameters in a data sheet are each a mean value for a large number of samples, but these values are not necessarily a consistent set. In other words no individual transistor could have exactly all the typical values. Although I don't find that entirely convincing, I have no better explanation, but anyway, if this problem was being discussed 50 years ago surely it has been resolved somewhere by now?

Here is one equivalent circuit showing the meaning of the common-emitter h-parameters:

So what is hre? Multiplied by Vce it is an ac signal appearing between base and emitter.This looks very much like the 'Vbe modulation' effect I investigated on my Common Mode Distortion page. There it was found that with 1V rms at 2kHz applied to the collector a BC560B has its Vbe modulated by 320uV. This is affected a little by DC collector current and voltage. The circuit used to detect this voltage is shown next, with an op-amp used to amplify the signal to a more easily observable level. Only pnp transistors were used for that test. The 600R resistor is just the internal resistance of my signal generator.

With the 1V signal applied in my test we expect from the equivalent circuit shown earlier a Vbe modulation of 1V x hre, = 3.5 x 10-4V = 350uV.
My measured value 320uV is surprisingly close. (But was measured at Ic = 1.14mA). The problem with this is that hre is specified as the ac base voltage with earthed emitter, and my method with earthed base is not measuring exactly the same thing. At least it appears that the two effects are very similar.

Anyway, if hoe is entirely determined by the collector-base resistance supplying base current what about the Vbe modulation effect? Surely that should also affect the output impedance? A quick estimate, if my measured 320uV is added to Vbe with gm = 1/26 for the 1mA collector current, then we should get a change in collector current equivalent to an output impedance of 81k, so the effect should be greater than the collector-base resistance effect, certainly not negligible. The question is whether the two effects are really different, or are just two ways of looking at the same effect.

Going back to the diagram above, suppose hre was zero, we would still expect some Vbe modulation because of the effect of the 100k corresponding to hoe. This feeds AC current into the emitter, and with Ic = 1mA there is an input impedance 26R at the emitter (given by hib, or by the inverse of gm). With 1V AC applied via 100k the voltage across that 26R will be 260uV. Almost, but not quite, enough to explain the Vbe modulation observed. What we have to remember however is that the h-parameters listed earlier are only 'typical' values, and can vary over a wide range, so exact agreement with measurements is not expected.
So, does this suggest that hre is superfluous, the whole effect being caused by hoe? But then we are back to the question of what happens with earthed emitter, and hre measured at the base.

This is getting confusing, maybe a better approach is to try to construct an equivalent circuit which at least agrees with the plausible 2 or 3 to 1 ratio of output impedance with source impedance varying from zero to infinity. One way to do this is shown next, starting from the small signal equivalent circuit I used in my 'Transistor Amplifier Design for Beginners' page where the base-emitter voltage is taken as a constant 600mV plus the effect of an internal resistor of value 1/gm. Adding further internal resistors Rcb and Rce can we work out their values?

It would be nicer if the circuit had at least a slight agreement with the known physics of transistor operation, but for now I only want to get a model that gives about the right answers, with no danger of negative resistances or other improbable results turning up. The transistor symbol should be understood as an idealised transistor with constant Vbe and infinite output impedance, the small signal variations in Vbe being contributed by re = 1/gm, and the output impedance is determined by Rce and Rcb. The current gain is still hfe.

To avoid effects from the base spreading resistance rbb' we can consider the 2SA1085E for which rbb' is very low, just a few ohms, and can therefore be ignored. From my common-mode distortion page, at 1.14mA collector current and 1V at 2kHz applied to the collector the emitter signal was measured at 203uV with earthed base, rising to 223uV with 1k from base to earth.
Then 1/gm is about 23R and so 23/Rce is the ratio of collector and emitter signal voltages, = 203/1000000. From this we find Rce = 23000000/203 = 113k. That seems reasonable, so onwards to find Rcb. With 1k base to earth there is an extra 20uV at the emitter, so 20/1000000 = 1000/Rcb, from which Rcb = 50M. Again a reasonable value.
The output impedance with earthed base is just Rce, with value 113k.
With open circuit base the current into the base is given by 1V across 50M, and with a current gain around 500 the collector current is then 10uA, which is what would be taken by an additional 100k from collector to emitter, so we now have 100k // 113k = 53k.
We therefore have output impedance 53k with open base and 113k with earthed base, a ratio of 2.13, about what we would expect from the Hobbs measurement mentioned earlier.

We have the expected ratio and agreement with my measurements, but we have only worked out resistor values for one value of Ic = 1.14mA. We could guess that the resistors are all inversely proportional to Ic, this is certainly true for the emitter resistor given by 1/gm.
I did measurements for the 2SA1085E at collector current 0.125mA, and doing the same calculations again this gives Rce = 1.07M and Rcb = 166M, and therefore the output impedances become 1.07M and 252k, now a ratio of 4.24, which suggests the ratio of output impedances for zero and infinite source impedances is a function of collector current. Note that Rce is almost perfectly inversely proportional to Ic but Rcb is about inversely proportional to the square root of Ic.
That may not be very reliable, I only have two data points and a small range of currents.

As I have said before none of this has much practical importance, if we look at the specifications of almost any transistor we find output admittance hoe either not specified at all or specified with a ratio of maximum to minimum values typically 5 or more. Designing circuits where such parameters are critical is clearly a bad idea, so the finer details of the effects of source impedance are of mostly academic interest.

The equivalent circuit proposed does have some advantage compared to the usual equivalents, it can be used with any external components. The h-parameters however are specified for more limited use, for example hoe is output admittance only for an open-circuit input and a common-emitter stage. My equivalent should be applicable for any source or load, also for common emitter, base or collector, or with any value emitter resistor Re. I may add a few examples eventually.

One final point, I mentioned that there could be some difference between the Vbe modulation I measured with earthed base compared to the Vbe modulation expected from hre which should be measured at the base with an earthed emitter. According to my equivalent circuit the first effect is caused by Rce and the second by Rcb, and the two effects will be equal only when the ratio of output impedances for zero and infinite source impedance is exactly 2. The fact that my measurement agreed with the hre value for a BC560B was therefore just a consequence of using Ic = 1.14mA, and would be very different at other currents. The surprising agreement was really just an accident, they are really two different things. The Hobbs article in Wireless World from 1964 refered to earlier has these two effects numerically equal, and also states that Rcb is inversely proportional to Ic, but my measurements suggest not.

An article by Hawksford includes an analysis of CE output impedance, and his equations 12 and 14 are for the zero and high source impedance cases. It is not immediately obvious what the ratio is but if we were to assume Zcb / Zce = Hfe and let the source impedance increase towards infinity then the ratio is close to 2 as expected. My own analysis was only concerned with the low frequency impedances, at higher frequencies it gets a little more complex.