# On Thevenin’s theorem – #2

On Thevenin’s theorem looked at a simple source network to demonstrate some key characteristics and limitations of Thevenin’s equivalent circuit.

The example network used was linear in V,I for all V,I combinations possible. Let’s now look at a network that is not linear for all V,I, but is sufficiently linear over a sub range to be usefully modelled using Thevenin’s equivalent circuit.

## Black Box for discussion

For the purpose of discussion, we have a Black Box with just two terminals and is a source of DC voltage and current, and the internal implementation is hidden from us.

A series of measurements is made with different load resistors attached and the voltage and current at the terminals is recorded and plotted uniformly stepped currents. The V,I characteristic is clearly non-linear, but on closer examination there are two fairly linear regions, from 0.008 to 0.060A and 0.08A to 0.1A. It is a device that is usually used in the region below the knee, and for our application, let us concentrate on 0.008 to 0.030A.

The voltage is fairly constant in that region, but falls with increasing output current. Let’s zoom in to get a better understanding and let’s fit a straight line to the data. Above, a closer look at the V,I characteristic of our Black Box in the region of interest.  Using Excel’s curve fitting, we find that the line V=4.7747-1.0944*I is a pretty good fit over this range of data. That equation tells us that a Thevenin equivalent circuit with Vth=4.7747V and Rth=1.0944Ω is a pretty good estimator of the observed V/I characteristic over the range of interest.

This is quite a useful equivalent circuit for predicting the voltage at different load currents over the range measured, and that simple equivalent circuit may be used in place of a quite complicated real circuit.

But I keep saying “over the range measured”. That is because without knowledge of the implementation (ie what’s inside the box) we cannot extrapolate those measurements to currents outside the measurement range used for the line fit.

For example, it would be wrong to predict that the current through a 0Ω load (a short circuit if you like) is Vth/Rth=4.7747/1.0944=4.36A. In fact, Isc for this Black Box is 0.12A, nothing like 4.36A. This illustrates a very common failure in application of a Thevenin equivalent circuit… use of it outside the range where it is an adequate model of the Black Box.

Whilst a Thevenin equivalent circuit may often be valid for all output currents from Ioc to Isc, Thevenin equivalent circuits that are valid over a lesser range can be used a very useful analysis tool over their valid range.

Thevenin equivalent circuits are often derived from measurement of Voc and Isc, and they may be correct, but that technique depends on whether or not the Black Box V/I characteristic is linear over that range.

Another operation commonly performed is to calculate system efficiency based on a Thevenin equivalent circuit and given load resistor.

For example, lets take Vth=10V, and Zth=1Ω and connect a load of 3Ω. There is a temptation to calculate load current 10/4=2.5A, Pout=3/4*10*2.5=18.75W, and Pin=10*2.5=25W for efficiency of 75%.

There is no justification for calculation of Pin above (and therefore efficiency), the Thevenin equivalent circuit does not imply the internal implementation of the Black Box and to make such an inference is invalid. Above is a plot of the V/I characteristic of our Black Box, and the efficiency. Note that efficiency increases with increasing current.

## What’s in the box

Readers could probably not resist the urge to guess what is in the Black Box. You cannot reliably infer that from the graphs above, and to continue to do so means you have not heard the message.

The Black Box is not some contrived magic or theoretical network, it is in fact a common Zener diode shunt regulator, a 12V ideal battery, 100Ω resistor and 1N750 Zener diode all in series, and the output terminals are connected across the Zener.

It is not a contrived example, a storage battery is quite similar in not being linear in absolute terms, but having a fairly linear V,I characteristic at normal discharge currents, the range of interest for prediction of V vs I, and it is very practical to model it using Thevenin’s equivalent circuit.

## Conclusions

• A Thevenin equivalent circuit is only valid over the range where Vout=Vth-Iout*Rth, and that range may be less than the full V/I range of the source.
• A Thevenin equivalent circuit of a source does not imply the internal implementation of the source.
• Calculation of system efficiency based on a Thevenin equivalent circuit is invalid as it implies / assumes the internal implementation of the source.
• Calculation of  Thevenin equivalent circuit Rth based on perceived efficiency is invalid as it implies / assumes the internal implementation of the source.