Thevenin Equivalents

Finding A Thevenin Equivalent Circuit

The TEC is a useful way of reducing complexity. If you have a complex circuit interacting with other circuits you don't want to look at all of the details of what is taking place inside the complex circuit - at least you don't often want that.

To illustrate some of the power of the TEC we'll get the TEC for a voltage divider.

The voltage divider isn't a very complex circuit, but it is more complex than the TEC. It can be represented with a TEC, and wherever you find a voltage divider you can replace it with the equivalent TEC. We need to establish values and/or formulas for two items in the voltage divider. Basically, we need two of the following:

• Open Circuit Voltage

• Internal Resistance

• Short Circuit Current.

Let us focus on Open Circuit Voltage, shown as V_oin the diagram below. We can use the voltage divider formula to compute the open circuit voltage:

So, we are able to compute one of the three quantities mentioned on the previous page.

Next, focus on theShort Circuit Current. Short circuit current, I_o is the current that would flow if the output of the circuit is short circuited. We can imagine shorting the Voltage Divider and shorting the TEC. Wires that short them are shown below and the short circuit current, I_sc is indicated.

For the TEC you should be able to see quickly that the short circuit current is just

V_o / R_o . For the voltage divider it's not any more difficult. There the short circuit current is just V_s / R_b. We can use both of these observations to compute the internal resistance of the voltage divider TEC.

Calculate the Short Circuit Current.

for the TEC model, and

for the voltage divider.

Combining we have

Continuing, we find:

So, we have the internal resistance, and it looks like the parallel formula. (But it really isn't!)

At this point you can compute both the open circuit voltge and the internal resistance for the voltage divider, so you have a TEC for the voltage divider. You can use that TEC any place you find the voltage divider. It doesn't matter where you find the voltage divider, you can replace it with its TEC, just like you can replace two parallel resistors by their parallel equivalent.

Q2 You have a battery with an open-circuit voltage of 12.6 volts, and an internal resistance of .05 W

. Is there a .05 W

resistor inside the battery?

Norton Equivalent Circuits

In electrical engineering there is a recurring duality. Voltage and current are duals, and where we find one variable appearing in an expression or a theorem, we usually find the other appearing in a dual expression or theorem. The dual of the Thevenin equivalent model is the Norton equivalent model, a sample of which is given in the figure below.

The relation between terminal voltage and load current for this circuit is interesting because it is indistinguishable from that of the Thevenin Circuit, and the two are electrically equivalent. Write KirchHoff's Current Law for the Norton circuit with a load attached. The result is:

V_terminal/R_o + I_load = I_o

So, we find:

If we identify R_oI_o with the V_o that appears in the Thevenin model, this is, as claimed, equivalent to the Thevenin model.

Note the following relations between the Thevenin and Norton equivalents.

• The internal resistance, R_o, is the same in both the Thevenin and Norton circuits.

• V_o = R_oI_o, where