Thevenin’s Theorem

There are times when it is advantageous to isolate a part of the circuit to simplify the analysis of the isolated part of the circuit. Rather than write loop or node equations for the complete circuit, and solving them simultaneously, Thevenin’s theorem enables us to isolate the part of the circuit we are interested in. We then replace the remaining circuit with a simple series equivalent circuit, thus Thevenin’s theorem simplifies the analysis. There are two theorems that do similar functions. The Thevenin theorem just described is the first, and the second is called Norton’s theorem. Thevenin’s theorem is used when the input source is a voltage source, and Norton’s theorem is used when the input source is a current source. Norton’s theorem is rarely used, so its explanation is left for the reader to dig out of a textbook if it is ever required. The rules for Thevenin’s theorem start with the component or part of the circuit being replaced. Referring to Figure 2–7, look back into the terminals (left from C and R3 toward point XX in the figure) of the circuit being replaced. Calculate the no load voltage (VTH) as seen from these terminals (use the voltage divider rule).

Figure 2–7. Original Circuit

Look into the terminals of the circuit being replaced, short independent voltage sources, and calculate the impedance between these terminals. The final step is to substitute the Thevenin equivalent circuit for the part you wanted to replace as shown in Figure 2–8.

Figure 2–8. Thevenin’s Equivalent Circuit for Figure 2–7

The Thevenin equivalent circuit is a simple series circuit, thus further calculations are simplified. The simplification of circuit calculations is often sufficient reason to use Thevenin’s theorem because it eliminates the need for solving several simultaneous equations. The detailed information about what happens in the circuit that was replaced is not available when using Thevenin’s theorem, but that is no consequence because you had no interest in it. As an example of Thevenin’s theorem, let’s calculate the output voltage (VOUT) shown in Figure 2–9A. The first step is to stand on the terminals X–Y with your back to the output circuit, and calculate the open circuit voltage seen (VTH). This is a perfect opportunity to use the voltage divider rule to obtain Equation 2–13.

Figure 2–9. Example of Thevenin’s Equivalent Circuit

Still standing on the terminals X-Y, step two is to calculate the impedance seen looking into these terminals (short the voltage sources). The Thevenin impedance is the parallel impedance of R1 and R2 as calculated in Equation 2–14. Now get off the terminals X-Y before you damage them with your big feet. Step three replaces the circuit to the left of X-Y with the Thevenin equivalent circuit VTH and RTH.

Note: Two parallel vertical bars ( || ) are used to indicate parallel components as shown in Equation 2–14.

The final step is to calculate the output voltage. Notice the voltage divider rule is used again. Equation 2–15 describes the output voltage, and it comes out naturally in the form of a series of voltage dividers, which makes sense. That’s another advantage of the voltage divider rule; the answers normally come out in a recognizable form rather than a jumble of coefficients and parameters.

The circuit analysis is done the hard way in Figure 2–10, so you can see the advantage of using Thevenin’s Theorem. Two loop currents, I1 and I2, are assigned to the circuit. Then the loop Equations 2–16 and 2–17 are written.

Figure 2–10. Analysis Done the Hard Way

Equation 2–17 is rewritten as Equation 2–18 and substituted into Equation 2–16 to obtain Equation 2–19.

The terms are rearranged in Equation 2–20. Ohm’s law is used to write Equation 2–21, and the final substitutions are made in Equation 2–22.

This is a lot of extra work for no gain. Also, the answer is not in a usable form because the voltage dividers are not recognizable, thus more algebra is required to get the answer into usable form.