INTRODUCTION TO ELECTRONICS: A Conceptual Analysis of Thevenin’s Theorem

A physical system would be meaningless without an observer. Conclusions about electrical systems are oftentimes made from the vantage point of the source ( V_s ), but this need not be the case. If a portion of a circuit is “ opened “, an observer can view the source and other components from the newly opened region. Since the vantage point has been changed, the portion of the circuit that is “ seen “ by an observer will be different; however, the physical characteristics of the open region itself are the same, regardless of the location from which it is observed. Consider the following circuit:

An observer at the source will see resistor R₁ in series with two resistors ( R₂ and R₃ ) that are parallel to one another. Let’s now open the region of the circuit that enables the R₃ current to reconnect with that from R₂:

An observer within the A-to-B open region will now observe resistor R₃ to be in series with two circuit branches, one of which contains R₂ with the other branch containing R₁ and V_s. We must now, however, take some fundamental laws of physics into account:

1. Law of Conservation of Energy
2. Law of Conservation of Charge
3. Law of Conservation of Mass

The opening between nodes A and B has now cut off current ( I ) that once flowed through R₃. In order for the circuit to retain its physical characteristics, this current must somehow or another be redirected elsewhere within the circuit; however, prior to establishing a new route for current to travel, we must take into account a physical parameter that has thus far been ignored: the voltage source’s internal resistance ( R_s ). Let’s redraw the circuit so that R_s is is taken into account:

We now have a circumstance where, for the sake of observing R_s, the source voltage was “ eliminated “. In order to preserve the circuit’s physical identity, we must now account for a current and voltage that are no longer present within our system. The first step requires recognizing that the internal resistance of a source, for all practical purposes, is zero. For this reason, we must short R_s out of the circuit:

Note: A shorted circuit has the exact opposite characteristics of an open circuit. An open circuit has zero current flowing through it, but a closed, shorted circuit has infinite current flowing through it. This is why a battery’s anode ( + ) and cathode ( – ) must never be connected via usage of a resistor-less conduction path.

We have now dealt with the issue of redirecting current within the circuit! From the vantage point of the open region, R₃ is now in series with a parallel circuit that contains the R₁ and R₂ resistors. If we redraw the circuit for clarity, our new circuit’s structure will be a reflection of the image we began with: 

The circuit would be an exact replica of what we began with if the resistor values were equal to one another. This will oftentimes not be the case, however. This is why emphasis is placed upon the circuit being derived as being an equivalent circuit. The complete restoration of the circuit’s physical characteristics will occur when a new voltage differential is present within the system. Recall that the first systemic observations were made from the vantage point of the source. As symmetry would have it, the equivalent voltage formed will appear between the A and B terminals of the open:

Note: It is of crucial and critical importance to realize that we’re no longer dealing with a circuit that has current flowing through it!!!

The demonstration above works well for a relatively simple circuit. An obvious mirror-image circuit may not be derived from more complex circuits, but the Thevenin Theorem still holds true. There is no preferred order in which the Thevenin resistance and voltage should be solved, but the following strategy will be advantageous in many scenarios:

1. Create an open from which new observations will be made.
2. Account for the open by shorting voltage sources.

As useful as this and other strategies may be, there is no substitute for old-fashioned problem solving that entails as many circuit types as possible.