3.5 Thevenin Equivalence
A circuit comprised of resistors and independent voltage and current sources can be substituted for its Thévenin equivalent circuit. Such a substitution can be helpful in circuit analysis as well as in circuit implementation. The Thévenin equivalent circuit is constructed from an ideal independent voltage source connected in series with a resistor as shown:
Pause and think about this for a minute: any circuit comprised of resistors and sources can be substituted for another circuit comprised of a voltage source in series with a resistor. The value of the voltage source and the resistor depend on the particulars of the original circuit, but the Thévenin circuit and the original circuit are equivalent to each other from the point of view of their respective output terminals. Each circuit could be enclosed within a box that obscures or hides the insides with only the pair of output terminals exposed, and one would not be able to tell the circuits apart by making measurements at the output terminals. They are equivalent.
The schematic diagram for the Thévenin equivalent circuit is an ideal independent voltage source of voltage volts in series with a resistor having resistance (in both cases, the subscript represents Thévenin voltage or resistance.) Later in this chapter, we describe an approach for determining these values. First we introduce an example to illustrate the utility of such a circuit.
Equivalent circuit for a real battery. Up until now, we have modeled batteries as ideal independent voltage sources. Recall that an ideal independent voltage source maintains a specified voltage between its terminals independent of anything connected to it. We know from experience that batteries can get warm and their voltage can drop, especially when they are loaded with small resistances that draw large currents. We can model a real battery as an ideal battery in series with a resistor. Modeling a real battery this way predicts some important behaviors, and this is a good illustration of the use of a Thévenin-type circuit.
Consider a 12V automobile battery. Such batteries typically have a small, but non-negligible internal resistance of ~ 0.05Ω. The equivalent circuit for such a battery is simply an ideal 12 battery in series with this small resistance, as shown in figure 3.37:
Such batteries are used to power many devices in a car, including the starter, the headlamps, radio, computer, etc… We will examine the impact of the internal resistance on these devices using the circuit shown in figure 3.38, which treats all devices attached to the battery (ie, starter, radio, computer, etc…) as a single load resistor that has an associated load current, . We are interested in determining the actual voltage that would applied to the headlamps and other devices in the car when powered by a 12V battery. What do you think the voltage will be?
Applying KVL around the circuit loop allows the load voltage to be determined:
(1)
or
(2)
Equation (