 # Unit 12: More Circuit Theorems

In Unit 11 you studied two powerful tools for circuit analysis. The first was the source-conversion theorem, which can make some circuit analysis tasks easier by letting you replace a voltage source with a current source, or vice versa. The second was the superposition theorem, which lets you analyze circuits containing more than one source. This unit will introduce two more useful theorems: Thevenin’s theorem, which can be used to simplify complicated circuits, and the maximum power transfer theorem, which lets you figure out how to deliver maximum power to a load.

##### The Black-Box Game
• Thevenin’s theorem is a powerful circuit-analysis theorem that can be difficult to understand when you’re first trying to learn it. There are different ways to state the theorem. Here I’ll use the concept of a black box to explain what the theorem says.
• By a "black box" I mean a box that contains a circuit and has two output terminals to which you can attach an external component or measuring instrument. Why do I call it a black box? Because I want you to think of a box whose contents you cannot see. Looking at the box, you can’t tell whether the circuit inside the box is very simple, with just a couple of resistors and a voltage source, or very complex, with perhaps hundreds of resistors and several voltage sources and current sources.
• Imagine a game in which someone gives you a black box containing a circuit and asks you to try to figure out what circuit is inside the box. You can’t see inside the box, and you’re not allowed to open the box. The only access you have to the circuit is through the box’s two output terminals. You’re allowed to attach anything you’d like to these output terminals.
• For instance, you might start by using a multimeter to measure voltage, current, and resistance at the output terminals when no external components are connected to the terminals.
• And then you might try connecting different resistors to the output terminals and measuring the current that flows out of the box through each of these resistors.
• You could even connect other kinds of components, such as capacitors or LEDs, to the output terminals, and make more observations and measurements.
• The question is, Using measurements such as these, how much will you be able to figure out about the circuit inside the box? You might initially think that by making lots of careful measurements, you may be able to figure out quite a bit about the circuit inside the box. But Thevenin’s theorem says that this is not true.
##### Thevenin’s Theorem
• Now we’ll use this idea of the black-box game to introduce Thevenin’s theorem.
• Suppose I build a black box (with two output terminals) containing a very complicated series-parallel circuit with hundreds of resistors and dozens of voltage sources. Thevenin’s theorem says that no matter how complicated I make my circuit, you will be able to build another black box (with two output terminals) containing just a single voltage source in series with a single resistor, and nobody playing the "black box game" will be able to tell our two boxes apart. Any measurements that the person makes at our boxes’ output terminals will give exactly the same answers for the two boxes.
• To introduce a fancy word for this, we’ll say that our two black boxes are equivalent. This does not mean that they contain the same circuit. It means that the two boxes will give the same result for any measurement or observation that can be made at the output terminals.
• To summarize, Thevenin’s theorem says that a two-terminal black box containing any number of resistors and any number of voltage sources or current sources is equivalent (in the sense just explained) to a two-terminal black box containing a single voltage source in series with a single resistor.
• So for any circuit with two output terminals, we can replace it by its Thevenin equivalent circuit that looks like this: • A and B are the Thevenin equivalent circuit’s output terminals. VTH is called the Thevenin equivalent voltage, and RTH is called the Thevenin equivalent resistance.
• Example: Consider the circuit shown below. Notice that this circuit has two output terminals labeled A and B. Thevenin’s theorem tells us that we can replace this circuit with a much simpler circuit containing a single voltage source in series with a single resistor, and that the two circuits will produce exactly the same effect on any measuring instrument, component, or group of components that we attach to terminals A and B. • Why is this useful? We’re not really interested in playing the black-box game. But we are interested in analyzing circuits. And Thevenin’s theorem is useful because it allows us to replace a very complicated circuit with a very simple circuit, which can make certain analysis tasks much easier.
##### Procedure for Applying Thevenin’s Theorem
• So far, we haven’t said how to find the values of VTH and RTH. There is a standard procedure by which you can find those values for any given circuit. Here is the procedure, along with the other steps that you would typically perform when using Thevenin’s theorem.
1. Open-circuit the terminals with respect to which the Thevenin circuit is desired. In other words, if the output terminals A and B originally have a load connected across them, remove that load so that terminals A and B are left open.
• Example: Suppose that you’re dealing with the circuit shown below. After you remove the load resistor, you’ll be left with this: 2. Determine the voltage across the open terminals A and B. This voltage is the circuit’s Thevenin equivalent voltage, VTH.
• Example: For the circuit shown just above, VTH = 4.08 V.
3. Remove all sources by short circuiting all voltage sources and open circuiting all current sources (just as you learned to do previously when you studied the superposition theorem). Then determine the resistance between the open terminals A and B. This resistance is the circuit’s Thevenin equivalent resistance, RTH.
• Example: For the circuit shown just above, RTH = 141 Ω.
4. Connect VTH and RTH in series, as shown in the diagram below. This is the Thevenin equivalent circuit of the original circuit. 5. If you removed a load resistor in Step 1, connect this load across terminals A and B of the Thevenin equivalent circuit. You can now easily compute the load current and load voltage, and these values are guaranteed to have the same values as the load current and load voltage in the original circuit.
• ##### Why Thevenin’s Theorem is Useful: Case 1
• Thevenin’s theorem is useful in a number of different situations. Here we’ll look at two cases where the theorem can save you a lot of work.
• Case 1: Suppose you need to compute the load current or load voltage for several different values of the load resistor in a circuit. For example, the diagram below shows the load resistor having a value of 1 kΩ. Suppose you need to find the load voltage for this circuit as shown, and you also need to find the load voltage for six other possible values of the load resistor. Which would you rather do: analyze that series-parallel circuit seven times (with seven different values of RL), or replace everything to the left of terminals A and B with a single voltage source in series with a single resistor, and then analyze that simplified circuit seven times (with seven different values of RL)?
• ##### Why Thevenin’s Theorem is Useful: Case 2
• Case 2: Suppose you need to find the load voltage or load current in a circuit that cannot be analyzed as a straightforward series-parallel circuit. The most common example of this is a loaded Wheatstone bridge circuit, as shown below. Notice that in this circuit, no resistor is connected in series or in parallel with any other resistor. So there’s no way that you can follow the usual procedure of finding the circuit’s total resistance, and then finding its total current, and then finding the individual voltages and currents, as you learned to do in Unit 4 of this course. Try it yourself–you won’t be able to do it!
• But here’s what you can do instead: remove the load resistor, so that you’ve got the circuit shown below. Then find the Thevenin equivalent circuit for this circuit, and then connect the load resistor to that Thevenin equivalent circuit, which will let you find the load resistor’s voltage and current. • • In many cases, it’s useful to think of an electrical system as containing a source of power and a load connected to that source. The source supplies the system’s electrical power, and the load uses that power to do some useful work.
• The source may be a generator, a power supply, an amplifier, or a similar device.
• The load may be a loudspeaker, a motor, a heating element, an antenna, or some other device that converts electrical energy to another form of energy.
• Often, we are interested in maximizing the power that is supplied to a particular load. In the case of a loudspeaker, this will mean getting the loudest possible volume level from our speaker. In the case of an electric motor, it will mean getting the greatest possible mechanical power. In the case of a heating element, it will mean getting the greatest heat output. And so on.
##### Maximum Power Transfer Theorem
• The maximum power transfer theorem says that maximum power is delivered to a load when the load resistance is equal to the Thevenin resistance of the source to which it is connected.
• So problems involving maximum-power transfer are essentially Thevenin problems. To determine what value of a particular load resistor will yield maximum power in a given circuit, you must find the Thevenin resistance of the rest of the circuit, and then set your load resistance equal to that same value.
• In some cases finding the source’s Thevenin resistance is very easy. For instance, recall from Unit 6 that a practical voltage source may be modeled as an ideal voltage source connected in series with an internal resistance, as shown below. Connecting a load resistor to such a source will result in the circuit shown below. In this very simple circuit, the Thevenin resistance RTH of the circuit to the left of terminals A and B is just the source resistance RS. So the maximum power transfer theorem tells us that maximum power will be delivered to the load when the load resistor RL is equal to the source’s internal resistance RS.
• In other cases, finding the source’s Thevenin resistance will involve much more work. In a stereo system, for instance, the load is a speaker and the source is the stereo’s amplifier, which is a complicated circuit containing many resistors, capacitors, transistors, and other components. Finding the Thevenin resistance of such a complicated circuit may be a difficult task.
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