 # Unit 11: Circuit Theorems

When you’re trying to analyze complicated circuits, a few tricks of the trade can greatly simplify your task. In this unit and the next one we’ll learn some of those tricks. The two main topics in this unit are source conversions and the superposition theorem. Source conversions let you replace a voltage source with a current source, or vice versa. In some cases, this can make a particular circuit easier to analyze. The superposition theorem tells you how to attack circuits that have more than one source–such as a circuit containing two voltage sources, or a circuit containing a voltage source and a current source.

This material is not easy to learn, so take it slow, and do plenty of practice problems to learn these techniques. These problems can be pretty challenging, and here’s where you really find out how well you learned the material in earlier units (such as Kirchhoff’s laws, the divider rules, and so on).

##### Ideal DC Voltage Sources
• An ideal voltage source is one that maintains a constant terminal voltage no matter how much current is drawn from it.
• Below is the familiar schematic symbol for an ideal dc voltage source. ##### Practical Voltage Sources
• It’s impossible to construct an ideal voltage source.
• Examples of real DC voltage sources (which some textbooks also call practical DC voltage sources) are flashlight batteries and the power supplies built into the trainers that we use in Sinclair’s electronics labs.
• The output voltage of a real voltage source will drop as you draw larger and larger currents from it.
• For instance, suppose you’ve got a real voltage source that is set to produce an output voltage of 10.0 V when no resistor is connected to its output terminals.
• Now suppose that you connect a 100 kΩ resistor across the voltage source’s output terminals. With such a large resistance, little current will flow–only about 100 µA, according to Ohm’s law–and connecting this resistor may not cause the voltage source’s output voltage to drop noticeably.
• Next, suppose you replace the 100 kΩ resistor with a smaller one, let’s say 1 kΩ. With this resistance, more current will flow–about 10 mA, according to Ohm’s law. This is still a fairly small current, and you may still find that connecting this resistor does not cause the voltage source’s output voltage to drop noticeably.
• But now suppose you replace the 1 kΩ resistor with a much smaller one, let’s say 1 Ω. With this very small resistance, a large current will flow–about 10 A, according to Ohm’s law. But the voltage source may not be physically capable of producing such a large current. For instance, the power supplies built into the trainers in our labs cannot produce a current that large. So what will happen instead? The voltage source’s voltage will have to drop to a smaller value, even though you haven’t reset it to another value. You may still have the knob set to 10.0 V, but the actual output voltage will drop to a lower value.
• This phenomenon is called loading of a voltage source. As you attach smaller and smaller load resistors to the voltage source, it must produce larger and larger currents, until eventually it cannot keep producing larger currents, at which point its output voltage will have to drop.
• In many cases we can safely ignore this phenomenon, and pretend that our real voltage source behaves like an ideal one. In particular, as we’ll see now, we can treat a voltage source as ideal if the external resistance that we’re connecting to the voltage source is much greater than the voltage source’s internal resistance.
##### Internal Resistance of a Voltage Source
• Why must a real voltage source’s output voltage drop as you draw large currents from it? One way of understanding this is to recognize that any real voltage source will have some internal resistance.
• An ideal voltage source, on the other hand, has zero internal resistance.
• In other words, we can view a real voltage source as being composed of an ideal voltage source connected in series with a resistor, as shown below: • In this diagram RS represents the source’s internal resistance. It is not an external resistor connected to the voltage source. In the diagram, everything inside the gray box is internal to the voltage source. Points A and B are the voltage source’s terminals to which you connect external resistors when you build a circuit. So VAB, the voltage between points A and B, is the source’s output voltage.
• Here’s the important point: Suppose that, as in the diagram above, no external component is connected to points A and B. Then we have what’s called an unloaded voltage source. In an unloaded voltage source, no current flows through RS, which means that there is no voltage drop across RS, which means that the source’s output voltage is equal to the internal voltage being produced by the source:

VAB = VS

But if we connect an external resistor to points A and B, as shown in the diagram below, we have a loaded voltage source. Then a current will flow through RS and the external resistor, which means that there will be a voltage drop across RS, which means that the source’s output voltage will not be equal to the internal voltage being produced by the source:

VABVS

And as more current flows through RS, more voltage will be dropped across RS, so the source’s output voltage will drop further from the original voltage. • As mentioned above, in many cases we can safely ignore a voltage source’s internal resistance, and treat the voltage source as an ideal source. In particular, we can ignore this internal resistance if the external resistance that we’re connecting is much greater than the internal resistance. The following self-test contains some examples to help you see this.
• ##### Ideal Current Source
• An ideal current source will supply the same current to any resistance connected across its terminals.
• Transistors can often be treated as current sources, as you’ll learn when you study transistors in later courses .
• Below is the schematic symbol for an ideal current source. ##### Practical Current Sources
• Just as it’s impossible to construct an ideal voltage source, it’s also impossible to construct an ideal current source.
• The output current from a real current source will drop as you increase the size of the load resistance connected to it.
• In many cases we can safely ignore this phenomenon, and pretend that our real current source behaves like an ideal one. In particular, as we’ll see now, we can treat a current source as ideal if the external resistance that we’re connecting to the current source is much smaller than the current source’s internal resistance.
##### Internal Resistance of a Current Source
• We saw above that real voltage sources can be considered to consist of an ideal voltage source connected in series with an internal resistance.
• Similarly, a real current source also has an internal resistance, but we treat this internal resistance as being connected in parallel with an ideal current source.
• An ideal current source, on the other hand, has infinite internal resistance.
• The figure below shows an unloaded current source. • In this diagram RS represents the source’s internal resistance. It is not an external resistor connected to the current source. In the diagram, everything inside the gray box is internal to the current source. Points A and B are the current source’s terminals to which you connect external resistors when you build a circuit.
• The figure below shows a loaded current source, with an external load resistor connected to the current source’s terminals. • Ideally, the internal resistance RS would be infinite, which means that the entire current IS would flow through the load RL. But if RS is not infinite (as it cannot be in any actual current source), then some current will flow through RS, decreasing the amount of current that flows through the external load RL.
• As mentioned above, in many cases we can safely treat a practical current source as an ideal source. In particular, we can ignore the internal resistance if the external resistance that we’re connecting is much smaller than the source’s internal resistance.
• ##### Source Conversions
• Sometimes when you’re analyzing a circuit, it can be convenient to replace a voltage source with an equivalent current source, or vice versa. When we do this, we say that we’re converting the one type of source into the other type of source. Keep in mind throughout this discussion that we’re simply talking about a mathematical conversion on paper (or in your head). We’re not physically converting one actual device into another actual device.
• Two sources are equivalent if, for any load resistor connected to the two sources, they produce the same voltage across that resistor and the same current through it.
• It turns out to be very easy to convert a practical voltage source into an equivalent practical current source, or vice versa. As you’ll see below, the equations that you use to do this look very much like Ohm’s law.
##### Converting a Voltage Source to a Current Source
• Suppose you’re analyzing a circuit that contains a practical voltage source with source voltage VS and internal resistance RS. You can replace this voltage source with a practical current source having the same internal resistance and having a source current of VS ÷ RS. In other words, the practical voltage source shown here can be replaced by the practical current source shown here as long as they both have the same internal resistance RS and as long as

IS = VS ÷ RS

• ##### Converting a Current Source to a Voltage Source
• Now let’s go in the other direction. Suppose you’re analyzing a circuit that contains a practical current source with source current IS and internal resistance RS. You can replace this current source with a practical voltage source having the same internal resistance and having a source voltage of IS × RS. In other words, the practical current source shown here can be replaced by the practical voltage source shown here as long as they both have the same internal resistance RS and as long as

VS = IS × RS

• ##### Superposition Theorem
• Up to this point we’ve been talking about the related topics of voltage sources, current sources, and how to convert between the two. Now we turn to a new topic.
• How do you analyze a circuit containing more than one voltage or current source? For example, we might have a circuit containing two or more voltage sources. Or a circuit might contain two or more current sources. Or, as in the diagram shown below, a circuit might contain a voltage source and a current source. • The superposition theorem says that you’ll get the right answer in such cases if you analyze the effect of each source acting alone, and then combine (superimpose) those effects.
• For instance, suppose you want to find the voltage across R1 in the circuit shown above. The superposition theorem tells you first to find the voltage that the voltage source produces across R1, then find the voltage that the current source produces across R1, and then combine those two voltages.
• In general, "combining" the voltages means either adding them or subtracting them. In the case of the circuit shown above, we’ll have to subtract them, because the voltage drop across R1 produced by the voltage source has the opposite polarity of the voltage drop across R1 produced by the current source.
##### Removing Voltage Sources or Current Sources
• Just below , we’ll look at a four-step procedure for using the superposition theorem. Before we do that, we must say what we mean by removing a voltage source or a current source from a circuit.
• To remove a voltage source from a circuit, replace the voltage source with a short circuit. For example, if we remove the voltage source from the two-source circuit shown above, we’ll get this circuit: • On the other hand, to remove a current source from a circuit, replace the current source with an open circuit. For example, if we remove the current source from the two-source circuit shown earlier, we’ll get this circuit: • It’s very important for you to remember that removing a voltage source is different from removing a current source; in the first case, you replace the source with a short, but in the second case you replace the source with an open.
##### Superposition Principle: Step-by-Step
• When analyzing a circuit that contains more than one source, follow these four steps:
1. Select one source in the circuit and remove the other sources by:
• replacing voltage sources with short circuits
• replacing current sources with open circuits.
2. Compute the desired voltage or current when the only source present is the one selected in step 1.
3. Repeat Steps 1 and 2 for each of the other sources.
4. Combine the values obtained from the steps above.
• As noted earlier, when Step 4 says to combine the values, that means adding the values if the voltage polarities or currents are in the same direction, but it means subtracting one value from another if the voltage polarities or currents are in opposite directions.
• For instance, in the example circuit shown above (and repeated below), suppose we’re trying to find the current through R1. The voltage source tries to push current through R1 from left to right. But the current source tries to push current through R1 from right to left. So in this case those two sources are working against each other, and we combine their values by subtracting rather than adding.
• On the other hand, in the same circuit, suppose we’re trying to find the current through R2. The voltage source tries to push current through R2 from top to bottom. And the current source also tries to push current through R2 from top to bottom. So in this case those two sources are working with each other, and we combine their values by adding rather than subtracting. • ##### Unit 11 Review

Congratulations! You’ve completed the e-Lesson for this unit.