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
which lets you analyze circuits containing more than one source. This
unit will introduce two more useful theorems: Thevenin's
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.
Unit 11 Review
- This unit will build on material that you studied in Unit
11. So let's
begin by taking this self-test to review what you learned in that unit.
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
- 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.
- Now we'll use this idea of the black-box game to introduce Thevenin's
- 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.
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.
- 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
- Example: Suppose that you're dealing with the circuit shown below.
After you remove the load resistor, you'll be left with this:
- 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.
- 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.
- Connect VTH and RTH
in series, as shown in the diagram below. This is the Thevenin equivalent
circuit of the original circuit.
- 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
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!
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.
Source and Load
- 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
- 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
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.
Matching Source and Load
- When the source and the load have the same resistance, they are said
to be matched.
- Example: You may know that stereo speakers are rated in terms of
their impedance, which is measured in ohms. For instance, you might
have 4 Ω speakers. Also, one of
the important specifications of a stereo amplifier is its output
impedance, also measured in ohms. For best results (maximum
power), your speakers should be properly matched
to your amplifier. For example, you would not want to connect 4 Ω speakers
to an amplifier whose output impedance is 16 Ω.
- In many practical applications, a major part of a circuit designer's
effort is to ensure that a load is matched to the source that is powering
Unit 12 Review
- This e-Lesson has covered several important topics, including:
- Thevenin's theorem
- the maximum power transfer theorem.
- To finish the e-Lesson, take this self-test to check your understanding
of these topics.
Congratulations! You've completed the e-Lesson for this unit.