Thevenin's Theorem
Thevenin's Theorem states that it is possible to simplify any
linear circuit, no matter how complex, to an equivalent circuit with
just a single voltage source and series resistance connected to a
load. The qualification of "linear" is identical to that found in
the Superposition Theorem, where all the underlying equations must
be linear (no exponents or roots). If we're dealing with passive
components (such as resistors, and later, inductors and capacitors),
this is true. However, there are some components (especially certain
gas-discharge and semiconductor components) which are nonlinear:
that is, their opposition to current changes with voltage
and/or current. As such, we would call circuits containing these
types of components, nonlinear circuits.
Thevenin's Theorem is especially useful in analyzing power
systems and other circuits where one particular resistor in the
circuit (called the "load" resistor) is subject to change, and
re-calculation of the circuit is necessary with each trial value of
load resistance, to determine voltage across it and current through
it. Let's take another look at our example circuit:
Let's suppose that we decide to designate R2 as the
"load" resistor in this circuit. We already have four methods of
analysis at our disposal (Branch Current, Mesh Current, Millman's
Theorem, and Superposition Theorem) to use in determining voltage
across R2 and current through R2, but each of
these methods are time-consuming. Imagine repeating any of these
methods over and over again to find what would happen if the load
resistance changed (changing load resistance is very common
in power systems, as multiple loads get switched on and off as
needed. the total resistance of their parallel connections changing
depending on how many are connected at a time). This could
potentially involve a lot of work!
Thevenin's Theorem makes this easy by temporarily removing the
load resistance from the original circuit and reducing what's left
to an equivalent circuit composed of a single voltage source and
series resistance. The load resistance can then be re-connected to
this "Thevenin equivalent circuit" and calculations carried out as
if the whole network were nothing but a simple series circuit:
. . . after Thevenin conversion . . .
The "Thevenin Equivalent Circuit" is the electrical equivalent of
B1, R1, R3, and B2 as
seen from the two points where our load resistor (R2)
connects.
The Thevenin equivalent circuit, if correctly derived, will
behave exactly the same as the original circuit formed by B1,
R1, R3, and B2. In other words, the
load resistor (R2) voltage and current should be exactly
the same for the same value of load resistance in the two circuits.
The load resistor R2 cannot "tell the difference" between
the original network of B1, R1, R3,
and B2, and the Thevenin equivalent circuit of EThevenin,
and RThevenin, provided that the values for EThevenin
and RThevenin have been calculated correctly.
The advantage in performing the "Thevenin conversion" to the
simpler circuit, of course, is that it makes load voltage and load
current so much easier to solve than in the original network.
Calculating the equivalent Thevenin source voltage and series
resistance is actually quite easy. First, the chosen load resistor
is removed from the original circuit, replaced with a break (open
circuit):
Next, the voltage between the two points where the load resistor
used to be attached is determined. Use whatever analysis methods are
at your disposal to do this. In this case, the original circuit with
the load resistor removed is nothing more than a simple series
circuit with opposing batteries, and so we can determine the voltage
across the open load terminals by applying the rules of series
circuits, Ohm's Law, and Kirchhoff's Voltage Law:
The voltage between the two load connection points can be figured
from the one of the battery's voltage and one of the resistor's
voltage drops, and comes out to 11.2 volts. This is our "Thevenin
voltage" (EThevenin) in the equivalent circuit:
To find the Thevenin series resistance for our equivalent
circuit, we need to take the original circuit (with the load
resistor still removed), remove the power sources (in the same style
as we did with the Superposition Theorem: voltage sources replaced
with wires and current sources replaced with breaks), and figure the
resistance from one load terminal to the other:
With the removal of the two batteries, the total resistance
measured at this location is equal to R1 and R3
in parallel: 0.8 Ω. This is our "Thevenin resistance" (RThevenin)
for the equivalent circuit:
With the load resistor (2 Ω) attached between the connection
points, we can determine voltage across it and current through it as
though the whole network were nothing more than a simple series
circuit:
Notice that the voltage and current figures for R2 (8
volts, 4 amps) are identical to those found using other methods of
analysis. Also notice that the voltage and current figures for the
Thevenin series resistance and the Thevenin source (total) do
not apply to any component in the original, complex circuit.
Thevenin's Theorem is only useful for determining what happens to a
single resistor in a network: the load.
The advantage, of course, is that you can quickly determine what
would happen to that single resistor if it were of a value other
than 2 Ω without having to go through a lot of analysis again. Just
plug in that other value for the load resistor into the Thevenin
equivalent circuit and a little bit of series circuit calculation
will give you the result.
- REVIEW:
- Thevenin's Theorem is a way to reduce a network to an
equivalent circuit composed of a single voltage source, series
resistance, and series load.
- Steps to follow for Thevenin's Theorem:
- (1) Find the Thevenin source voltage by removing the load
resistor from the original circuit and calculating voltage across
the open connection points where the load resistor used to be.
- (2) Find the Thevenin resistance by removing all power sources
in the original circuit (voltage sources shorted and current
sources open) and calculating total resistance between the open
connection points.
- (3) Draw the Thevenin equivalent circuit, with the Thevenin
voltage source in series with the Thevenin resistance. The load
resistor reattaches between the two open points of the equivalent
circuit.
- (4) Analyze voltage and current for the load resistor
following the rules for series circuits.
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