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 gasdischarge 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 recalculation 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 R_{2}
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 R_{2} and current
through R_{2}, but each of these methods are
timeconsuming. 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 reconnected 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 B_{1}, R_{1}, R_{3},
and B_{2} as seen from the two points where our load
resistor (R_{2}) connects.
The Thevenin equivalent circuit, if
correctly derived, will behave exactly the same as the
original circuit formed by B_{1}, R_{1}, R_{3},
and B_{2}. In other words, the load resistor (R_{2})
voltage and current should be exactly the same for the same
value of load resistance in the two circuits. The load
resistor R_{2} cannot "tell the difference" between
the original network of B_{1}, R_{1}, R_{3},
and B_{2}, and the Thevenin equivalent circuit of E_{Thevenin},
and R_{Thevenin}, provided that the values for E_{Thevenin}
and R_{Thevenin} 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" (E_{Thevenin})
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 R_{1}
and R_{3} in parallel: 0.8 Ω. This is our "Thevenin
resistance" (R_{Thevenin}) 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 R_{2} (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.
