THEVENIN’S THEOREM INTRODUCTION THEVENIN’S EQUIVALENT CIRCUIT ILLUSTRATION OF THEVENIN’S THEOREM FORMAL PRESENTATION OF THEVENIN’S THEOREM PROOF OF THEVENIN’S THEOREM WORKED EXAMPLE 2 WORKED EXAMPLE 3 WORKED EXAMPLE 4 SUMMARY INTRODUCTION Thevenin’s theorem is a popular theorem, used often for analysis of electronic circuits. Its theoretical value is due to the insight it offers about the circuit. This theorem states that a linear circuit containing one or more sour ces and other linearelements can be represented by a voltage source and a resistance. Using this theorem, a model of the circuit can be developed based on its output chara cteristic. Let us try to find out what Thevenin’s theorem is by using an investigative approach. THEVENIN’S EQUIVALENT CIRCUIT In this section, the model of a circuit is derived based on its output chara cateristic. Let a circuit be represented by a box, as shown in Figure 8. Its output chara cteris tic is also displayed. As the load resistor is varied, the load current m varies. The load current is bounded between two li mits, zero and I, and the load voltage is bounded between limits, EVolts and zero volts. When the load resistoris infinite, it is an open circu it. In this case, the load voltage is at its highest, which is Evolts and the load current is zero. This is the point at which the output characteristic intersects with the Y axis. When the load resistor is of zero value, there is a short circuit acr oss the output terminals of the circuit and in this in stance, m the load current is maximum, specified asIand the load voltage is zero. It is the point at which the output characteristic intersects with the X axis.
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Thevenin’s theorem is a popular theorem, used often for analysis of electronic
circuits. Its theoretical value is due to the insight it offers about the circuit. This
theorem states that a linear circuit containing one or more sources and other linear
elements can be represented by a voltage source and a resistance. Using this
theorem, a model of the circuit can be developed based on its output characteristic.
Let us try to find out what Thevenin’s theorem is by using an investigative
approach.
THEVENIN’S EQUIVALENT CIRCUIT
In this section, the model of a circuit is derived based on its output characateristic.Let a circuit be represented by a box, as shown in Figure 8. Its output
characteristic is also displayed. As the load resistor is varied, the load current
mvaries. The load current is bounded between two limits, zero and I , and the load
voltage is bounded between limits, E Volts and zero volts. When the load resistor
is infinite, it is an open circuit. In this case, the load voltage is at its highest,
which is E volts and the load current is zero. This is the point at which the output
characteristic intersects with the Y axis. When the load resistor is of zero value,
there is a short circuit across the output terminals of the circuit and in this instance,
mthe load current is maximum, specified as I and the load voltage is zero. It is the point at which the output characteristic intersects with the X axis.
Equation (22) defines the expressions for Thevenin’s voltage and Thevenin’s
resistance. They are obtained from equation (21).
From the expression for the load current, we can obtain a circuit and this circuit
is presented in Figure 12. We can now ask what Thevenin’s voltage andThevenin’s resistance represent? How do we obtain them in a simpler way? They
can be obtained as shown next.
Thevenin’s voltage is the voltage across the load terminals with the load resistor removed. In other words, the load resistor is replaced by an open circuit. In this
3instance, the load resistor is R and it is replaced by an open circuit. Then
2Thevenin,s voltage is the open circuit voltage, the voltage across resistor R . This
voltage can easily be obtained by using the voltage division rule. The voltage
division rule states the division of source voltage is proportionate to resistance.
Thevenin’s resistance is the resistance, as viewed from the load terminals, with
both the load resistor and the sources in the circuit removed. Here removal of the
voltage source means that it is replaced by a short circuit, and the load resistor is
replaced by an open circuit. Thevenin’s resistance is the parallel value of resistors1 2 R and R . Next Thevenin’s theorem is presented in a formal manner.
Thevenin’s theorem represents a linear network by an equivalent circuit. Let
a network with one or more sources supply power to a load resistor as shown in
Fig. 14. Thevenin’s theorem states that the network can be replaced by a single
equivalent voltage source, marked as Thevenin’s Voltage or open-circuit voltage
and a resistor marked as Thevenin’s Resistance. Proof of this theorem is presented below. Thevenin’s theorem can be applied to linear networks only.
Thevenin’s voltage is the algebraic sum of voltages across the load terminals, due
to each of the independent sources in the circuit, acting alone. It can be seen that
Thevenin’s theorem is an outcome of superposition theorem.
Thevenin’s equivalent circuit consists of Thevenin’s voltage and Thevenin’s
resistance. Thevenin’s voltage is also referred to as the open-circuit voltage,
meaning that it is obtained across the load terminals without any load connected
to them. The load is replaced by an open-circuit and hence Thevenin’s voltage is
called as the open-circuit voltage.
Figure 15 shows how Thevenin’s voltage is to be obtained. Here it is assumed that
we have a resistive circuit with one or more sources. As shown in Fig. 15,Thevenin’s voltage is the open-circuit voltage across the load terminals. The
voltage obtained across the load terminals without the load being connected is the
open-circuit voltage. This open-circuit voltage can be obtained as the algebraic
sum of voltages, due to each of the independent sources acting alone. Given a
circuit, Thevenin’s voltage can be obtained as outlined below.
Figure 16 shows how Thevenin’s resistance is to be obtained. Thevenin’s
resistance is the resistance as seen from the load terminals. To obtain this
resistance, replace each independent ideal voltage source in the network by a
short circuit, and replace each independent ideal current source by an open circuit.If a source is not ideal, only the ideal part of that source is replaced by either a
short circuit or an open circuit, as the case may be. The internal resistance of the
source, reflecting the non ideal aspect of the circuit, is left in the circuit, as it is
where it is. A voltage source is connected across the load terminals. Then
Thevenin’s resistance is the ratio of this source voltage to its current, as marked
in Fig. 16. A few examples are presented after this page to illustrate the use of
Thevenin’s theorem.
PROOF OF THEVENIN’S THEOREM
The circuit in Fig. 17 can be used to prove Thevenin’s theorem. Equation (1) inYthe diagaram expresses an external voltage V connected to the load terminals, as
Ya function of current I and some constants. It is valid to do so, since we are
dealing with a linear circuit. Let us some that the internal independent sources
Y Yremain fixed. Then, as the external voltage V is varied, current I will vary, and
Y Ythe variation I with V is accounted for by provision of a coefficient , named as
1 1k in equation (1). It can be seen that k reflects resistance of the circuit as seen by
Y 2external voltage source V . Coefficient k reflects the contribution to terminal
voltage by internal sources and components of the circuit. It is valid to do so,
since we are dealing with a linear circuit, and a linear circuit obeys the principleof superposition. Each independent internal source within the circuit contributes
2its part to terminal voltage and constant k is the algebraic sum of contributions of
Yinternal sources. Adjust external voltage source such that current I becomes
2zero. As shown by equation (2), the coefficient k is Thevenin’s voltage. To
determine Thevenin’s resistance, set external source voltage to zero. If the internal
Ysources are such as to yield positive Thevenin’s voltage, current I will be negative1and coefficient k is Thevenin’s resistance, as shown by equation (3). This
concludes the proof of Thevenin’s theorem.
The step involved in the application of Thevenin’s theorem are summarized below.
WORKED EXAMPLE 2
A problem has been presented now. For the circuit in Fig. 18, you are asked to
obtain the load current using ThevEnin’s theorem. We have already looked at this
circuit, but the purpose here is to show, how to apply Thevenin’s theorem.
You are asked to obtain the Thevenin’s equivalent of the circuit in Fig. 22. This
problem is a bit more difficult, since it has dependent sources. The Thevenin’s
theorem can be applied to circuits containing dependent sources also. The only
constraint in applying Thevenin’s theorem to a circuit is that it should be a linear
circuit.
Steps involved can be listed as follows:
C Obtain the Thevenin’s Voltage.
C Obtain the Thevenin’s Resistance.
C Draw the Thevenin’s equivalent circuit.
Given a circuit with dependent sources, it may at times be preferable to obtain the
open circuit voltage and the short circuit current, and then obtain Thevenin’s
resistance as the ratio of open circuit voltage to the short circuit current. The short
circuit current is obtained by replacing the load resistor by a short circuit, and itis the current that flows through the short circuit. This technique has been used
in the proof of Thevenin’s theorem.
2Since there is no load connected to the output terminals, voltage V is the open
circuit voltage, which is the same as the Thevenin’s voltage. To obtain the opencircuit voltage, the following equations are obtained.
2Equation (26) expresses the voltage across resistor R . The current through
2 2resistor R is ten times current I , and the value of resistor R is 100W. Equation
(27) is written for the loop containing the independent source voltage. The
1independent source voltage is 10 Volts. The value of resistor R is 10W, and the
current through it can be obtained as shown by equation (27). Equation (28) is2obtained by replacing voltage V in equation (27) by its corresponding expression
in equation (26).
On simplifying, we can obtain the value of current I , and the Thevenin’s voltage,
as illustrated by equation (29).
The second step is to obtain Thevenin’s resistance. The circuit in Fig. 23 is used
for this purpose.
To obtain Thevenin’s resistance of a circuit with dependent source, it is preferable
to obtain the short circuit current and then obtain Thevenin’s resistance as the ratio
of Thevenin’s voltage to short circuit current. The circuit in Fig. 23 is used to
obtain the short circuit current.
Equations (30) to (33) are obtained from the circuit in Fig. 23. When the output
terminals are shorted, the short circuit current, known also as the Nortons current,
is ten times current I , as shown by equation (30). Note that the source voltage is
10 Volts. When the output voltage is zero, current I is the ratio of source voltage
1to resistor R and it equals one Ampere, as displayed by equation (31). Equations