Basic Circuit Backgrounds II
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Department
Year
Electrical &
Electronic
Engineering
Senior
Student ID
Class
Team
Name
1. Objective
Understand the purpose of Thevenin’s and Norton’s equivalent circuit and its method.
Know the importance of transmission of energy in DC circuit, and learn how to maximize it by
reducing power loss. Understand equilibrium bridge circuit and conduct actual experiment.
2. Theory
 Thevenin’s and Norton’s equivalent circuit
1) Describe Thevenin’s and Norton’s theorem.
a) Thevenin’s Theorem1 :
In circuit theory, Thevenin’s theorem for linear electrical networks states that
any combination of voltage sources, current sources, and resistors with two terminals is
electrically equivalent to a single voltage source V and a single series resistor R. For
single frequency AC systems the theorem can also be applied to general impedances, no
just resistors. The theorem was first discovered by German scientist Hermann von
Helmholtz in 1853, but was then rediscovered in 1883 by French telegraph engineer
Leon Charles Thevenin (1857 – 1926).
source in series with an ideal resistor.
This theorem states that a circuit of
voltage sources and resistors can be converted
into a Thevenin equivalent, which is a
simplification technique used in circuit analysis.
The Thevenin equivalent can be used as a good
model for a power supply or battery (with the
resistor representing the internal impedance
and the source representing the electromotive
force). The circuit consists of an ideal voltage
b) Norton’s Theorem2 :
Norton’s theorem for linear electrical
networks, known in Europe as the MayerNorton theorem, states that any collection of
voltage sources, current sources, and resistors
with two terminals is electrically equivalent to
an ideal current source, I, in parallel with a
single resistor, R. For single-frequency AC
systems the theorem can also be applied to
general impedances, not just resistors. The
Norton equivalent is used to represent any
network of linear sources and impedances, at a
given frequency. The circuit consists of an ideal
current source in parallel with an ideal impedance (or resistor for non-reactive circuits).
Norton’s theorem is an extension of Thevenin’s theorem and was introduced in
1926 separately by two people : Siemens & Halske researcher Hans Ferdinand Mayer
(1895 – 1980) and Bell Labs engineer Edward Lawry Norton (1898 – 1983).
2) Solve the equivalent circuit problems by using Thevenin’s and Norton’s
theorem.
- Thevenin’s Theorem
In the example, calculating equivalent voltage :
(Notice that R1 was not taken into consideration, as
above calculations are done in an open circuit
condition between A and B, where there flows no
current through this part, which also means there is
no current through R1 and no voltage drop along
too)
Calculating equivalent resistance :
Therefore, the Thevenin equivalent circuit is like
below :
- Norton’s Theorem:
From the result of Thevenin’s theorem, the current source I is :
𝐼1 =
𝑉𝑇𝐻
𝑅𝑒𝑞
So the equivalent circuit
=
7.5 𝑉
2 𝑘𝛺
= 3.75 [𝑚𝐴]
is like below,
3) Get the Thevenin’s and Norton’s equivalent circuit each for the circuit in fig
2-6.
a) Thevenin’s equivalent circuit
𝑅𝑇𝐻 = 200 𝛺 + 300 Ω 470 Ω
= 6.104 [𝑉]
b) Norton’s equivalent circuit
10 [𝑉]
= 0.0227 [𝐴] = 22.7 [𝑚𝐴]
300 𝛺 + 470 𝛺 200 𝛺
470 𝛺
𝐼=
𝐼
= 0.0159 [𝐴] = 15.9 [𝑚𝐴]
470 𝛺+ 200 𝛺 𝑇𝑜𝑡𝑎𝑙
𝑉𝑜𝑐
𝑖𝑠𝑐 =
𝑅𝑒𝑞
𝐼𝑇𝑜𝑡𝑎𝑙 =
4) When RL = 30Ω, calculate the each voltage and current for the circuit in fig
2-6.
30
𝑉𝐿 = 610 ×
= 0.443 [𝑉]
383.12 + 30
0.443
𝐼𝐿 =
= 0.0148 [𝐴] = 14.8 [𝑚𝐴]
30
 Maximum power transmission
1) Explain how to get maximum power transmission for VDC.3
In electrical engineering, the maximum power transfer theorem states that, to
obtain maximum external power from a source with a finite internal resistance, the
resistance of the load must be equal to the resistance of the source as viewed from the
output terminals.
If the load resistance is smaller than the source resistance, then most of the
power ends up being dissipated in the source, and although the total power dissipated is
higher, due to a lower total resistance, it turns out that the amount dissipated in the load
is reduced. The theorem states how to choose (so as to maximize power transfer) the
load resistance, once the source resistance is give, not the opposite. It does not say how
to choose the source resistance, once the load resistance is give. Given a certain load
resistance, the source resistance that maximizes power transfer is always zero,
regardless of the value of the load resistance.
The theorem can be extended to AC circuits that include reactance, and states
that maximum power transfer occurs when the load impedance is equl to the complex
conjugate of the source impedance.
In the diagram opposite, power is being
transferred from the source, with voltage V and fixed
source resistance Rs, to a load with resistance RL, resulting
in a current I. By Ohm’s law, I is simply the source voltage
divided by the total circuit resistance :
The power PL dissipated in the load is the square of the
current multiplied by the resistance :
The value of RL for which this expression is a maximum could be calculated by
differentiating it, but it is easier to calculate the value of R L for which the denominator
𝑅𝑆2 ⁄𝑅𝐿 + 2𝑅𝑆 + 𝑅𝐿
is a minimum. The result will be the same in either case. Differentiating the denominator
with respect to RL :
𝑑
(𝑅 2 ⁄𝑅 + 2𝑅𝑆 + 𝑅𝐿 ) = − 𝑅𝑆2 ⁄𝑅𝐿2 + 1
𝑑𝑅𝐿 𝑆 𝐿
For a maximum or minimum, the first derivative is zero, so
or
𝑅𝑆2 ⁄𝑅𝐿2 = 1
𝑅𝑆 = ±𝑅𝑆
In practical resistive circuits, RS and RL are both positive, so the positive sign in the above
is the correct solution. To find out whether this solution is a minimum or a maximum,
the denominator expression is differentiated again :
𝑑2
(𝑅𝑆2 ⁄𝑅𝐿 + 2𝑅𝑆 + 𝑅𝐿 ) = 2𝑅𝑆2 ⁄𝑅𝐿3
2
𝑑𝑅𝐿
This is always positive for positive values of RS and RL, showing that the denominator is a
minimum, and the power is therefore a maximum, when
𝑅𝑆 = 𝑅𝐿
2) Calculate RL for maximum power transmission in fig 2-5 and its power Pmax
After source transformation,
𝑅𝑇 ≈ 3.5 [𝛺], 𝑉𝑇 ≈ 0.84 [𝑉]
 Equilibrium bridge circuit
1) Describe the principle of equilibrium bridge circuit4
A wheatstone bridge is an electrical circuit
used to measure an unknown electrical resistance by
balancing two legs of a bridge circuit, one leg of
which includes the unknown component. Its
operation is similar to the original potentiometer. It
was invented by Samuel Hunter Christie in 1833 and
improved and popularized by Sir Charles Wheatstone
in 1843. One of the Wheatstone bridge’s initial uses
was for the purpose of soils analysis and comparison.
In the figure, Rx is the unknown resistance to
be measured; R1, R2 and R3 are resistors of known resistance and the resistance of R2 is
adjustable. If the ratio of the two resistances in the known leg (R2/R1) is equal to the
ratio of the two in the unknown leg (Rx/R3), then the voltage between the two midpoints
(B and D) will be zero and no current will flow through the galvanometer V g. If the bridge
is unbalanced, the direction of of the current indicates whether R2 is too high or too low.
R2 is varied until there is no current through the galvanometer, which then reads zero.
Detecting zero current with a galvanometer can be done to extremely high
accuracy. Therefore, if R1, R2 and R3 are known to high precision, then Rx can be
measured to high precision. Very small changes in Rx disrupt the balance and are readily
detected.
At the point of balance,
𝑅2 ⁄𝑅1 = 𝑅𝑥 ⁄𝑅3
Therefore
𝑅𝑥 = (𝑅2 ⁄𝑅1 ) ∙ 𝑅3
Alternatively, if R1, R2, and R3 are known, but R2 is not adjustable, the voltage difference
across or current flow through the meter can be used to calculate the value of R x, using
Kirchhoff’s circuit laws (also known as Kirchhoff’s rules). This setup is frequently used in
strain gauge and resistance thermometer measurements, as it is usually faster to read a
voltage level off a meter than to adjust a resistance to zero the voltage.
2) Take an example how this circuit is used in the field.
Wheatstone bridge circuit is used in electric scale(early), analog tester, and
finding telephone line disconnected.
3. PSPICE Simulation
References
1. ‘Thevenin’s Theorem’ in Wikipedia, http://en.wikipedia.org/wiki/Th%C3%A9venin's_theorem
2. ‘Norton’s Theorem’ in Wikipedia, http://en.wikipedia.org/wiki/Norton's_theorem
3. ‘Maximum power transfer theorem’ in Wikipedia,
http://en.wikipedia.org/wiki/Maximum_power_transfer_theorem
4. ‘Weatstone bridge’ in Wikipedia, http://en.wikipedia.org/wiki/Wheatstone_bridge