Chapter 2
Engr228
Circuit Analysis
Dr Curtis Nelson
Chapter 2 Objectives
• Understand symbols and behavior of the following circuit
elements:
– Independent voltage and current sources;
– Dependent voltage and current sources;
– Resistors.
• Be able to use Ohm’s and Kirchhoff’s laws to analyze circuits.
• Know how to calculate the power for each element in a circuit
and determine whether or not the power balances.
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Circuit Element
• A circuit element most often has
two terminals.
• The relationship between the
voltage v across the terminals and
the current i through the device
defines the circuit element model.
Various Circuit Elements
•
Electric sources
– Independent Sources – voltage, current;
– Dependent Sources – voltage, current.
•
•
Resistors, inductors, capacitors
Measurement devices
– Ammeters (current);
– Voltmeters (volts);
– Ohmmeters (resistance).
• Electric wire
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Independent Voltage Sources
• An ideal voltage source is a circuit element that will maintain
the specified voltage vs across its terminals.
A Battery as an Independent Voltage Source
• An “ideal” battery is an example of an independent voltage
source.
- A “real-world” battery has a maximum power that it can
deliver.
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Independent Current Sources
• An ideal current source is a circuit element that maintains the
specified current flow is through its terminals.
Dependent Voltage and Current Sources
• A dependent voltage or current
source establishes a voltage or
current whose value depends on a
voltage or current elsewhere in
the circuit.
• Dependent sources are sometimes
called controlled sources.
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Active and Passive Elements
• Active elements are capable of generating electric energy:
– Voltage sources;
– Current sources.
• Passive elements that cannot generate electric energy:
–
–
–
–
Resistors;
Capacitors;
Inductors;
Transistors.
Interconnections of Ideal Sources
• Which of the following are valid connections?
a)
b)
c)
d)
e)
Valid
Valid
Not Valid
Not Valid
Valid
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The Concept of Ground
• Ground is commonly referred to as a reference point.
• Ground is said to be at a potential of 0.00 volts. In other
words, Ground has zero voltage.
• Ground symbols:
Ground
1 Volt
1V
1Ω
0 Volts
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Open Circuit
• An open circuit means no
current can flow (i = 0).
• Voltage across an open
circuit: any value.
• An open circuit is equivalent
to a resistance of ∞ Ω.
• Open circuit summary:
1V
1Ω
– Infinite resistance;
– Zero current;
– Voltage can be any value.
Short Circuit
• A short circuit means the
voltage is zero (v = 0).
• Current through a short
circuit: any value.
• A short circuit is equivalent
to a resistance of 0 Ω.
• Short circuit summary:
1V
1Ω
- Zero resistance;
- Zero voltage;
- Current can be any value.
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Current
• Current is the same in all elements connected in Series.
1A
1V
1Ω
1A
1A
Voltage
• Voltage is the same for all elements connected in Parallel.
x + 1 Volts
1V
1Ω
x Volts
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Resistance (R) and Conductance (G)
• Resistance (R) is the capacity of a material to impede the
flow of current.
• More current will flow if there is less resistance.
• Conductance (G) is the inverse of resistance.
• The unit of resistance is the ohm (R), has the symbol Ω, and
has units of (volts/amp).
• The circuit symbol used for a resistor of R ohms is:
Ohm’s Law
• The relationship between voltage, current, and resistance is
defined by Ohm’s law which states that
V = IR
where
V = the voltage in volts;
I = the current in amps;
R = the resistance in ohms.
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Other Forms of Ohm’s Law
• Ohm’s law can be expressed in any one of three forms,
depending on which quantities are known:
V = IR
I = V/R
R = V/I
(I and R known)
(V and R known)
(V and I known)
Ohm’s Law Illustrated
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Example 2.3
Using Ohm’s law, calculate the values of v and i in the examples
below and determine the power dissipated in each resistor.
a)
b)
c)
d)
va = 8V
ib = 10A
vc = -20V
id = -2A
p = 8W
p = 500W
p = 20W (positive)
p = 100W
Wire Gauge and Resistivity
• The resistance of a wire is determined by the resistivity of the
conductor as well as the geometry:
R=ρl/A
• In most cases, the resistance of wires can be assumed to be 0 Ω.
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Resistors
Resistor Color Code Chart
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Ohm’s Law Example
Calculate the value of R from the information shown in the
graph below:
R is the inverse of the slope of the line:
R = V/I = (8V)/(4A) = 2Ω
Power and its Alternate Forms
• Resistors always absorb power:
p = vi = v2/R = i2R
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Power Example
Calculate the power absorbed or generated by both elements
in the circuit below:
2.5mA
10V
4KΩ
• DC source generates power: 10V * -2.5mA = - 25mW
• Resistor absorbs power: 10V * 2.5mA = 25mW
• Sum of all power in a circuit must = 0
Note: Resistors always absorb power but DC sources can
either generate or absorb power.
Textbook Problem 2.10 (Nilsson 10E)
Find the total power developed in the circuit if vo = 5 V.
P9A = -135W
P20V = 180W
P10va = -75W
Pvg = 27W
P6A = 3W
Ptotal = 210W and generated power = absorbed power
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Nodes, Paths, Loops, Branches
• These two circuits are equivalent.
• There are three nodes and five branches:
– Node: a point at which two or more elements have a common connection
point;
– Branch: a single path in a circuit composed of one simple element and the
node at each end of that element.
• A path is a sequence of nodes.
• A loop is a closed path.
Kirchhoff’s Current Law
• Kirchhoff’s Current Law (KCL) states that the algebraic sum
of all currents entering a node is zero.
iA + iB + (−iC) + (−iD) = 0
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KCL: Alternative Forms
• Current IN is positive:
iA + iB + (−iC) + (−iD) = 0
• Current OUT is positive:
(-iA )+ (-iB ) + iC + iD = 0
• Current IN = Current OUT:
iA+ iB = iC + iD
KCL Application
Find the current through resistor R3 if it is known that the voltage
source supplies a current of 3 A.
Answer: i = 6 A
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Kirchhoff’s Voltage Law
• Kirchhoff’s Voltage Law (KVL) states that the algebraic sum
of the voltages around any closed path is zero.
-v1 + v2 + -v3 = 0
KVL: Alternative Forms
• Sum of RISES is zero (clockwise from B):
v1 +(- v2 ) + v3 = 0
• Sum of DROPS is zero (clockwise from B):
(-v1 ) + v2 + (-v3 ) = 0
• Sum of RISES is equal to sum of DROPS (clockwise from B):
v1 + v2 = v3
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KVL Application
Find the current ix and the voltage vx
Answer: vx= 12 V and ix =120 mA
Textbook Problem 2.19 - Nilsson 9E
Find the values of ia, ib, vo, and the power absorbed or generated
by all circuit elements.
ia = 2A
ib = 0.5A
vo = 40V
P4 = 25W
P20 = 80W
P80 = 20W
P50v = -125W (delivered)
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Circuit Analysis with Dependent Sources
• Circuits that contain dependent sources can be analyzed using
Ohm’s and Kirchhoff’s laws.
• A dependent source generally adds another equation to the
solution process.
Textbook Example - Figure 2.22 (Nilsson 9E)
A) Use Kirchhoff’s and Ohm’s laws to find the voltage vo
B) Show that the total power developed equals the total power
dissipated.
A) vo = 480 V
B) Power developed = - 21.7 W
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Example
Compute the power absorbed in each element of the circuit below:
+ -
30Ω
120V
Answer: P120 =
P30 =
Pdep =
P15 =
2VA
15Ω
VA
+
-960W
1920W
-1920W
960W
Chapter 2 Summary
• Understand symbols and behavior of the following circuit
elements:
– Independent voltage and current sources;
– Dependent voltage and current sources;
– Resistors.
• Defined and used Ohm’s and Kirchhoff’s laws to analyze
circuits.
• Illustrated how to calculate power for each element in a
circuit and determine whether or not power balances.
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