Chapter 3: Resistive Circuits
3.1 Introduction 3.2 Nodes, Branches, and Loops
3.3 Kirchhoff’s Laws
3.4 Series Resistors and Voltage Division
3.5 Parallel Resistors and Current Division
3.6 Series Voltage Sources and Parallel Current Sources
3.7 Wye‐Delta Transformations
3.8 Summary
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3.1 Introduction (1)
In this chapter, we will do the following:
• Write equations using Kirchhoff’s laws.
• Analyze simple electric circuits, using only Kirchhoff’s laws and the constitutive equations of the circuit elements.
• Analyze two very common circuit configurations: series resistors and parallel resistors.
• Determine equivalent circuits for series voltage sources and parallel current sources.
• Determine the equivalent resistance of a resistive circuit. 2
3.1 Introduction (2)
Fig. (a) An electric circuit. (b) The same circuit, redrawn using straight lines and horizontal and vertical elements. (c) The circuit after labeling the nodes and elements.
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3.2 Nodes, Branches, and Loops (1)
• A branch represents a single element such as a voltage source or a resistor. • A node is the point of connection between two or more branches. • A loop is any closed path in a circuit. Ex. How many branches, nodes and loops are there?
Original circuit
Equivalent circuit
• A network with b branches, n nodes, and l independent loops will satisfy the fundamental theorem of network topology:
b  l  n 1
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3.2 Nodes, Branches, and Loops (2)
Example: How many branches, nodes and loops are there?
Should we consider it as one branch or two branches?
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3.3 Kirchhoff’s Laws (1)
• Kirchhoff’s current law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero.
Mathematically, The total current entering the closed surface is equal to the total current leaving the surface
N
i
n 1
n
0
The sum of the currents entering a node is equal to the sum of the currents leaving the node. Example:
Determine the current I.
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3.3 Kirchhoff’s Laws (2)
• Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero. Mathematically, M
v
m 1
n
0
Example: Applying the KVL equation for the circuit.
va-v1-vb-v2-v3 = 0
V1 = IR1 v2 = IR2 v3 = IR3
 va-vb = I(R1 + R2 + R3)
I
va  vb
R1  R2  R3
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2.3
3.3 Kirchhoff’s Laws (3)
Example: Find i0 & v0.
Example: Find Vo / Vs in terms of , R1, R2, R3, & R4. If R1 =
R2 = R3 = R4, what value of 
will produce |Vo / Vs| = 10?
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3.4 Series Resistors and Voltage Division (1)
• Series: Two or more elements are in series if they are cascaded or connected sequentially and consequently carry the same current.
• The equivalent resistance of any number of resistors connected in a series is the sum of the individual resistances. N
Req  R1  R2      R N   Rn
n 1
• The voltage divider can be expressed as
Rn
vn 
v
R1  R2      R N
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3.4 Series Resistors and Voltage Division (2)
Example:
Example:
10V and 5Ω
are in series
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3.5 Parallel Resistors and Current Division (1)
• Parallel: Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them.
• The current divider can be • The equivalent resistance of a expressed as: circuit with N resistors in parallel: 1
1
1
1


  
Req R1 R2
RN
v iReq
in 

Rn Rn
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3.5 Parallel Resistors and Current Division (2)
Example:
Example:
2Ω, 3Ω and 2A
are in parallel
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3.5 Parallel Resistors and Current Division (3)
Example: Calculate the equivalent resistance Rab.
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3.5 Parallel Resistors and Current Division (4)
Example: Calculate the equivalent resistance Rab.
Example: Calculate the equivalent conductance Geq.
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3.6 Series Volt. and Parallel Curr. Sources (1)
(a) A circuit containing series voltage sources
(b) an equivalent circuit.
3.6
(a) A circuit containing parallel current sources
(b) an equivalent circuit.
3.6 Series Volt. and Parallel Curr. Sources (2)
3.7 Wye‐Delta Transformations (1)
Delta ‐> Wye
R1 
Rb Rc
( Ra  Rb  Rc )
Wye ‐> Delta
R1 R2  R2 R3  R3 R1
Ra 
R1
R1 R2  R2 R3  R3 R1
R2
Rc Ra
R2 
( Ra  Rb  Rc )
Rb 
Ra Rb
R3 
( Ra  Rb  Rc )
R1 R2  R2 R3  R3 R1
Rc 
R3
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3.7 Wye‐Delta Transformations (2)
Example: Obtain the equivalent resistance Req. 18
3.7 Wye‐Delta Transformations (3)
Example: Obtain the equivalent resistance Rab and use it to find i. 19
Prob. 2.52
3.8 Summary