Oluwayomi Adamo Department of Electrical Engineering
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EENG 2610: Circuits AnalysisClass 2: Kirchhoff’s Laws, Single-Loop Circuits, Single-Node Pair Circuits
Oluwayomi AdamoDepartment of Electrical EngineeringCollege of Engineering, University of North Texas
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Some Important Concepts Lumped-Parameter Circuit
Wires in circuits are assumed perfect conductor Interconnections in circuits have zero resistance Wire doesn’t consume energy; energy in circuits is
lumped in each circuit element. Node
A point of connection of two or more circuit elements. A node is one end of a circuit element together with
all the perfect conductor that are attached to it. Loop
A loop is any closed path through the circuit in which no node is encountered more than once.
Branch A branch is a portion of a circuit containing only
a single element and the nodes at each end of the element.
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Kirchhoff’s Current Law (KCL) KCL
The algebraic sum of the currents ENTERING any node is zero:
0)(1
N
jj ti
Example 2.5: Write all KCL equations
Our sign convention for KCL:- The algebraic sign of the current is ‘plus’ if the current is entering the node- The algebraic sign of the current is ‘minus’ if the current is leaving the node
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Closed Surface as Super-Node Super-Node
If some set of elements are completely contained within a surface that is interconnected, the surface is called super-node.
Generalized KCL for Super-Node The algebraic sum of the currents entering any closed surface
(or super-node) is zero.
Write KCL equations for super-nodes
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Kirchhoff’s Voltage Law (KVL) KVL
The algebraic sum of the voltages around any loop is zero:
Voltage is defined as the difference in energy level of a unit positive charge located at each of the two points. KVL is based on the conservation of energy: the work required to move a unit charge around any loop is zero.
Our Sign Convention for KVL As moving around a loop, the algebraic sign of voltage is
positive in KVL equation if encounter the plus sign first, and the algebraic sign of voltage is negative in KVL equation if encounter the minus sign first.
0)(1
N
jj tv
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Example 2.9: Using KVL equation to find3R
V
= 18 V
= 12 V
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Convention for Voltage Notation Double-subscript notation
Vab = Va – Vb
+ and – notation
Arrow notation Use an arrow between two
points, pointing from negative node to positive node.
KVL can be applied to a closed path even if part of the closed path is the arrow notation.
V = Vab
a
b
Vab
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Example 2.11: Use KVL to find Vae and Vec
aeVecV
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Single-Loop Circuits
Single-loop circuits Elements are connected
in series. All elements carry same
current. Voltage Divider
Source v(t) is divided between two resistors in proportion to their resistance
2
1
)(
)(
0)(
2
1
21
Rtiv
Rtiv
vvtv
R
R
RR
)(
)(
21
2
21
1
2
1
tvRR
Rv
tvRR
Rv
R
R
21
)()(
RR
tvti
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Multiple Voltage Sources in Single-Loop Circuit
04321 Rvvvvv
1v
2v
3v 4v
RRv+
_
Sv RRv
+
_
0 RS vv
4321 vvvvvS
EquivalentTransform
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Multiple Resistors in Single-Loop Circuit
iRv iRi
0...)(321
NRRRR vvvvtv
NR
R
R
Rtiv
Rtiv
Rtiv
N)(
...
)(
)(
2
1
2
1
)(tv SR
)(ti
EquivalentTransform
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Single-Node Pair Circuits Single-Node Pair Circuits
Elements are connected in parallel Elements have the same voltage across them
Current division
)()(21
21 tiRR
RRtv
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Multiple Current Sources in Single-Node Pair Circuits
EquivalentTransform
)(2 ti)(5 ti
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Multiple Resistors in Single-Node Pair Circuits
)()()()(
)()(ti
R
Rti
R
tvti
tiRtv
Ok
pK
kk
OP
General current divider:
)(tio)(tv+
_
PR
EquivalentTransform
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Equivalent circuit
Example 2.19: Find the current IL