Transcript Node Voltage Method

ECE 221
Electric Circuit Analysis I
Chapter 7
Node Voltage Method
Herbert G. Mayer, PSU
Status 10/15/2015
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Syllabus
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Studying ECE 221
Parallel Resistor
Definitions
General Circuit Problem
Samples
Node Voltage Method (No Vo Mo)
Node Voltage Method Steps
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Studying ECE 221
 You learned how to construct simple circuit models
 Learned the 2 Kirchhoff Laws: KCL and KVL
 You know constant voltage and constant current
sources
 You know dependent current and dependent voltage
sources
 When a dependent voltage source depends on a*ix
then that factor defines the voltage, i.e. the amount
of Volt generated, NOT the current!
 Ditto for dependent current source, depending on
some voltage: A current is being defined, through a
numeric voltage factor of b*vx
 We’ll repeat a few terms and laws
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Parallel Resistors
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Resistors R1 and R2 are in a parallel circuit:
What is their resulting resistance?
How to derive this?
Hint: think about Siemens!
Not the engineer, the conductance 
This seems trivial, yet will return again with
inductivities in the same way
 And in a similar (dual) way for capacities
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Definitions
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Node (Nd):
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Essential Nd: ditto, but 3 or more circuit elements come together
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Path:
trace of >=1 basic elements w/o repeat
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Branch (Br):
circuit path connecting 2 nodes
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Essential Br:
path connecting only 2 essential nodes, not more
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Loop:
path whose end-node equals start-node w/o repeat
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Mesh:
loop not enclosing any other loops
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Planar Circuit: circuit that can be drawn in 2 dimensions without
crossing lines
where 2 or more circuit elements come together
Cross circuit exercise in class and pentagon to test planarity!
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General Circuit Problem
 Given n unknown currents in a circuit C1
 How many equations are needed to solve the
system?
 Rhetorical question: We know n equations!
 If circuit C1 also happens to have n nodes,
can you solve the problem of computing the
unknowns using KCL?
 Also rhetorical question: We know that n
nodes alone will not suffice using only KCL!
 How then can we compute all n currents?
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General Circuit Problem
 Given n unknowns and n nodes: one can
generate n-1 independent equations via KCL
 But NOT n equations, as electrical units at
the nth node can be derived from the other n1 equations: would be redundant
 Redundant equations do not help solve
unknowns
 But one can also generate equations using
KVL, to compute the remaining currents –or
other unknowns
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Circuit for Counting Nodes etc.
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Sample: How Many of Each?
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Nodes:
5
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Essential Nodes:
3
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Paths:
large number, since includes sub-paths
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Branches:
7
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Essential Branches:
5
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Loops:
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Meshes:
3
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Is it Planar:
yes
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Node Voltage Method (No Vo Mo)
 Nodes have no voltage! So what’s up? What does this
phrase mean?
 Nodes are connecting points of branches
 Due to laws of nature, expressed as KCL: nodes have
a collective current of 0 Amp!
 So why discuss a Node Voltage Method (No Vo Mo)?
 No Vo Mo combines using KCL and Ohm’s law across
all paths leading to any one node, from all essential
nodes to a selected reference node
 The reference node is also an essential node
 Conveniently, we select the essential node with the
largest number of branches as reference node
 Is not necessary; works with any essential node
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Node Voltage Steps
 Interestingly, the No Vo Mo –i.e. Node
Voltage Method– applies also to non-planar
circuits!
 The later to be discussed Mesh-Current
Method only applies to planar circuits
 We’ll ignore this added power of No Vo Mo
for now, and focus on planar circuits
 Here are the steps:
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Node Voltage Steps
 Analyze your circuit, locate and number all essential
nodes; we call that number of essential nodes ne
 For now, view only planar circuits
 From these ne essential nodes, pick a reference node
 Best to select the one with the largest number of
branches; simplifies the formulae
 Then for each remaining essential node, compute the
voltage rises from the reference node to the selected
essential node, using KCL
 For ne essential nodes we can generate n-1 Node
Voltage equations
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Sample 1
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Node Voltage Steps for Sample 1
1. Using KCL:
2. Analyze the circuit below, and generate 2
Node Voltage equations
3. Enables us to compute 2 unknowns
4. We see that v1 and v2 are unknown
5. Once v1 and v2 are known then we can
compute all currents
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Node Voltage Sample 1
 In the Sample 1 circuit, use the Node Voltage Method
to compute v1 and v2
 There are 3 essential nodes
 Pick the lowest one as the reference node, since it
unites the largest number of branches
 Once voltages v1 and v2 are known, the currents in
the 5 and 10 Ohm resistors are computable
 Using KCL and Ohm’s Law, all other currents can be
computed
 Note: the current through the right-most branch of
Sample 1 is known to be 2 A
 Here is the Sample 1 circuit:
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Node Voltage Sample 1
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Node Voltage Sample 1
For node n1 compute all currents using KCL:
(v1 - 10)/1 + v1/5 + (v1 - v2)/2 = 0
For node n2 compute all currents using KCL:
V2/10 + (v2 - v1)/2 - 2
Students compute v1 and v2
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= 0
Node Voltage Sample 1: Compute
v1 = 100 / 11
= 9.09 V
v2 = 120 / 11
= 10.91 V
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Sample 2
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Node Voltage Sample 2
 In the following sample circuit, use the Node Voltage
Method to compute v1, ia, ib, and ic
 There are 2 essential nodes
 Hence we need just 1 equation to compute v1
 We pick the lowest one as the reference node, since
it has the largest number of branches
 Once voltage v1 is known, the currents are
computable
 Using KCL, all other current can be computed
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Node Voltage Sample 2
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Node Voltage Sample 2
For node n1 compute all currents using KCL:
v1/10 + (v1-50)/5 + v1/40 - 3 = 0
Students compute v1, ia, ib, and ic
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Node Voltage Sample 2: Compute
v1/10 + (v1-50)/5 + v1/40 - 3 = 0 // *40 +3
4*v1 +8*v1 - 50*8 + v1
= 3 * 40
v1*( 4 + 8 + 1 ) - 400
= 120
13*v1
= 520
v1 = 40 V
ia = ( 50 – 40 ) / 5 = 2 A
ib = 40 / 10 = 4 A
ic = 40 / 40 = 1 A
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