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ECE 221Electric Circuit Analysis I

Chapter 7Node Voltage Method

Herbert G. Mayer, PSUStatus 2/2/2015

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Syllabus

Current Status Parallel Resistor Definitions General Circuit Problem Samples Node Voltage Methodology (No Vo Mo) Node Voltage Methodology Steps

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Current Status You have 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 voltage of b*vx

We’ll repeat a few terms and laws

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Parallel Resistors

Two 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 Node (Nd): where 2 or more circuit elements come together

Essential Nd: ditto, but 3 or more circuit elements come together

Path: trace of >=1 basic elements w/o repeat

Branch (Br): path of circuit element connecting 2 nodes

Essential Br: path connecting only 2 essential nodes, not more

Loop: path whose end-node equals start-node w/o repeat

Mesh: loop not enclosing any other loops

Planar Circuit: circuit that can be drawn in 2 dimensions without crossing lines

Cross circuit exercise in class! And pentagon!

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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 that n

equations are needed 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 can we compute all n currents?

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General Circuit Problem

Given n unknowns and n nodes: one can generate only n-1 equations using KCL

But NOT n equations, as electrical units at the nth node can be derived from the other n-1 equations, so this would be redundant

Redundant equations do not help solve unknowns

But one can also generate equations using KVL, to compute the remaining currents –or unknowns

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Circuit for Counting Nodes etc.

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Sample: How Many of Each?

Nodes: 5

Essential Nodes: 3

Paths: large number, since includes sub-paths

Branches: 7

Essential Branches: 5

Loops: 6

Meshes: 3

Is it Planar: yes

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Node Voltage Methodology (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 Methodology– 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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Node Voltage Steps for Sample1

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 Sample1 In the following sample 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 current can be computed

Note: the current through the right-most branch is known to be 2 A

Here is the sample circuit:

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Node Voltage Sample1

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Node Voltage Sample1

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 = 0

Students compute v1 and v2

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Node Voltage Sample1: Compute

v1 = 100 / 11 = 9.09 V

v2 = 120 / 11 = 10.91 V

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Node Voltage Sample2

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 Sample2

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Node Voltage Sample2

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 Sample2: Compute

v1/10 + (v1-50)/5 + v1/40 - 3 = 0//*40

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