EECE251

Circuit Analysis ILecture Integrated Program

1SMEECE 251, Set 2

Set 2: Methods of Circuit Analysis

Shahriar MirabbasiDepartment of Electrical and Computer Engineering

University of British Columbiashahriar@ece.ubc.ca

Methods of Circuit Analysis

• Two popular and powerful techniques for analyzing circuits are:

– Nodal analysis: a general procedure to find all the nodevoltages in a circuit. It is based on KCL and Ohm’s Law.

– Mesh analysis: another general approach to find meshcurrents which circulate around closed paths in the circuit. It

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currents which circulate around closed paths in the circuit. Itis based on KVL and Ohm’s Law.

• Yet there is another more general! and powerful! techniquewhich we call:– Modified Nodal Analysis (MNA)– Though more powerful it is not as popular of the first two

(Almost all books don’t even have it!)

More Terminology

• Reference node or ground: a node that is assumed to have azero potential.– If the reference node is not explicitly indicated on the circuit

one can arbitrarily choose any node as the ground. We willsoon see how to choose a good ground node.

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• Node voltage is the voltage difference/drop from a given nodeto the reference node.

Nodal Analysis

• Steps to determine the node voltages for a circuit with nofloating voltage source:

1. Select a reference node. A floating voltage source is a voltagesource that neither of its terminals is connected to the referencenode.

2. Assign voltages to other nodes. These node voltages are

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2. Assign voltages to other nodes. These node voltages arereferenced to the reference node.

3. Write KCL for all unknown non-reference nodes. When possibleuse Ohm’s law to relate the branch currents to node voltages

4. Solve the resulting system of equations for unknown nodevoltages.

Nodal Analysis Example

• Let’s analyze the following circuit using nodal analysis:

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Nodal Analysis

• Consider node a:

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• Consider node b:

Floating Voltage Sources

• Problem: The current through the floating voltage source cannotbe written as function of its two terminal voltages!

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• Solution: Form a supernode which is formed by enclosing thefloating voltage source (independent or dependent) and anyelements in parallel with it in a closed boundary.

Floating Voltage Sources

• Since there are two nodes (two terminals of the floating voltagesource) are enclosed in the supernode, two equations areneeded for each supernode:

– KCL at supernode gives one equation

– The other equation is the relationship between the voltages

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– The other equation is the relationship between the voltagesof the two nodes enclosed in the supernode

• For example for the supernode in the previous slide we canwrite the following two equations:

Nodal Analysis Example

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• KCL at Node a:

• For the supernode we have:

Example

• Find v and i in the circuit below:

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Mesh Analysis

• Mesh analysis is a special case of a more general techniquecalled loop analysis.

• A mesh is a loop that does not contain any other loops within it.• Mesh analysis is not quite as general as nodal analysis since it

can only be applied to planar circuits• A planar circuit is a circuit that can be drawn in a plane with no

branches crossing one another.

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branches crossing one another.• Example of non-planar circuits:

Mesh Analysis

• Steps to calculate mesh currents for a given circuit in which nocurrent source is shared between two meshes:

1. Assign mesh currents to each mesh

2. Write KVL for each of the meshes and use Ohm’s law toexpress the voltages of the elements in the mesh in terms of

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express the voltages of the elements in the mesh in terms ofmesh currents

3. Solve the resulting systems of linear equations for unknownmesh currents

Mesh Analysis Example

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• For mesh 1:

• For mesh 2:

• Mesh 3?

or

or

Example

• In the following circuit, use mesh analysis to find Io:

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Mesh Analysis

• Steps to calculate mesh currents for a given circuit in whichsome current sources are shared between two meshes:

1. Assign mesh currents to every mesh in the circuit.

2. Define a supermesh when two (or more) meshes have a(dependent or independent) current source(s) in common.

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3. Write KVL for each regular mesh.

4. Apply both KVL and KCL to suppermeshes.

5. Solve the resulting system of equations.

Example

• In the following circuit, find the mesh currents:

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Example

• Write the mesh equations for the following circuit:

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Notes

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On the Way to Modified Nodal Analysis

• For modified nodal analysis (MNA) we need some moredefinitions!

• We identify five general types for branches:Types of branches (we have seen these before, and here weare just formalizing them by giving them proper names!):

R branch

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R branchRV branchI branchRI branchV branch (also known as evil branch!)

Let’s see if we can calculate the current of these branchesbased on the end-point node voltages!

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R Branch

• A branch that consist of only a resistor (or seriescombination of resistors that can be represented bytheir equivalent resistors)

I

RVV

I ba −=

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• How about the current in the other direction!

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RI =

I

RVV

I ab −=

RV Branch

• A branch that consist of a resistor (or series combination ofresistors that can be represented by their equivalent resistors) inseries with a voltage sourse (or a series combination of voltagecoursed that sources that can be represented by theirequivalent voltage source)

Ia b

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a b

RVVV

R)VV(V

I sbasba −−=+−=

I Branch and IR Branch

• I branch: A branch that consists of only a current source!

I

sII =

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• IR branch: A branch that consists of a resistor (or equivalentresistor) in series with a current source:

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sII =

V Branch (Evil Branch)

• A branch that consists of only one voltage source:

I

a b ?I =

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• However, the good news is:

• Note: The sources in V, RV, I, and RI branch can be eitherdependent (controlled) or independent sources

EECE 251, Set 2

a b ?I =

bas VVV −=

Modified Nodal Analysis (MNA)

• A general technique to solve a circuit (i.e., to find voltage,current and power of every element in the circuit).

• Unknowns:1) controlling variables (for dependent sources)2) current in V branches (evil branches)3) voltage of each true node

• MNA steps:

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• MNA steps:1. Identify every true node of the circuit.2. Choose one of them as a reference node (node whose voltage

is zero).3. Write one equation per controlling current or voltage of

dependant sources.4. Write the relationship between the two nodes of the V branch .5. Write one KCL per true node.

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Example (MNA)

• Solve the following circuit using MNA.

+_

5 3

2

10V 4ix

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+_

+_

+_

7

6

10V

7V

5A

4i

5ix

x

ix

Notes

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A Note on Series and Parallel Capacitors

• The equivalent capacitance of series-connected capacitors isthe reciprocal of the sum of the reciprocals of the individualcapacitances. Why?

neq CCCC

1111

21

+++= L

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• The equivalent capacitance of parallel capacitors is the sum ofthe individual capacitances. Why?

neq CCCC +++= L21

A Note on Series and Parallel Inductors

• The equivalent inductance of series-connected inductors is thesum of the individual inductances. Why?

neq LLLL +++= L21

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• The equivalent inductance of parallel inductors is the reciprocalof the sum of the reciprocals of the individual inductances. Why?

neq LLLL

1111

21

+++= L