ee 1105 : introduction to ee freshman seminar

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015

EE 1105: Introduction to EEFreshman Seminar

Lecture 4: Circuit AnalysisNode Analysis, Mesh Currents

Superposition


Circuits• Abstraction describing how (the topology) electrical or electronic

modules are interconnected.• Closely related to a GRAPH.• Nomenclature:

– Nodes, Extraordinary nodes, Supernodes (adjacent nodes sharing a voltage source)

– Edges(Branches)– Paths (collection of edges with no node appearing twice), – Loops (closed paths)– Meshes (loop containing no other loop), Supermeshes (adjacent meshes

sharing a current source)


Kirchhoff’s Voltage Law

• The sum of the voltage drops around a closed path is zero.

• Example: -120 + V1 + V2 + V3 + V4 = 0

120 V

0.25 2.57

2.57

144

+ V1 - + V2 -

- V4 +

+V3

-


Kirchhoff’s Current Law

• A node is a point where two or more circuit elements are connected together.

• The sum of the currents leaving a node is zero.

I1

I2I3

I4

1 2 3 4 0I I I I


Using Loops to Write Equations

KVL @Loop a:

KVL @ Loop b:

KVL @ Loop c:

Loop c equation same as a & b combined.

va

R2

vb

R1 R3

+ v2 -

+v1

-

+v3

-a b

c

2 1 0av v v

3 1 0bv v v

2 3 0a bv v v v


Using Nodes to Write Equations

KCL @ Node x:

KCL @ Node y:

KCL @ Node z:KCL @ Node w: <== Redundant

va

R2

vb

R1 R3

+ v2 -

+v1

-

+v3

-

xy z

w

ia

i2 i2 ib ib

iai3

i1

i1

i3

2 1 0bi i i

2 0ai i

3 0bi i 1 3 0ai i i


Combining the Equations

• There are 5 circuit elements in the problem.• va and vb are known.• R1, R2 and R3 are known.• v1, v2 and v3 are unknowns.• ia, ib, i1, i2 and i3 are unknowns.• There are 2 loop (KVL) equations.• There are 3 node (KCL) equations.• There are 3 Ohm’s Law equations.• There are 8 unknowns and 8 equations.


Example 1 (1/3)

By KCL:

By Ohm’s Law:

50 V

20 A

25

30 A

50 V10

+ Va - + Vb -+ Vc

-

+Vd

-

Ie

If

Ic Id

20 , 30 , 30 , 10e d f ci A i A i A i A

25 250 , 10 300c c d dV I V V I V


Example 1 (2/3)

By KVL:

Power:

50 V

20 A

25

30 A

50 V10

+ Va - + Vb -+ Vc

-

+Vd

-

Ie

If

Ic Id

300 , 600a bV V V V

300 20 6.0aP V A kW

600 30 18.0bP V A kW


Example 1 (3/3)

50 V

20 A

25

30 A

50 V10

+ Va - + Vb -+ Vc

-

+Vd

-

Ie

If

Ic Id

250 10 2.5cP V A kW 300 30 9.0dP V A kW 50 20 1.0eP V A kW 50 30 1.5fP V A kW


Example 2 (1/4)

Find Source Current, I, and Resistance, R.

1

84 V4

12

8

12 R

8

3 A

I


Example 2 (2/4)

Ohm’s Law: 36 V KVL: 48 V Ohm’s Law: 6 A

1

84 V4

12

8

12 R

8

3 A

I+

36 V-

+48 V

-

6 A


Example 2 (3/4)

KCL: 3 A Ohm’s Law: 12 V KVL: 60 V

1

84 V4

12

8

12 R

8

3 A

I+

36 V-

+48 V

-

6 A

3 A -12 V+

+ 60 V-


Example 2 (4/4)

Ohm’s Law: 3 A KCL: 6 AOhm’s Law: R=3 KCL: I=9 A

KVL: 24 V

1

84 V4

12

8

12 R

8

3 A

I+

36 V-

+48 V

-

6 A

3 A -12 V+

+ 60 V-

+ 24 V -

3 A

6 A


Superposition Principle

• Fundamental Property of Linear Circuits

• Replace all but one source in the circuit with a short (voltage source) or an open (current sources).

• Apply analysis to find nodal voltages.

• Repeat for all sources

• Add all nodal voltages to find the total result.


Combining Voltage Sources

10 V

15 V

25 V=

a

a

b

b

Voltage sources are

added algebraically



10 V

15 V

5 V=

a

a

b

b

Voltage sources are

added algebraically



5 V 10 V

b

a

Don’t do this.

Why is this illogical?

Whose fundamental circuit

law is violated by this?


Combining Current Sources

5 A 10 A

b

a

15 A

a

b

=

Current sources are

added algebraically



5 A 10 A

b

a

5 A

a

b

=

Current sources are

added algebraically



5 A

10 A

b

aDon’t do this.

Why is this illogical?

Whose fundamental circuit

law is violated by this?


Nodal Analysis

• Identify all extraordinary nodes. Select one as ground reference and assign node voltages to the other ones.

• Write KCL at the non-zero voltage nodes in conjunction with Ohm’s law.

• Solve a system of simultaneous equations• In the case of a supernode, apply KVL along

the connection, and ignore any resistors in parallel to a voltage source.


Example

EG1 EG214V 14V

R1 R2

R3

30K 15K

7.5KVAB

A

B

N4


More complex example

Image Source: Textbook


Example

• Replace EG1 with a short (zero). Solve resulting circuit for Va1.

• Replace EG2 with a short. Solve resulting circuit for Va2.

• Total Va=Va1+Va2

• Exercise in Lab – you should obtain the same result as in the previous case.

EG1 EG214V 14V

R1 R2

R3

30K 15K

7.5KVAB

A

B

N4


Mesh Analysis

• Identify all meshes and assign them an unknown current, clockwise.

• Write KVL on each mesh

• Solve a system of simultaneous equations

• In the case of a supermesh, add an extra equation with the dependence between the currents


Example

• Two meshes with currents I1 and I2.• KVL:

• Resulting current through R3 is I1-I2.

EG1 EG214V 14V

R1 R2

R3

30K 15K

7.5KVAB

A

B

N4


More complex example

Image Source: Textbook


Mesh Analysis by Inspection• Applies only if all sources be independent

voltage sources

• Same procedure to assign mesh currents.

• Define Rij – resistances as follows: – Rii – sum of all resistances connected to mesh I– Rij=Rji – minus sum of all resistances shared between mesh I and J

• Define total voltages from voltage sources along mesh I as Vi.

• Write and solve matrix equation RI=V, in which R=(Rij), V=(Vi), I=(Ii).


Nodal Analysis By Inspection• Applies only if all sources be independent

current sources

• Same procedure to assign node voltages.

• Define Gij – conductances as follows: – Gii – sum of all conductances connected to node I– Gij=Gji – minus sum of all conductances connected between node I and J

• Define currents from current sources entering node I as Ii.

• Write and solve matrix equation GV=I, in which G=(Gij), V=(Vi), I=(Ii).


Solving Linear Systems of Equations

AX=Ba

11x1

+ a12x2

+ · · · + a1nxn = b1

a21x1

+ a22x2

+ · · · + a2nxn = b2

am1x1

+ am2x2

+ · · · + amnxn = bm

Methods to solve:

1)Elimination

2)Substitution

3)Cramer’s rule

4)Matrix inverse


Homework 4 due next class!!Available online at course website

Acknowledgements: Dr. Bill Dillonhttp://tuttle.merc.iastate.edu/ee201/topics/equivalent_circuits/thevenin.pdf

Questions?

32

ee 1105 : introduction to ee freshman seminar

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