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EE 616 Computer Aided Analysis of Electronic Networks

Lecture 2

Instructor: Dr. J. A. Starzyk, ProfessorSchool of EECSOhio UniversityAthens, OH, 45701

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Review and Outline

Review of the previous lecture

-- Class organization

-- CAD overview Outline of this lecture

* Review of network scaling

* Review of Thevenin/Norton Analysis

* Formulation of Circuit Equations

-- KCL, KVL, branch equations

-- Sparse Tableau Analysis (STA)

-- Nodal analysis

-- Modified nodal analysis

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Network scaling

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Network scaling (cont’d)

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Network scaling (cont’d)

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Voc

Thevenin equivalent circuit

ZTh

+–

Norton equivalent circuit

ZThIsc

Note: attention to the voltage and current direction

Review of the Thevenin/Norton Analysis

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1. Pick a good breaking point in the circuit (cannot split a dependent source and its control variable).

2.Replace the load by either an open circuit and calculate the voltage E across the terminals A-A’, or a short circuit A-A’ and calculate the current J flowing into the short circuit. E will be the value of the source of the Thevenin equivalent and J that of the Norton equivalent.

3. To obtain the equivalent source resistance, short-circuit all independent voltage sources and open-circuit all independent current sources. Transducers in the network are left unchanged. Apply a unit voltage source (or a unit current source) at the terminals A-A’ and calculate the current I supplied by the voltage source (voltage V

across the current source). The Rs = 1/I (Rs = V).

Review of the Thevenin/Norton Analysis

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Modeling

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Formulation of circuit equations (cont’d)

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Ideal two-terminal elements

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Ideal two-terminal elements

Topological equations

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Determined by the topology of the circuit

Kirchhoff’s Current Law (KCL): The algebraic sum of all the currents leaving any circuit node is zero.

Kirchhoff’s Voltage Law (KVL): Every circuit node has a unique voltage with respect to the reference node. The voltage across a branch eb is equal to the difference between the positive and negative referenced voltages of the nodes on which it is incident

KVL and KCL

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Unknowns– B branch currents (i)– N node voltages (e)– B branch voltages (v)

Equations– KCL: N equations– KVL: B equations– Branch equations: B equations

Formulation of circuit equations (cont’d)

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Determined by the mathematical model of the electrical behavior of a component– Example: V=R·I

In most of circuit simulators this mathematical model is expressed in terms of ideal elements

Branch equations

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Matrix form of KVL and KCL

N equations

B equations

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55

4

3

2

1

5

4

3

2

1

4

3

2

1

0

0

0

0

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01

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sii

i

i

i

i

v

v

v

v

v

R

R

GR

Kvv + i = is B equations

Branch equation

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1 2 3 j B

12

i

N

branches

nodes (+1, -1, 0)

{Aij = +1 if node i is terminal + of branch j-1 if node i is terminal - of branch j0 if node i is not connected to branch j

PROPERTIES•A is unimodular•2 nonzero entries in each column

Node branch incidence matrix

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– Sparse Table Analysis (STA) Brayton, Gustavson, Hachtel

– Modified Nodal Analysis (MNA) McCalla, Nagel, Roher, Ruehli, Ho

Equation Assembly for Linear Circuits

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Sparse Tableau Analysis (STA)

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Advantages and problems of STA

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1. Write KCLA·i=0 (N equations, B unknowns)

2. Use branch equations to relate branch currents to branch voltagesi=Yv (B equations, B unknowns)

3. Use KVL to relate branch voltages to node voltagesv=ATe (B equations, N unknowns)

Yne=insN equationsN unknowns

N = # nodesNodal Matrix

Nodal analysis

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Nodal analysis

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Spice input format: Rk N+ N- Rkvalue

kk

kk

RR

RR11

11N+ N-

N+

N-

N+

N-

iRk

sNNk

others

sNNk

others

ieeR

i

ieeR

i

1

1KCL at node N+

KCL at node N-

Nodal analysis – Resistor “Stamp”

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Spice input format: Gk N+ N- NC+ NC- Gkvalue

kk

kk

GG

GGNC+ NC-

N+

N-

N+

N-

Gkvc

NC+

NC-

+

vc

-

sNCNCkothers

sNCNCkothers

ieeGi

ieeGi KCL at node N+

KCL at node N-

Nodal analysis – VCCS “Stamp”

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Nodal analysis- independent current sources “stamp”

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Rules (page 36):

1. The diagonal entries of Y are positive and

admittances connected to node j

2. The off-diagonal entries of Y are negative and are given by

admittances connected between nodes j and k

3. The jth entry of the right-hand-side vector J is

currents from independent sources entering node j

Nodal analysis- by inspection

jky

jjy

jJ

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Example of nodal analysis by inspection

ExerciseFormulate nodal equations by inspection

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Example of nodal analysis by inspection

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Example of nodal analysis by inspection

ExerciseFormulate nodal equations by inspection

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Example of nodal analysis by inspection

Exercise

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Nodal analysis (cont’d)

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Modified Nodal Analysis (MNA)

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Modified Nodal Analysis (2)

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0

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5

7

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88

88

433

32

32

1

ES

i

i

i

e

e

e

e

EE

RR

RR

RRR

RG

RG

R

s

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Modified Nodal Analysis (3)

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General rules for MNA

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Example 4.4.1(p.143)

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Advantages and problems of MNA

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Analysis of networks with VVT’s & Op Amps

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Example 4.5.2 (p.145)

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Example 4.5.5 (p. 148)

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Example 4.5.5 (cont’d)