ELECTRIC CIRCUITSECSE-2010Spring 2003

Class 21

ASSIGNMENTS DUE• Today (Tuesday/Wednesday):

• Experiment #7 Report Due (1 Per Team)• Will do Computer Project #3 in Class• Activities 21-1, 21-2 (In Class)

• Thursday:• Activities 21-3, 22-1 (In Class)

• Next Week – Spring Break:• No Classes!!

• Monday, March 17:• Homework #8 Due (1 Per Team)• Computer Project #3 Report Due (1 Per Student)• Activities 23-1, 23-2 (In Class)

AC STEADY STATE

1x(t) A cos ( t ) F 2y (t) B cos ( t )

nd2 Order Circuit

Input x(t) Output y(t)

Ny (t) Under, Over or Critically Damped

REVIEW

• AC Steady State:• Input = x(t) =

• Steady State Output = yF(t) = yss(t)

• yss(t) =

• Phasors; Express x(t) as Phasor X:

1 1Acos ( t ) X cos ( t )

2 2Bcos ( t ) Y cos ( t )

1

r i

1

j

X X jX Rectangular Form; j 1

X X / Polar Form

X X e Euler Form

REVIEW

A

1

Complex Space

Real

Imaginary1X A/ Polar Form

r 1X Acos

i 1X Asin

r iX X jX Rectangular Form

1je

1jX Ae Euler Form

1

COMPLEX MATH

• Addition:• A + B = (ar + j ai) + (br + j bi)

= (ar + br) + j (ai + bi)

• Subtraction:• A - B = (ar + j ai) - (br + j bi)

= (ar - br) + j (ai - bi)

• => Do Addition/Subtraction in Rectangular Form

COMPLEX MATH• Multiplication:

• Difficult to do in Rectangular Form• Use Euler or Polar Form

• Division:

• Do Multiplication/Division in Polar Form

1 2 1 2j j j( )

1 2 1 2

AA B Ae /Be e

BA

A/ / B/ /( )B

1 2 1 2j j j( )

1 2 1 2

A x B Ae x Be AB e

A/ x B/ AB/( )

RATIONALIZATION

• Division in Rectangular Form:r i

r i

a jaC A B

b jb

r i r i

r i r i

a ja b jbMultiply by

b jb b jb

r iWant to express as C C jC

r r i i i r r i2 2 2 2r i r i

r i

a b a b a b a bC j

b b b b

C j C

COMPLEX CONJUGATES

• A = ar + j ai

• A* = ar - j ai = Complex Conjugate of A

• A x A* = ar2 + ai

2 =

• Angle of A* = - (Angle of A)

2A

*/ A / A

ACTIVITY 21-1

A j5 B 4 j2 C 1 j3

Find D A x B j5 x ( 4 j2) 2j20 j 10

2j 1 D 10 j20

ACTIVITY 21-1

*

DFind E

C D 10 j20

C 1 j3 *C 1 j3

10 j20E

1 j3

1 j3

x 1 j3

10 j30 j20 60

1 9

50 j50

5 j510

ACTIVITY 21-20j120A 4e B 26 j15

0C 2/ 150 0D 2/ 30

*A x BFind E

C

ACTIVITY 21-2

0 oE 60 (angle of 240 ) 60/240

*o 0A 4/ 120 , A 4/ 120

Convert All to Polar Form

* 0 0

0

A B 4/120 30/ 30E

C 2/ 150

oB 26 j15 30 / 30

ACTIVITY 21-2oE 60/240

60 Real

Imag

030

30

52

E 30 j52

ACTIVITY 21-2

3 oF D E 55 25 j60 65 /112.6

3F D E 55

Convert All to Rectangular Form

3 0 3 0D (2/ 30 ) 8/ 90 j8

0E 60/240 30 j52

55 55 j0

ACTIVITY 21-2

Real

Imag

25

j60

oF 25 j60 65 /112.6

F

067.4

65

K’S LAWS FOR PHASORS

• Circuit in AC Steady State:

• Can Express All v’s and i’s in Circuit as Phasors:

Input x(t) X cos( t )

Express x(t) as a Phasor X X /

1 1 2 2

1 1 2 2

v (t) V ; v (t) V

i (t) I ; i (t) I

etc.

K’S LAWS FOR PHASORS

• KCL:• If i1 + i2 = i; => I1 + I2 = I

• KVL:• If v1 + v2 = v; => V1 + V2 = V

• K’s Laws Work for Phasors!• Complex Addition, not Simple Addition

• Will do Activity 21-3 Tomorrow

• Will do Computer Project #3 Today

COMPUTER PROJECT 3

inv

1R2R

50

10 nF

10 nF

x2 v

outv

xv

1

2

R 10,000 x b

R 10,000/b

inv Pulse; 1V; 1msec

Op Amp Circuit

outFind v (t)

0 t 3ms

b .25, .5, 1.0, 1.5

COMPUTER PROJECT 3

inv (t)

Volts

t (ms)

1

1 1.01

Piece-Wise Linear Model

1m,1

1.01m,0

COMPUTER PROJECT 3

• Transient Simulation of 2nd Order Ckt:

• Model of an Op Amp Ckt:• Real Messy to do this Analytically!• Input = Rectangular Pulse; 1 V, 1 msec

duration• Use a Piece-Wise Linear model of Input

• R1 = 10,000 x b; R2 = 10,000 / b

• b = .25, .5, 1.0, 2.5• Transient Analysis for 0 < t < 3 msec• Identify Over, Under and Critical Damping

COMPUTER PROJECT 3

COMPUTER PROJECT 3

• Piece-Wise Linear Model – Schematics:• Use Source Named VPWL• Doubleclick on VPWL• Set T1, V1; T2, V2, T3, V3, etc

PSPICE TRANSIENTS

• Transient Analysis - Schematics:• Click on Setup Analysis• Check Transient• Doubleclick on Transient • Set Print Step = Final Time• Set Final Time• Save• Click on Simulate

PSPICE TRANSIENTS

• Transient Analysis - Schematics:• Also Need to Set Initial Conditions on L

and C• Create Circuit using L and c• Doubleclick on L and c • Set Value• Set Initial Conditions (Amps, Volts)

PSPICE TRANSIENTS

• Transient Analysis – Circuit File:• .tran Statement

• .tran tf tf uic

• Start time for analysis is t = 0, End time = tf

• uic = use initial conditions

PSPICE TRANSIENTS

• Transient Analysis with PSpice:• Need to put in initial conditions for L, C• C1 2 3 4u IC = 4Capacitor C1 connected between nodes

2 and 3, value = 4 uF, vC (0+) = 4 Volts

COMPUTER PROJECT 3

• Piece-Wise Linear Model – Circuit File:• Vin 1 0 pwl 1m,1 1.01m,0

• Resistors:• R1 1 2 {10000*b}• R2 2 3 {10000/b}

• Initial Conditions:• IC = 0 for both C’s

COMPUTER PROJECT 3• Circuit FileComputer Project 3 Spring 2003

vin 0 1 pwl 1m,1 1.01m,0

R1 1 2 {10000*b}

R2 2 3 {10000/b}

C25 2 5 10n ic=0

R34 3 4 50

C40 4 0 10n ic=0

E50 5 0 (3,0) 2

.param b=.25

.step param b list .25 .5 1 2.5

.tran 3m 3m uic

.probe

.end