ENE 104 Electric Circuit Theory - KMUTTwebstaff.kmutt.ac.th/~dejwoot.kha/CalenderYear/Year2554/ene104/... · ENE 104 Electric Circuit Theory ... ( 5 90 10 ) cos( 5 10 ) 1 1 1 1 q

ENE 104

Electric Circuit Theory

Lecture 09:

Sinusoidal Steady-State Analysis

http://webstaff.kmutt.ac.th/~dejwoot.kha/

(ENE) Mon, 12 Mar 2012 / (EIE) Wed, 21 Mar 2012

Week #10 : Dejwoot KHAWPARISUTH



Objectives : Ch10

Week #10

Page 2

ENE 104

• sinusoidal functions

• impedance and admittance

• use phasors to determine the forced response of

a circuit subjected to sinusoidal excitation

• Apply techniques using phasors

Sinusoidal Steady-State Analysis

Introduction: Page 3

The sinusoidal function v(t) = Vm sin wt is plotted

(a) versus wt and (b) versus t.

Characteristics of Sinusoids

tVtv m wsin)(

Tf

1 w 2T fw 2

Week #10 ENE 104 Sinusoidal Steady-State Analysis

Lagging and leading: Page 4

The sine wave Vm sin (wt + q) leads Vm sin wt by q rad.



Converting Sines to Cosines

)1005sin(

)10905sin(

)105cos(

1

1

11

+

++

+

tV

tV

tVv

m

m

m

)305sin(22 tVv m


In electrical engineering, the phase angle is commonly given in degrees, rather than radian.



Two sinusoidal waves whose phase are to be

compared must:

1. Both be written as sine waves, or both as

cosine waves.

2. Both be written with positive amplitudes.

3. Each have the same frequency.

Normally, the difference in phase between two

sinusoids is expressed by that angle which is

less than or equal to 180 degree in magnitude

Practice: 10.1 Page 7


Find the angle by which 𝑖1 lags 𝑣1 if

𝑣1 = 120cos(120𝜋𝑡 − 40𝑜) V and 𝑖1 equals

(a) 2.5 cos 120𝜋𝑡 + 20𝑜 A; (b) 1.4 𝑠𝑖𝑛 120𝜋𝑡 − 70𝑜 A;

(c) Acos 100𝑡 + 𝐵𝑠𝑖𝑛100𝑡 = 𝐶𝑐𝑜𝑠(100𝑡 +𝜙).



Find A, B, C, and 𝜙 if 40 cos 100𝑡 − 40° −20 sin 100𝑡 + 170° = 𝐴𝑐𝑜𝑠100𝑡 + 𝐵𝑠𝑖𝑛100𝑡 =𝐶𝑐𝑜𝑠(100𝑡 + 𝜙).



Forces Response: Page 10

Forces Response to sinusoidal Functions:

The Steady-State Response

tVRidt

diL m wcos+



tVRidt

diL m wcos+






Example: Page 14

Find the current Li


Example: Page 15

Find the current Li

) )

) .3.5610cos222

20

103010tan10cos

10301020

8

tancos)(

3

3313

2332

1

222

mAt

t

R

Lt

LR

Vti m

+

+

ww

w




Let 𝑣𝑠 = 40𝑐𝑜𝑠8000𝑡 V in the circuit. Use Thevenin’s

theorem where it will do the most good, and find the

value at t = 0 for (a) 𝑖𝐿; (b) 𝑣𝐿; (c) 𝑖𝑅; (d) 𝑖𝑠





The Complex Forcing Fn: Page 19
















Evaluate and express the result in rectangular form:

(a) 2∠30° 5∠ − 110° 1 + 2𝑗 ; (b) 5∠ − 200° +4∠20°. Evaluate and express the result in polar form: (c)

(2 − 𝑗7)/(3 − 𝑗); (d) 8 − 𝑗4 + 5∠80° / 2∠20° .





If the use of the passive sign convention is

specified, find the (a) complex voltage that

results when the complex current 4𝑒𝑗800𝑡 A is

applied to the series combination of a 1-mF

capacitor and a 2-Ω resistor; (b) complex current

that results when the complex voltage100𝑒𝑗2000𝑡 V is applied to the parallel combination of a 1-mH

inductor and a 50-Ω resistor.





The Phasor: Page 31

A phasor transformation:

0)cos()( mm VtVtv w

w + mm ItIti )cos()(




Transform each of the following functions of time

into phasor form: (a) −5sin(580𝑡 − 110°); (b)

3cos600t − 5sin(600𝑡 + 110°); (c)8𝑐𝑜 𝑠 4𝑡 − 30° +4sin(4𝑡 − 100°). Hint: First convert each into a single cosine function

with a positive magnitude







Let 𝜔 = 2000 rad/s and t = 1 ms. Find the

instantaneous value of each of the currents given

here in phasor form: (a) 𝑗10 A; (b) 20 + 𝑗10 A; (c)

20 + 𝑗(10∠20°) A.



Phasor Relationship: Page 37







The Capacitor: dt

dvCti )(






Phasor: Page 43


Phasor: Page 44




In the circuit, let 𝜔 = 1200 rad/s, 𝐈𝐶 = 1.2∠28° A,

and 𝐈𝐿 = 3∠53° A. Find (a) 𝐈𝒔; (b) 𝐕𝒔; (c) 𝑖𝑅(𝑡).



Impedance: Z Page 47

A resistance

A reactance




With reference to the network shown, find the input

impedance 𝐙𝑖𝑛 that would be measured between

terminals: (a) a and g; (b) b and g; (c) a and b



Example: find i(t) Page 50

9040)903000cos(40)3000sin(40 tt

kj

jLj

13

13000w

kj

j

Cj

26

13000

1

w


Example: Page 51

+

+

+

kj

jj

jjZtotal

87.365.25.12

21

)21()(5.1

.9.12616.87.365.2

9040mAmA

Z

VI

total

s

.)9.1263000cos(16 mAt




In the frequency-domain circuit, find (a) 𝐈1; (b) 𝐈2;

(c) 𝐈3



Admittance: Y Page 54

G = the conductance

B = the susceptance

R = the resistance

X = the reactance … immittance



Determine the admittance (in rectangular form)

of (a) an impedance 𝐙 = 1000 + 𝑗400 ; (b) a

network consisting of the parallel combination of

an 800- resistor, a 1-mH inductor, and a 2-nF

capacitor, if 𝜔 = 1 Mrad/s; (c) a network

consisting of the series combination of an 800-

resistor, a 1-mH inductor, and a 2-nF capacitor, if

𝜔 = 1 Mrad/s




Nodal and Mesh Analysis: Page 57

Example: find v1(t) and v2(t)

0101105105

212111 jj

VV

j

VV

j

VV+

+

+

+

5.0)05.0(105105

221212 jV

j

V

j

VV

j

VV++

+




Use nodal analysis on the circuit to find 𝐕1 and 𝐕2.



Nodal and Mesh Analysis: Page 60

Example: find I1 and I2

09015)(3)(5010 211 +++ IIIj

0020)(4)(39015 212 ++ IjII


'

1V '

2V

Superposition: Page 61

Example: find V1

''

1V''

2V


'

1V '

2V


Example: find V1

5561.01951.0

2540

)25()40(50

20'

1

j

j

jj

Y

IV

+



Example: find V1

''

1V''

2V

I

mj

IV

50

''

1

)m

mjmjm

mjI 9050

25

1

50

1

40

1

25

1

+

+




For the circuit, find the (a) open-circuit voltage 𝐕𝑎𝑏;

(b) downward current in a short circuit between a

and b; (c) Thevenin-equivalent impedance 𝐙𝑎𝑏 in

parallel with the current source.




Determine the current 𝑖 through the 4- resistor




Phasor Diagram: Page 68

1.5310861 + jV



(a) A phasor diagram showing the

sum of V1 = 6 + j8 V and V2 = 3

– j4 V, V1 + V2 = 9 + j4 V =

9.8524.0o V.

(b) (b) The phasor diagram shows

V1 and I1, where I1 = YV1 and Y

= 1 + j S = 1.445o S. The

current and voltage amplitude

scales are different.








Example: Page 73

Select V =

; IR =

IL = IC =

01

01)2.0(

901.001)1.0( j

903.001)3.0( j


Example: Page 74




Select some convenient reference value for 𝐈𝐶 in

the circuit, draw a phasor diagram showing 𝐕𝑅,

𝐕2, 𝐕1, and 𝐕𝑠, and measure the ratio of the

lengths of (a) 𝐕𝑠 to 𝐕1; (b) 𝐕1 to 𝐕2; (c) 𝐕𝑠 to 𝐕R



Ex: Page 77

31.25 nF

2 kΩVg

500 mHVO

+

-

The circuit is operating in the sinusoidal steady

state. Find the steady-state expression for vo(t) if vg(t) = 64 cos 8000t V.


Ex: Page 78


Hw: Page 79


Reference: Page 80

W.H. Hayt, Jr., J.E. Kemmerly, S.M. Durbin, Engineering Circuit Analysis, Sixth Edition.

Copyright ©2002 McGraw-Hill. All rights reserved.

Circuit Analysis and Electrical Engineering

Documents

ENE 104 Electric Circuit Theory - KMUTTwebstaff.kmutt.ac.th/~dejwoot.kha/CalenderYear/Year2554/ene104/... · ENE 104 Electric Circuit Theory ... ( 5 90 10 ) cos( 5 10 ) 1 1 1 1 q