Lecture 04:Lecture 04: AC SERIES/AC SERIES/PARALLEL PARALLEL CIRCUITS, CIRCUITS, VOLTAGE/VOLTAGE/CURRENT CURRENT DIVIDERDIVIDER

OBJECTIVESOBJECTIVES1.1. Compute the total circuit impedance for Compute the total circuit impedance for

series RLC circuits.series RLC circuits.

2.2. Explain the effects of frequency changes Explain the effects of frequency changes on RLC circuit response.on RLC circuit response.

3.3. Determine circuit voltages and currents Determine circuit voltages and currents in series RLC circuits.in series RLC circuits.

4.4. Compute voltage drops across Compute voltage drops across components in an RLC circuit using the components in an RLC circuit using the voltage divider formula.voltage divider formula.

Example #1: Example #1: Parallel CircuitsParallel Circuits

V S

50V<0 o XC=150 X L=100

R=100

iS

iC

iRL

(a) Find total impedance(a) Find total impedance• The circuit given is The circuit given is

already in freq already in freq domain:domain:

100 0

150 90

100 90

oR

oC

oL

Z

Z

Z

100

150

100

j

j

V S

50V<0 o XC=150 X L=100

R=100

iS

iC

iRL

Circuit simplificationCircuit simplification

60180

18190

57.268.111

4521213

50100

4521213

100100150

4542.14190150

4542.14190150

10010090150

9010010090150

j

j

jj

j

ZZZZZZ RLCLRCtotal

(b) Draw Impedance (b) Draw Impedance TriangleTriangle

190 18 (180 60)o j Z

ZZTTXXCC

RR

ZZTT

j

- j

RR

(c) Find is, i(c) Find is, iCC, i, iRLRL

mAZ

vi

mAZ

vi

RL

SRL

C

SC

453534542.141

050

9033390150

050

mAZ

vi

T

ss

1826318190

050

Vvvvv LRCS 050

(d) Using Current Divider(d) Using Current Divider

mA

mAiZ

Zi S

C

Tc

90333

1826390150

18190

mA

mAiZ

Zi S

RL

TRL

45353

1826390100100

18190

Example #2: Find i and Example #2: Find i and vvcc

Elements in Frequency Elements in Frequency domaindomain

=4 rad/s

Time domainTime domain Freq domainFreq domain

tvs 4cos10

5R

FC 1.0 5.2)1

(X jC

jC

5R

010Vs

Circuit in Freq Circuit in Freq DomainDomain

(a) The current, i(a) The current, i• The current in series circuit,The current in series circuit,

• The total impedance is;The total impedance is;

T

s

Z

VI

56.2659.55.25

5.25ZT

j

j

Aj 8.06.1

56.2679.1

56.2659.5

010

Z

VI

T

s

(b) The capacitor voltage, (b) The capacitor voltage, VVCC

V

j

44.6348.4

905.256.2679.1

5.256.2679.1

IZV CC

Example #3: Find iExample #3: Find iXX (time domain)(time domain)

63.522000cos4607.0)( ttiX

Solution:Solution:

Example #4: Find ZExample #4: Find Z1 1 and and ZZ22

SolutionSolution::

896.8535.1

213

2

1

jZ

jZ

Example #5: Find i(t)Example #5: Find i(t)

Different waveform of Different waveform of VVSS

VVS1S1 and and VVS2S2 are different: are different:

When analyzing ac circuit, it is expedient to When analyzing ac circuit, it is expedient to express both signals as either Sine or express both signals as either Sine or Cosine.Cosine.

904cos201 tvs

154sin102 tvs

Change the waveform Change the waveform of Vof VS1S1

• Using identity:Using identity:

180sinsin

180coscos

90cossin

90sincos

tt

tt

tt

tt

Changing Cosine to Sine:Changing Cosine to Sine:

tv

tv

tv

s

s

s

4sin20

90904sin20

904cos20

1

1

1

Example #6: (a) Find Example #6: (a) Find i(t)i(t)

Different waveform of Different waveform of VVSS

VVS1S1 and and VVS2S2 are different: are different:

When analyzing ac circuit, it is expedient to When analyzing ac circuit, it is expedient to express both signals as either Sine or express both signals as either Sine or Cosine.Cosine.

904cos201 tvs

154sin102 tvs

Change the waveform of Change the waveform of VVS1S1

• Using identity:Using identity:

180sinsin

180coscos

90cossin

90sincos

tt

tt

tt

tt

Redraw the circuitRedraw the circuit

Elements in Frequency Elements in Frequency domaindomain

=4 rad/s=4 rad/s

Time domainTime domain Freq domainFreq domain

154sin102 tvs

60R

mFC 10 25)1

(X jC

jC

60R

1510V 2s

HL 5

tvs 4sin201 020V 1s

20X jLjL

Find The CurrentFind The Current

321 ZZZZT 21 ssT vvV

T

T

Z

VI

Solution:Solution:

31.5617.1089060 jZT

VjVT 99.477.2959.266.29

Atti

AI

3.614sin28.0)(

3.6128.0

(b) Find the v(b) Find the vcc(t), v(t), vLL(t)(t)

Vttvtv

VVV

LC

LC

7.284sin28)()(

7.2828

:Answer

(c) Find the i(c) Find the icc(t), i(t), iLL(t)(t)

Atti

Atti

AI

AI

L

C

L

C

3.614sin4.1)(

7.1184sin12.1)(

3.614.1

7.11812.1

:Answer

(d) Sketch the waveform for v(d) Sketch the waveform for vcc(t), (t), vvLL(t), i(t), icc(t), i(t), iLL(t)(t)

(e) Explain the phase relationship (e) Explain the phase relationship between V and I for capacitor and between V and I for capacitor and inductor.inductor.

Example #7: Find ZExample #7: Find ZTT, I, ITT, , VVC2C2, P, P

kR

kX

kR

kX

kR

VV

C

C

S

11

10

8

12

10

010

3

2

2

1

1R 1

R 2

R 3

X c1

X c2

V s Z2

• Find the total impedance:Find the total impedance:

• Then, the total current is:Then, the total current is:

SolutionSolution

4.427.21

6.2356.6

211

2232

kZXRZ

kXRRZ

CT

C

4.42461

4.427.21

010A

k

V

Z

VI

T

ST

SolutionSolution

8.1936.28.1803.3108

9010

8.1803.30104.427.21

6.2356.6

322

22

2

2

322

2

VVkjk

kV

Z

ZV

VVk

kV

Z

ZV

RCRCR

CC

ST

RCR

P = VI cos (T) = (10V)(461 A) cos(-42.4o) = 3.40 mW

Ex: Find ZEx: Find ZTT, I, IL3L3, P , P

R 1

R 2

R 3

X L1

X L2

E

R 4

X L3

kR

kX

kR

kX

kR

kX

kR

VV

L

L

L

S

8

3

7

2

6

4

5

010

4

3

3

2

2

1

1

Z3Z2

ZZ22, Z, Z33, Z, ZTT, and I, and ITT

4.3187.9

2.1841.8

38.1861.3

3

2

kZ

kZ

kZ

T

4.3101.1 mAZ

VI

T

ST

IIR3R3, I, IL3L3, P, P

8.513.4072.31435903

4.6981.2

2.314354.3101.12.1841.8

4.1861.3

3

3

43

3

3

3

2

AAk

kI

Z

ZI

AmAk

kI

Z

ZI

RL

RLL

TR

P = EI cos (T) = (10V)(1.01 mA) cos(31.4o) = 8.62 mW

TOOLSTOOLS• Note from the previous examples, all Note from the previous examples, all

of our tools from DC work:of our tools from DC work:– Ohm’s LawOhm’s Law– Voltage DividerVoltage Divider– Current DividerCurrent Divider– Kirchoff’s Current LawKirchoff’s Current Law– Kirchoff’s Voltage LawKirchoff’s Voltage Law

• They are the same – but we use They are the same – but we use COMPLEX NUMBERSCOMPLEX NUMBERS..