Filter & Transfer Function

21

2

ZZ

Z

V

VH

in

outV

Z1

Z2VinVout

To determine filter type, look how HV as a function of frequency

Z1 and Z2 could include multiple components

RLC at Resonance

CL ZZ

CjLj

CL

1

1

when 0

R

V

CLR

VI

CLjR

V

ZZZ

jVjI

CLR

22 1

1

)(

At resonance:• current is max.• Zeq =R• current and voltage are in phase.• the higher Q, the narrower the resonant peak.

•The average ac power (Pav) is the power dissipated on the load resistor. • 0cos1, dependent on the complex load.• ideal power factor: cos=1, Z=R, pure resistive load

Exam II Review: AC Power

RI

ZI

VItP

2

2

~

cos~

cos~~

)(

tIti

tVtv

p

p

cos)(

cos)(

rms value

Average power

AC Power: instantaneous and average power

tVtv p cos)(

sign thedetermine to thisuse ;tan

cos

22

R

X

Z

R

Z

Z

Xj

Z

RXR

jXRZ

tIti

IZ

VZVI

p

pp

cos)(

/

Complex Power

II

VV

~~

0~~

jQP

XXIjP

Z

XIVjP

IVjIV

IVIVS

av

CLav

av

)(~

~~

sin~~

cos~~

~~*

~~

2

• real power Pav(unit: watts): power absorbed by the load resistance.• reactive power Q (unit: volt-amperes-reactive, VAR): exchange of energy between the source and the reactive part of the load. No net power is gained or lost during the process. Q=QL-QC, if Q<0, the load is capacitive, Q>0, the load is inductive• Apparent power: |S|=|Vrms||Irms| (unit: volt-amperes, VA): computed by measuring the rms load voltage and currents without regard for the phase angle.

Example 7.8, P345: power factor correction

sec/377)(602 ;0117~

radHzVVS

))(047.1(17.1047.1100

0117~

~

047.11007.8650

AZ

VI

jZ

LL

L

)(5.1184.68

047.117.10117*~~

Wj

IVS LL

FX

C

Q

VX

C

C

L

C

1.23115377

11

1155.118

117~

22

Transformer

Transformation of Voltage:

Transformation of Resistance:

Residential Wiring

GFCI

An electrical engineering major has designed and constructed a motor for your sophomore design project. It has been wound with an inductance of 16 mH and a resistance of 3.0 ohm. You had specified that it be optimized for the best possible power factor at 60 Hz; consequently, a 354 microFarad capacitor was added in parallel with the motor, as shown below.

a. Operated at 120 V, what is the current through the motor when operated at 60 Hz? b. What power is delivered by the voltage source? c. What power is dissipated by the motor? Your complete project was so innovative that your engineering professor entered it in an international competition. You arrive in Germany for the competition before remembering that the line frequency there is 50 Hz. d. What is the power factor of this circuit at 50 Hz? e. What capacitance do you need to replace 354 mF capacitor in order to correct the power factor back to unity?

Example: Midterm 1998

Example Cont.

))(4.63(9.174.637.6

0120~

~

4.637.6636023

AZ

VI

jLjZ

M

M

WRI

IRIIVPP

M

MMRRRs

96039.17~

~~~~1cos Since

22

**

a. Operated at 120 V, what is the current through the motor when operated at 60 Hz?

b. What power is delivered by the voltage source?

d. What is the power factor of this circuit at 50 Hz?

93.022cos

22sin214822cos2148222148~~

))(22(9.17225.10

0120~

~

93.022cos

225.10535

317.52

43

909598.5

90991.1

1

502

1

598.5535023

*

pf

jIVS

AZ

VI

elyAlternativ

pf

jZ

jjCj

Z

jLjZ

LL

LL

C

M

e. What capacitance do you need to correct the power factor back to unity?

FC

CjZ

VVIVj

j

j

IVS

AZ

VI

jLjZ

C

CCCC

MM

M

MM

M

470

5021

120~~

~~2129

21291279

59sin248459cos2484

597.200120~~

))(59(7.20598.5

0120~

~

598.5535023

2

*

**

*

Note: you calculate S only on the motor part!

Negative just means Capacitor

Example 3: 1st Fall 199An electrical engineering firm has designed a motor for your sophomore design

project. You specified that it should be wound with an inductance of 16 mH and a resistance of 3.0  ohm. You had specified that it be optimized for the best possible power factor at 60 Hz; consequently, a 354 F capacitor was added in parallel with the motor. Unfortunately the motor was actually wound with an inductance of 61 mH and a resistance of 30 ohm because of typographical errors in the FAX to the motor manufacturer.

                                                                                                                         

              

a. What power is delivered by the voltage source? b. What power is dissipated by the motor? c. What capacitance do you need to replace 354 mF capacitor in order to correct

the power factor back to unity?

Example Cont.

VARjWIVS

AZ

VI

LL

M

MM

2323025.3717.30120~~

)(5.3717.3)5.37(8.37

0120~

~

*

VARjWIVS

AZ

VI

jZ

jjCj

Z

jLjZ

LL

LL

C

M

16503065.79140120~~

)(5.7914)5.79(6.8

0120~

~

)5.79(6.8277.33

)5.52(291

3.1530

)90(7.75.378.37

)90(7.77.713.0

1

602

1

5.378.37233060230

*

a. What power is delivered by the voltage source?

b. What power is dissipated by the motor?

d. What capacitance do you need to correct the power factor back to unity?

FC

CjZ

VVIVj

C

CCCC

43

6021

120~~

~~230

2

*

**