lecture 03: ac response ( reactance n impedance )

Lecture 03:AC RESPONSE( REACTANCE N

IMPEDANCE )

OBJECTIVES Explain the relationship between AC voltage and AC

current in a resistor, capacitor and inductor. Explain why a capacitor causes a phase shift between

current and voltage (ICE). Define capacitive reactance. Explain the relationship

between capacitive reactance and frequency. Explain why an inductor causes a phase shift between

the voltage and current (ELI). Define inductive reactance. Explain the relationship

between inductive reactance and frequency. Explain the effects of extremely high and low

frequencies on capacitors and inductors.

AC RESISTOR

AC V AND I IN A RESISTORAC V AND I IN A RESISTOROhm’s Law still applies even though the

voltage source is AC.The current is equal to the AC voltage

across the resistor divided by the resistor value.

Note: There is no phase shift between V and I in a resistor.

( )( ) R

R

v ti t

R

vR(t)

( )( ) R

R

v ti t

R

AC V AND I IN A RESISTORAC V AND I IN A RESISTOR

PHASE PHASE ANGLE ANGLE FOR R, FOR R, =0=0

AC CAPACITOR

CURRENT THROUGH A CAPACITORCURRENT THROUGH A CAPACITOR

The faster the voltage changes, the larger the current.

dt

dvCi c

c

The phase relationship between “V” and “I” is established by looking at the flow of current through the capacitor vs. the voltage across the capacitor.

PHASE RELATIONSHIP

Graph vC(t) and iC(t)

90°90°

vvcc(t)(t)

iicc (t)(t)

NoteNote: Phase : Phase relationship relationship of I and V in of I and V in a capacitora capacitor

dt

dvCi c

c

In the Capacitor (C), Voltage LAGS charging current by 90o or Charging Current (I) LEADS Voltage (E) by 90o

I. C. E. V C

IC

90

PHASE RELATIONSHIP

CAPACITIVE REACTANCE

In resistor, the Ohm’s Law is V=IR, where R is the opposition to current.

We will define Capacitive Reactance, XC, as the opposition to current in a capacitor.

CX IV

CAPACITIVE REACTANCE

XC will have units of Ohms.

Note inverse proportionality to f and C.

1 1

2CX

fC C

Magnitude of XC

Ex.Ex.

Ex: f = 500 Hz, C = 50 µF, XC = ?

V S

C 1

Capacitive reactance also has a phase angle associated with it.

Phasors and ICE are used to find the angle

PHASE ANGLE FOR XPHASE ANGLE FOR XCC

IV

XC

PHASE ANGLE FOR XPHASE ANGLE FOR XCC

If If V is our reference wave: is our reference wave:

90

900 _Z

IV

CX

I.C.E

AC INDUCTOR

The phase angle for Capacitive Reactance (XC) will always = -90°

XC may be expressed in POLAR or RECTANGULAR form.

ALWAYS take into account the phase angle between current and voltage when calculating XC

90_CX CjXor

VOLTAGE ACROSS AN INDUCTORVOLTAGE ACROSS AN INDUCTOR

Current must be changing in order to create the magnetic field and induce a changing voltage.

The Phase relationship between VL and IL (thus the reactance) is established by looking at the current through vs the voltage across the inductor.

dtdi

Lvind

Graph vL(t) and iL(t)

Note the phase relationship

vvLL(t)(t)

iL(t)90°90°

In the Inductor (L), Induced Voltage LEADS current by 90o or Current (I) LAGS Induced Voltage (E) by 90o.

E. L. I. VC

IC90

INDUCTIVE REACTANCE

We will define Inductive Reactance, XL, as the opposition to current in an inductor.

LX IV

INDUCTIVE REACTANCE

XL will have units of Ohms ().

Note direct proportionality to f and L.

2LX fL L

Magnitude of XL

Ex1.

f = 500 Hz, L = 500 mH, XL = ?

V S

L

PHASE ANGLE FOR XPHASE ANGLE FOR XLL

If If V is our reference wave: is our reference wave:

90

90

0L Z

I

VX

E.L.I

The phase angle for Inductive Reactance (XL) will always = +90°

XL may be expressed in POLAR or RECTANGULAR form.

ALWAYS take into account the phase angle between current and voltage when calculating XL

90LXLjXor

COMPARISON OF XL & XC

XL is directly proportional to frequency and inductance.

XC is inversely proportional to frequency and capacitance.

2LX fL L

1 1

2CXfC C

SUMMARY OF V-I RELATIONSHIPS

ELEMENT TIME DOMAIN FREQ DOMAIN

RiV

dt

diLV

dt

dvCi

RIV

IjV L

Cj

Cj II

V

R

L

C

Extreme Frequency effects on Capacitors and Inductors

Using the reactances of an inductor and a capacitor you can show the effects of low and high frequencies on them.

2LX fL L

1 1

2CX

fC C

Frequency effects

At low freqs (f=0): an inductor acts like a short circuit. a capacitor acts like an open circuit.

At high freqs (f=∞): an inductor acts like an open circuit. a capacitor acts like a short circuit.

Ex2.

Represent the below circuit in freq domain;

REVIEW QUIZ- What is the keyword use to remember the

relationships between AC voltage and AC current in a capacitor and inductor

- .- What is the equation for capacitive reactance?

Inductive reactance?

- T/F A capacitor at high frequencies acts like a short circuit.

- T/F An inductor at low frequencies acts like an

open circuit.

IMPEDANCE

IMPEDANCEThe V-I relations for three passive elements;

The ratio of the phasor voltage to the phasor current:

CjjR

IV LI,V I,V

CjjR

1

I

V L,

I

V ,

I

V

From that, we obtain Ohm’s law in phasor form for any type of element as:

Where Z is a frequency dependent quantity known as IMPEDANCE, measured in ohms.

IZVor I

VZ

IMPEDANCE

Impedance is a complex quantity:

R = Real part of Z = Resistance

X = Imaginary part of Z = Reactance

jXRZ

Impedance in polar form:

where;

θZjXRZ

R

XXR 122 tan,Z

θsin ZX θ, cosZR

IMPEDANCES SUMMARY

Impedance Phasor form: Rectangular form

ZR R+j0

ZL 0+jXL

ZC 0-jXC

90oLX

90oCX

0oR

ADMITTANCE

ADMITTANCE

The reciprocal of impedance.Symbol is Y Measured in siemens (S)

V

I

Z

1Y

ADMITTANCE

Admittance is a complex quantity:

G = Real part of Y = Conductance

B = Imaginary part of Y = Susceptance

jBGY

Z AND Y OF PASSIVE ELEMENTS

ELEMENT IMPEDANCE ADMITTANCE

RZ

Lj

Cj1

Z

R

1Y

Lj1

Y

CjY

R

L

C

TOTAL IMPEDANCE FOR AC CIRCUITS

To compute total circuit impedance in AC circuits, use the same techniques as in DC. The only difference is that instead of using resistors, you now have to use complex impedance, Z.

TOTAL IMPEDANCE FOR PARALLEL CIRCUIT

1 2

1 1

1 2

1 1 1 1 1

1 1 1 1

total x x

totalx x

Z Z Z Z Z

ZZ Z Z Z

As a conclusion, in parallel circuit, the impedance can be easily computed from the admittance:

xtotal

totaltotal

total

YYYY

YY

Z

...

1

21

1

Ex3: SERIES CIRCUIT

R=20Ω

L = 0.2 mH

C = 0.25μF

V6010sin10V 5s t