ac circuits chapter 23. ac circuits capacitive reactance phasor diagrams inductive reactance rcl...

AC Circuits

Chapter 23

AC Circuits

Capacitive Reactance Phasor Diagrams Inductive Reactance RCL Circuits Resonance

Resistive Loads in AC Circuits

Ohm’s Law:

R is constant – does not depend on frequency No phase difference between V and I

ftIftR

V

R

VI

ftVVR

VI

R

VI

tt

t

rmsrms

2sin2sin

2sin

00

0

Capacitive Reactance

At the moment a capacitor is connected to a voltage source:

Current is at its maximum Voltage across capacitor is zero

+

-V C

I 0

V = 0

+

-

Capacitive Reactance

After a long time, the capacitor is charged:

Current is zero Voltage is at its maximum (= supply voltage)

V C V

+

-

+

-

Capacitive Reactance

Now, we reverse the polarity of the applied voltage:

Current is at its maximum (but reversed) Voltage hasn’t changed yet

V C V

+

-+

- +

-

I

Capacitive Reactance

Time passes; the capacitor becomes fully charged:

Current is zero Voltage has reversed to match the applied polarity

V C V

+

-

+

-

+

-

Capacitive Reactance

Apply an AC voltage source:

an AC current is present in the circuit a 90° phase difference is found between the voltage

and the current

V(t) = V0sin(2f t)

C+

-

I (t) = I 0sin(2f t+/ 2)

Capacitive Reactance

We want to find a relationship between the voltage and the current that we can use like Ohm’s Law for an AC circuit with a capacitive load:

We call XC the capacitive reactance, and calculate it as:

units of capacitive reactance: ohms ()

Crmsrms XIV

fCXC 2

1

Capacitive Reactance

A particular example:

2.12 F 107.5Hz 1002

1

2

14-fC

XC

V0 = 50 V

+

- C = 750 mFf = 100 Hz

Capacitive Reactancevoltage vs. time

-50

-40

-30

-20

-10

0

10

20

30

40

50

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

time, s

volt

age,

V

ftVVt 2sin0

Capacitive ReactanceVoltage and Current vs. Time

-50

-40

-30

-20

-10

0

10

20

30

40

50

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

time, s

volt

age,

cu

rren

t (V

, A

)

voltage

current

22sin

2sin

0

0

ftII

ftVV

t

t

Capacitive ReactanceVoltage, Current, and Power vs Time

-60

-40

-20

0

20

40

60

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

time, s

volt

age,

cu

rren

t, p

ow

er (

V,

A,

10W

)

voltage

current

power

ttt

t

t

IVP

ftII

ftVV

22sin

2sin

0

0

Capacitive Reactance

Power is zero each time either the voltage or current is zero

Power is positive whenever V and I have the same sign

Power is negative whenever V and I have opposite signs

Power spends equal amounts of time being negative and positive

Average power over time: zero

Capacitive Reactance

The larger the capacitance, the smaller the capacitive reactance

As frequency increases, reactance decreases DC: capacitor is an “open circuit” and

high frequency: capacitor is a “short circuit”

and

CX

0CX

fCXC 2

1

Phasor Diagrams

Consider a vector which rotates counterclockwise with an angular speed :

This vector is called

a “phasor.” It is a

visualization tool.

f 2

t=2f t

V0V0 sin(2f t)

V

Phasor Diagrams

For a resistive load: the current is always proportional to the voltage

t=2f t

V0V0 sin(2f t)

V, I

I 0 sin(2f t)

I 0 = V0 / R

voltage phasor

current phasor

Phasor Diagrams

For a capacitive load: the current “leads” the voltage by /2 (or 90°)

t=2f t

V0 sin(2f t)

V, I

I 0 sin(2f t + / 2)

voltage phasor

current phasor

Inductive Reactance

A coil or inductor also acts as a reactive load in an AC circuit.

AC

Inductive Reactance

For a coil with a self-inductance L:

AC

t

ILEMF

Inductive Reactance

As the current increases through zero, its time rate of change is a maximum – and so is the induced EMF

AC

t

ILEMF

Inductive Reactance

As the current reaches its maximum value, its rate of change decreases to zero – and so does the induced EMF

AC

t

ILEMF

Inductive Reactance

The voltage “leads” the current in the inductor by /2 (or 90°)

AC

t

ILEMF

Inductive Reactance

The inductive reactance is the Ohm’s Law constant of proportionality:

units of inductive reactance:

ohms ()

AC

fLX

XIV

L

Lrmsrms

2

Inductive Reactance

The voltage-current relationship in an inductive load in an AC circuit can be represented by a phasor diagram:

t=2f t

V0 sin(2f t)

V, I

I 0 sin(2f t - / 2)

voltage phasor

current phasor

Inductive Reactance

Mnemonic for remembering what leads what:

“ELI the ICEman”

EMF (voltage)

inductor (L)

current (I)

current (I)

EMF (voltage)

capacitor (C)

Inductive Reactance

Larger inductance: larger reactance (more induced EMF to oppose the applied AC voltage)

Higher frequency: larger impedance (higher frequency means higher time rate of change of current, which means more induced EMF to oppose the applied AC voltage)

fLX L 2

RCL Circuit

Here is an AC circuit containing series-connected resistive, capacitive, and inductive loads:

The voltages across the loads at any instant are different, but a common current is present.

AC

R

C

L

RCL Circuit

The current is in phase with voltage in the resistor.

The capacitor voltage trails the current; the inductor voltage leads it.

We want to calculate the entire applied voltage from the generator.

VR

VL

VC

I

RCL Circuit

We will add the voltage phasors as vectors (which is what they are.)

We start out by adding the reactive voltages (across the capacitor and the inductor).

This is easy because those phasors are opposite in direction. The resultant’s magnitude is the difference of the two, and its direction is that of the larger one.

VR

VL - VC I

RCL Circuit

Now we use Pythagoras’ Theorem to add the VL – VC phasor to the VR phasor.

VR

VL - VC I

V = VR2 + (VL - VC)

2

RCL Circuit

The current phasor is unaffected by our addition of the voltage phasors.

It now makes an angle with the overall applied voltage phasor.

I

V = VR2 + (VL - VC)

2

RCL Circuit

We can make Ohm’s Law substitutions for the voltages:

22

2222

22222

CL

CL

CLCLR

RLLCC

XXRIV

XXIRIV

IXIXRIVVVV

IRVIXVIXV

RCL Circuit

Our result:

suggests an Ohm’s Law relationship for the combined loads in the series RCL circuit:

Z is called the impedance of the RCL circuit.

SI units: ohms ()

22CL XXRIV

22

22

CL

CL

XXRZ

XXRIIZV

RCL Circuit -- Power

If the load is purely resistive, the average power dissipated is

We can use the phasor diagram to relate R to Z trigonometrically:

VR

VL - VC I

V = VR2 + (VL - VC)

2

RIP rms2

cos

cos

cos

22 ZIRIP

ZR

Z

R

ZI

RI

V

V

rmsrms

rms

rmsR

“power factor”

RCL Circuit -- Resonance

Series-connected inductor and capacitor:

C L

+

-

Capacitor is initially charged.

When connection is made, discharge current I flows through inductor. I nduced EMF opposes and limits discharge current.

I

RCL Circuit -- Resonance

C L

+

-

When capacitor is discharged, current I slows and stops.

Decrease of magnetic flux in inductor induces EMF that opposes the decrease (and continues the current I ).

I

RCL Circuit -- Resonance

C L+

-

I nduced current charges C with the opposite polarity to its original state. When the capacitor is charged, the current I is stopped.

RCL Circuit -- Resonance

C L+

-

Now the capacitor begins to discharge through the inductor again – this time in the opposite direction (new discharge current = -I ).

Opposite induced EMF across inductor again limits this new discharge current.

The cycle continues.

-I

RCL Circuit -- Resonance

Energy is alternately stored in the capacitor (in the form of the electrical potential energy of separated charges) and in the inductor (in its magnetic field). When the magnetic field collapses, it charges the capacitor; when the capacitor discharges, it builds the magnetic field in the inductor.

C L

RCL Circuit -- Resonance

This “LC oscillator” or “tuned tank circuit” oscillates at a natural or resonant frequency of

LCfres 2

1 C L

RCL Circuit -- Resonance

At the resonant frequency, how are the inductive and capacitive reactances related?

The reactances are equal to each other.

CresL

resC

res

XC

LC

LC

LCL

LC

L

LCLLfX

C

LC

LC

CCfX

LCf

2

122

2

21

2

1

2

1

RCL Circuit -- Resonance

At the resonant frequency, when the inductive and capacitive reactances are equal, what is the situation in the circuit?

VR

VL

VC

IVR

I

RCL Circuit -- Resonance

At the resonant frequency, when the inductive and capacitive reactances are equal, what is the impedance of the circuit?

At resonance, the circuit’s impedance is simply equal to its resistance, and its voltage and current are in phase.

If the resistance is small, the current may be quite large.

RRXXRZ CL 222