Inductors in AC Circuits

Inductors

• An inductor affects a circuit whenever current (I) is changing.

• The magnetic field generated by the inductor acts to induce an opposing current (Lenz’s Law).

• The ideal inductor stores energy in its magnetic field which is then returned to the circuit as electrical energy, the only energy loss is from the resistance of the circuit.

Inductors in AC

• In an AC circuit current is constantly changing so inductors play an important role

• The current opposing ability of inductors is called reactance and given the symbol XL

• Like XC the units are Ohms

Voltage and Current Phase Differences

• In a circuit composed only of an inductor and an AC power source, there is a 90° phase difference between the voltage and the current in the inductor.

• For an inductor the current lags the voltage by 90°, so it reaches its peak ¼ cycle after the voltage peaks.

Relationship between V and I

• Because the inductor acts to oppose the change in current, as current increases a clear relationship with voltage can be seen

Inductor Voltage/Current Graph

0

5

10

15

20

25

30

35

40

0 5 10 15

Current (mA)

Vo

lta

ge

(m

V)

LL IXV

I

VX LL

~

VL

A

6V AC

Examples

1. Find the inductor voltage of an AC circuit with a reactance of 2.4 and a current of 0.18A

0.43V2. An inductor has a voltage of 8.2V AC and a

reactance of 54. Calculate the current of the circuit.

0.15A3. Calculate the reactance of a circuit with an

inductor voltage of 16V and a current of 1.2A13

Factors Affecting Reactance (XL )

• Increasing the size of the inductor (L) will induce a higher opposing voltage and therefore increase XL

• Increasing frequency increases induced current (increasing reactance). This is because more frequent creation and collapse of magnetic field produces greater opposing current

• The reactance of a capacitor with a supply frequency f;

LX L

fX L

LXfLX LL or 2

Examples

1. A 0.5H inductor is connected to a 6V 50Hz AC supply.

a) Calculate the reactance of the inductor157

b) The RMS current in the circuit0.038A

2. What size inductor is needed to give an reactance of 25 in a 18V 60Hz circuit?

66 mF

• VL as ¼ cycle ahead of resistive voltage

• Because VL is maximum where VR is changing most (gradient steepest)

• Note: the value of VR and VL are not always equal as in this example

Resistor and Inductor Phase Differences

-1.5

-1

-0.5

0

0.5

1

1.5

0 200 400 600 800

Time (ms)

Vo

lta

ge

(m

V)

ResistorInductor

Phase Differences in LR Circuits

VL

VR

The Effect of Phase Differences in LR Circuits

• In DC circuits the voltages across components in a circuit add up to the supply voltage

• In AC Inductor/Resistor (LR) circuits the same does not appear to apply (at first glance) just like RC circuits

VS

VCVR

0.50H100

12V

6.4V

10V

The Effect of Phase Differences in LR Circuits

• However if we consider the phase differences, we see that this is a vector problem

VL

LRS VVV~~~

VR

VS

22

222

LRS VVV

BAC

From

s; PythagoruVL

VR

VS

VS

VCVR

0.50H100

12V

6.4V 10V

0 100 200 300 400 500 600 700 800

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Supply Voltage of Resistor/Inductor Ciruits

Induc-tor

Resis-tor

Supply Voltage

Time (ms)

Vo

lta

ge

(m

V)

In an LR circuit;• At any instant

Note the graph• But when considering

the rms voltages the phase differences are important

The Effect of Phase Differences in LR Circuits

LRS VVV

LRS VVV~~~ 22

LRS VVV

Exercises1. Find the AC supply voltage of an LR circuit where the

resistor voltage is 3.4V and the inductor voltage is 1.5V3.7V

2. Calculate the voltage across the resistor in an AC circuit with a supply voltage of 8.5V and a inductor voltage of 2.4V

8.2V3. Calculate the voltage across the inductor in an 12V AC

circuit with a voltage of 8.5V across the resistor. 8.5V

4. Find the supply voltage of an 60Hz AC circuit with a 120V across a 2k resistor and an inductor voltage of 0.80V

120V

Impedance

• As with LR circuits impedance relates supply voltage to current.

• Using Pythagoras from the addition of phasorsI

VZ

and

IZV

S

S

so;

22LXRZ

Examples

1. Calculate the impedance of an LR circuit with a resistance of 75 and a reactance of 15

76 2. An LR circuit has an impedance of 65 and has a

resistance of 24 . What is the reactance of the circuit?

60 3. Find the resistance of an LR circuit with 25

impedance and 12 reactance.22

Inductors in DC c.f. AC

• Both circuits have the same components but behave quite differently because of their power supplies;1. Find the resistance of the resistor2. What assumption did you make in 1?3. Calculate the reactance of the circuit4. What is the impedance of the circuit?5. Calculate the current in the AC circuit

AA400mH 400mH

0.15A

18V DC 18V AC 50Hz

The LCR Series Circuit

• The LCR circuit has some interesting and useful properties.

• The current and voltage in the circuit vary considerably as frequency changes

• The voltage across each component will depend on the resistance or reactance of each component

Variable Frequency AC

A

fLIXV

fCIX

IRV

LL

C

R

2

21

L

CC

X

X V

constant isR

𝑉=𝐼𝑅𝑉 𝐶=𝐼𝑋 𝐶

𝑉 𝐿=𝐼 𝑋𝐿

LRC Phase Differences

• Phase differences are the same as the individual RC and LR circuits combined

• Inductor voltage (VL ) leads resistor voltage (VR) by 90 and VR leads capacitor voltage (VC ) by 90

• In LCR circuits inductor and capacitor voltages have an opposite phase, so fully or partially cancel each other

VL

VR

VC

0 100 200 300 400 500 600 700 800

-1.5

-1

-0.5

0

0.5

1

1.5

LCR Voltages

Resis-torCapac-itorInduc-torSource

Time (ms)

Volt

age

(V)

LCR Phasors• In most cases the L, C and R

phasors will be different lengths• Most commonly voltage and

reactance/resistor phasors are considered

• In either case remember to calculate the differences between the two opposite phasors before calculating VS or Z

VL

VR

VC

VL-VC

VR

XL

RXC

XL-XCR

VS

ZLCT XXX CLeffectiveLorC VVV

or;

Supply Voltage in LCR Circuits

• Calculations of the supply voltage must take the into account the differences of the components

VL

VR

VC

VL-VCVS

𝑉 𝑆=√(𝑉 ¿¿𝐶−𝑉 𝐿)2+𝑅2 ¿

0 100 200 300 400 500 600 700 800

-1.5

-1

-0.5

0

0.5

1

1.5

LCR Voltages

Resis-torCapac-itorInduc-torSource

Time (ms)Vo

ltag

e (V

)

Examples

1. Calculate the supply voltage of an LCR circuit where the capacitor voltage is 12V, the resistor voltage is 18V and the inductor voltage is 6V

19V 2. Calculate the resistor voltage of an LCR circuit where the

supply voltage 240V, the capacitor voltage is 85V and the inductor voltage is 220V

198 3. Find the inductor voltage of an LCR circuit where the

supply voltage is 12V, the resistor voltage is 9.8V and the capacitor voltage is 4.5V

2.4V

Impedance in LCR Circuits

22 RXXZ LC )(

• Impedance is a measure of the combined opposition to alternating current of the components of a circuit.

• It describes not only the relative amplitudes of the voltage and current, but also the relative phases the components in the circuit.

• Impedance has the symbol Z and units Ohms

XL

RXC

XL-XCR

Z

Examples

1. Calculate the impedance of an LCR circuit where the capacitor reactance is 25, the resistance is 50 and the inductor reactance is 15

51 2. Calculate the resistance of an LCR circuit where the

impedance 110 is capacitor reactance is 64 and the inductor reactance is 25

100 3. Find the inductor reactance of an LCR circuit where the

impedance is 120 , the resistance is 110 and the capacitor reactance is 30

120

Resonance• Because reactance is

dependant on supply frequency and directly proportional for inductors and inversely proportional for capacitors at a certain frequency (resonant frequency fO) these reactances cancel each other out

• At this frequency current in the circuit reaches a maximum and the circuit is said to be tuned

fC

fL

21

2

C

L

X

X

LC XX

fo

Resonant frequency

Curr

en

t

(A)

Resonant Frequency• Because at resonance;

so;Cf

Lfo

o

21

2

LC XX

LCfo 2

1

Note that the resonant frequency is independent of the resistance

Examples

1. Calculate the resonance frequency of an LRC circuit with a 200F capacitor and a 0.5H inductor.

2. Find the size of the capacitor needed for resonance in an LRC with a resonant frequency of 50Hz and an inductor of 0.20H

Voltage at Resonance

• At resonance;

• And because Z = RCL

CL

CL

VV

soI

V

I

V

XX

;

IRVV RS

And cancel each other out

Examples

Exercises

ESA Pg 282Activity 16E, 16F, 16G, 16H

ABA Pg 186-196