RESISTOR & INDUCTORS RESISTOR & INDUCTORS IN SERIESIN SERIES

This diagram shows the waveform relationship that would be

generated when you connect a Resistor and Inductor in series.

OhmsL

VR

IT

VL

When you place a resistor in an AC circuit the voltage and

current are in phase.

An Inductor however, produces a phase shift between its voltage

and current of 900, with the current lagging the volts

CIRCUIT CURRENTCIRCUIT CURRENT

In a DC series circuit, the current is the same value for all components In a DC series circuit, the current is the same value for all components connected in that circuit.connected in that circuit.

The series AC circuit has this same relationship.The series AC circuit has this same relationship.

So the current flowing through the Resistor and Inductor are the same So the current flowing through the Resistor and Inductor are the same value.value.

This current must be supplied from the power source, so the source This current must be supplied from the power source, so the source current must also be equal to this value.current must also be equal to this value.

100Ω

OhmsL

0.320A

IT = IR = IL

Again you use Ohm’s Law and component Again you use Ohm’s Law and component values from either the Resistor or Inductor values from either the Resistor or Inductor

to find current.to find current. IR = VR R IL = VL XL

IT = IR = IL

CIRCUIT VOLTAGECIRCUIT VOLTAGE

When you use a Phasor diagram in a series circuit, the When you use a Phasor diagram in a series circuit, the CURRENT is used as the REFERENCE because it is common to all components in because it is common to all components in

that circuit. that circuit.

The reference is always on the X axis at 00.

VR

IT

VL

VR

VL

IT

A Resistor has a phase relationship of 0A Resistor has a phase relationship of 000 between its current and between its current and voltage i.e. the are in phasevoltage i.e. the are in phase..

The Inductor phase relationship is different, with the Inductor voltage The Inductor phase relationship is different, with the Inductor voltage leading the Inductor current by 90leading the Inductor current by 9000. .

Because this is a RL circuit, the Because this is a RL circuit, the circuit phasor diagram will also be a will also be a combination the combination the Resistor phasor and the and the Inductor phasor..

So the two voltage phasors are combined, with the circuit current So the two voltage phasors are combined, with the circuit current phasor being used as the reference.phasor being used as the reference.

We now have two voltages We now have two voltages ((VR and and VL) operating at 90) operating at 9000 to each other. to each other.

VR

VL

IT

This is the phase relationship for this circuit

When you have two voltages operating at 90When you have two voltages operating at 9000, the resultant voltage , the resultant voltage (VT, or circuit voltage) is equal to the phasor addition of the is equal to the phasor addition of the X axis

voltage voltage (VR) and the and the Y axis voltage voltage (VL)

22CRT VVV VR

VL

IT

VT

You can use Pythagoras Theorem to find the resultant (Hypotenuse)

You can also use Pythagoras Theorem to find the X and Y values (Adjacent & Opposite)

Both the resistor and inductor produce a Both the resistor and inductor produce a voltage drop across themselves voltage drop across themselves

proportional to their opposition and the proportional to their opposition and the current flowing through them. current flowing through them.

(OHMS LAW) (OHMS LAW)

VR = IR x R VL = IL x XL

PROBLEMS (PROBLEMS (all circuits 50Hz)all circuits 50Hz)

1. A series RL circuit has a resistor volts of 60v and an inductor volts of 100v. The supply voltage is?

2. In a series RL circuit the supply volts is 240v and the resistor volts is 180v. The inductor voltage is?

3. A RL circuit has a supply voltage of 1500v and the inductor has a voltage of 560v. The resistor voltage is?

116.6 V

158.7 V

1391.6 V

4. An RL circuit has a resistance of 150 and the circuit current is 4.0A. The resistor voltage is?

5. An RL circuit has a reactance of 60 and the circuit current is 1.5A. The inductor voltage is?

6. An RL circuit has a current of 15A and the inductor voltage is 400V. The inductor value is?

7. An RL circuit has a current of 7A and the resistor voltage is 150V. The resistor value is?

600 V

90 V

84.9mH

21.48 Ω

CIRCUIT IMPEDANCECIRCUIT IMPEDANCE

Resistors and Inductors both provide opposition to circuit current flow.

Resistors have Resistance measured in Ohms.

Inductors have Inductive Reactance (XL) also in Ohms.

VL

IT R

XL

VR

This diagram shows the phase relationship that exists between the Resistance and the Inductive

Reactance.

It is the same phase relationship that the component voltages have.

Both components provide opposition to the circuit current. Both components provide opposition to the circuit current.

The issue is the Resistance is 90The issue is the Resistance is 9000 out of phase with the Inductive out of phase with the Inductive reactance (Xreactance (XLL).).

Because of this phase angle, you can only find the total opposition Because of this phase angle, you can only find the total opposition by the phasor addition of R and Xby the phasor addition of R and XLL

R

XL

When any AC circuit has a combination of different components connected in it, the circuit total opposition is called IMPEDANCE.

R

XL

So this circuit has some value of R (due to the Resistor) and some value of XL (due to the Inductor), but the total circuit

opposition is the phasor addition of R and XL.

IMPEDANCE (Z) measured in Ohms.

Z

The circuit Impedance is found by the The circuit Impedance is found by the phasor addition of these two values.phasor addition of these two values.

R

XL Z

As with the voltage phasor you can also use Pythagoras Theorem to find the X and Y values (Adjacent & Opposite)

The circuit impedance can also be found byThe circuit impedance can also be found by Z = VT IT

so

IT = VT Z

And

VT = Z x IT

The value of the Resistance and Inductive Reactance can be found by The value of the Resistance and Inductive Reactance can be found by using component values.using component values.

R = VR IR XL = VL IL

PROBLEMS (PROBLEMS (all circuits 50Hz)all circuits 50Hz)

1. A series RL circuit has a resistor of 60 and a reactance of 100. The circuit impedance is?

2. In a series RL circuit the impedance is 240 and the resistor is 180. The inductor value is?

3. An RL circuit has an impedance of 1500 and the inductor has a reactance of 560. The resistor value is?

116.6

505 mH

1391.6

4. An RL circuit has an impedance of 150 and the circuit voltage is 400V. The circuit current is?

5. An RL circuit has a circuit current of 15A and the circuit voltage is 200V. The circuit impedance is?

6. An RL circuit has a resistance of 180 and the resistor voltage is 40V. The circuit current is?

7. An RL circuit has a circuit current of 7.5A. The circuit voltage is 400V and the resistor voltage is 200V . The value of the resistor and inductor are?

2.67 A

0.22 A

R=26.7 L=147mH

13.33

PHASE ANGLEPHASE ANGLE

The phase angle in any circuit is determined by the value of the The phase angle in any circuit is determined by the value of the components in the circuit. components in the circuit.

When you use Phasors to determine values When you use Phasors to determine values in a series circuit, the phase angle in the Voltage Phasor is the same angle in the

Impedance Phasor.

VR

VL

IT

VT

R

XL Z

Phase angle anywhere between near 00 and near 900

Phase angle anywhere between near 00 and near 900

The phase angle can vary anywhere between The phase angle can vary anywhere between 0 degrees (large R 0 degrees (large R value) and value) and 90 degrees (large inductor value) 90 degrees (large inductor value) depending on the depending on the

component values in the circuit. component values in the circuit.

Phase angle anywhere between near 00 and near 900

VR

VL VT

IT

Note:- is the same angle for the impedance and voltage phasor.

If (VT or Z) and (VL or XL) are known.

We can use Sin

Sin-1 is used to determine the angle in degrees from the known sin value.

Sin = VL VT

VL VT

IT

XL Z

Sin = XL Z

If (VT or Z) and (VR or R) are known.

We can use Cos We can use Cos

Cos-1 is used to determine the angle in degrees from the known Cos value.

Cos = VR VT

VR

VT

IT R

Z

Cos = R Z

If (If (VR or R) and (VL or XL) are known. are known.

We can use Tan We can use Tan

Tan-1 is used to determine the angle in degrees from the known Tan value.

Tan = VL VR

VR

VL

IT R

XL

Tan = XL R

POWER FACTORPOWER FACTOR

Power factor indicates Power factor indicates how efficient the circuit current is converted to how efficient the circuit current is converted to true power and is determined by true power and is determined by the phase relationship that exists the phase relationship that exists

between the supply voltage and current in a circuit.between the supply voltage and current in a circuit.

Power factor = Cos

Phase angle anywhere between near 00 and near 900

VR

VL VT

IT

Remember, voltage is always used as the reference

The phase angle can vary anywhere between The phase angle can vary anywhere between 0 degrees 0 degrees (large R) and (large R) and 90 degrees (large X 90 degrees (large XLL value). value).

The component values in the circuit determine this. The component values in the circuit determine this.

Therefore the power factor can be between Therefore the power factor can be between 0 and 0 and 1(unity) 1(unity) lagging. lagging.

The word lagging tells use that the circuit current is lagging the circuit voltage.

VR

VL VT

IT

Power factor near 1(unity)

Power factor near 0 lag So the power factor can

be between these limits

POWERPOWER

The power dissipated by a combined circuit is determined only by the resistive component in the circuit.

Pure Inductors consume no power.

P = IT x VT x COS (Circuit values)

Because the power of a circuit is determined by the resistive component only, the phasor representation for power is drawn on

the X AXIS.

Power

The power can also be found by using resistor component values.

P = IR2

x R

P = VR2 R

P = IR x VR

REACTIVE POWERREACTIVE POWER

Reactive power is the wattless power taken by the reactive components

measured in var.

The reactive power is drawn by the inductor, therefore the phasor

representation for Q is drawn on the Y AXIS.

QL

The inductor is a reactive component which consumes energy to establish its magnetic field

during the first quarter cycle but returns the energy back to the supply in the next quarter

cycle when the inductor discharges.The power required to achieve this is called

Reactive Power.

The power equations you have been using can be used The power equations you have been using can be used to determine VAR by substituted P with Q and R being to determine VAR by substituted P with Q and R being

substituted with substituted with XXLL

QL = IL2 x XL ; QL = VL

2 XL ; QL = IL x VL(Note:- these formula are used with component values only)

QT = IT x VT x sin (Note:- this formula is used with circuit values only)

APPARENT POWERAPPARENT POWER

When an inductor and resistor are connected into a circuit, When an inductor and resistor are connected into a circuit, there will be Power and VAR developed in the circuit.there will be Power and VAR developed in the circuit.

The phase relationship is such that the power is always on The phase relationship is such that the power is always on the X axis while the VAR is on the Y axis.the X axis while the VAR is on the Y axis.

Power

Reactive power

We now have an X and Y axis We now have an X and Y axis components. components.

When we have this relationship, we can When we have this relationship, we can find the Phasor sum of these to find the Phasor sum of these to

determine the Total power.determine the Total power.

In AC circuits, the phasor addition of these In AC circuits, the phasor addition of these Powers is called the Powers is called the

APPARENT POWER APPARENT POWER (S)(S)

measured in VA (volt x amps).measured in VA (volt x amps). P

QL

True Power

Reactive

Pow

er

SApparent Power

The Apparent power in any circuit can by The Apparent power in any circuit can by also found byalso found by

S = IS = ITT x V x VTT

Note:- Note:- is the same value for the is the same value for the Impedance, Power and Voltage phasorImpedance, Power and Voltage phasor..

Phase angle anywhere between near 00 and near 900

R Values

L ValuesCircuit Values

IT

EFFECTS OF CHANGING EFFECTS OF CHANGING CIRCUIT CONDITIONSCIRCUIT CONDITIONS

The frequency of the supply will determine the value of inductive reactance in the circuit.

Therefore if the inductive reactance changes with frequency, then the circuit impedance, power, var, phase angle, current and voltage drops will also

change.

For a RL series connected circuitFor a RL series connected circuit

Indicate what will happen to these if the frequency is reduced?

Current VR VL Power Power Factor

Indicate what will happen to these if the resistance is increased?

Current VR VL Power Power Factor

Indicate what will happen to these if the inductor value is increased?

Current VR VL Power Power Factor

For Resistor and Inductors connected in series,For Resistor and Inductors connected in series,

The current value is the same through all componentsThe current value is the same through all components The Resistor voltage is in phase with the circuit current

The Inductor voltage is leading the circuit current by 900

The supply voltage is found by the phasor addition of VR and VL

The Resistance value is always drawn on the X axis.

The Inductive Reactance is at 900 to the resistance and is drawn on the Y axis.

The total opposition is called Impedance and is measured in Ohms.

The Impedance is found by the phasor addition of R and XL.

REVIEW QUESTIONS REVIEW QUESTIONS ((all circuits 50Hz)all circuits 50Hz)

2.. A circuit draws a current of 6.8 amps when connected to a 400v supply. If the A circuit draws a current of 6.8 amps when connected to a 400v supply. If the reactive power of the circuit is 2.2 kVAR the power factor is ?reactive power of the circuit is 2.2 kVAR the power factor is ?

3. A circuit develops 5kVAR when connected to a 200v supply with a power factor of A circuit develops 5kVAR when connected to a 200v supply with a power factor of 0.75 lag. The current drawn by this circuit is?0.75 lag. The current drawn by this circuit is?

4. A circuit has a resistance of 5 ohms and a reactance of 6 ohms when a 50hz A circuit has a resistance of 5 ohms and a reactance of 6 ohms when a 50hz supply is connected. Determine circuit impedance and angle by which load current supply is connected. Determine circuit impedance and angle by which load current would lag the voltage.would lag the voltage.

5. A winding has 50 ohms of resistance and 0.2h of inductance. It is connected A winding has 50 ohms of resistance and 0.2h of inductance. It is connected across a 240v, 50hz supply. Calculate the:across a 240v, 50hz supply. Calculate the:(a) inductive reactance(a) inductive reactance(b) impedance(b) impedance(c) circuit current(c) circuit current(d) angle by which the current lags the voltage.(d) angle by which the current lags the voltage.(e) power consumed(e) power consumed(f) power factor(f) power factor

5184 VAR

0.59 Lag

37.88 A

7.8 Ω @ 50.20

1. A circuit is connected to a 240v 50hz supply and draws a current of 27 amps. If the power factor is 0.6 lagging ,what is the VAR of the circuit?

80.3 Ω

51.480

62.83 Ω

2.99 A

447 W

0.623 Lag

47.17 Ω

55.86 Ω4.3 A57.520

553.8 W

2.25 Ω7.9 mH

6. A winding has 30 ohms of resistance and 0.15h of inductance. It is to be . A winding has 30 ohms of resistance and 0.15h of inductance. It is to be connected to a 240v, 50hz supply. Calculate the:connected to a 240v, 50hz supply. Calculate the:(a) inductive reactance(a) inductive reactance(b) impedance(b) impedance(c) circuit current(c) circuit current(d) angle by which the current lags the voltage.(d) angle by which the current lags the voltage.(e) power(e) power

7. A RL series circuit takes 2000w of power and has a power factor of 0.67lag . A RL series circuit takes 2000w of power and has a power factor of 0.67lag when it is connected to a 100V, 50 Hz supply. Calculate.when it is connected to a 100V, 50 Hz supply. Calculate.(a) the value of the resistor(a) the value of the resistor(b) the value of the inductor(b) the value of the inductor

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