BASIC ELECTRICITY

AC Generators

Basic Generator A basic generator consists of a magnetic field, an armature, slip

rings, brushes and a resistive load. The magnetic field is usually

an electromagnet. An armature is any number of conductive

wires wound in loops which rotates through the magnetic field.

For simplicity, one loop is shown. When a conductor is moved

through a magnetic field, a voltage is induced in the conductor.

As the armature rotates through the magnetic field, a voltage

is generated in the armature which causes current to flow. Slip

rings are attached to the armature and rotate with it. Carbon

brushes ride against the slip rings to conduct current from the

armature to a resistive load.

Pole Piece

Magnetic Field

Armature

Brush

Slip Ring

R1

Basic Generator Operation An armature rotates through the magnetic field. At an initial

position of zero degrees, the armature conductors are moving

parallel to the magnetic field and not cutting through any

magnetic lines of flux. No voltage is induced.

R1

45

Generator Operation from The armature rotates from zero to 90 degrees. The conductors

Zero to 90 Degrees cut through more and more lines of flux, building up to a

maximum induced voltage in the positive direction.

R1

90

Degrees

Generator Operation from The armature continues to rotate from 90 to 180 degrees,

90 to 180 Degrees cutting less lines of flux. The induced voltage decreases from a

maximum positive value to zero.

S

R1

180

Degrees

Generator Operation from The armature continues to rotate from 180 degrees to 270

degrees. The conductors cut more and more lines of flux, but

in the opposite direction. voltage is induced in the negative

direction building up to a maximum at 270 degrees.

R1

270

Degrees

46

Generator Operation from The armature continues to rotate from 270 to 360 degrees.

270 to 360 Degrees Induced voltage decreases from a maximum negative value to

zero. This completes one cycle. The armature will continue to

rotate at a constant speed. The cycle will continuously repeat as

long as the armature rotates.

S

R1

One Revolution

360

Degrees

47

Frequency

The number of cycles per second made by voltage induced in

the armature is the frequency of the generator. If the armature

rotates at a speed of 60 revolutions per second, the generated

voltage will be 60 cycles per second. The accepted term for

cycles per second is hertz. The standard frequency in the United

States is 60 hertz. The following illustration shows 15 cycles in

1/4 second which is equivalent to 60 cycles in one second.

1/4 Second

Four-Pole AC Generator The frequency is the same as the number of rotations per

second if the magnetic field is produced by only two poles.

An increase in the number of poles, would cause an increase

in the number of cycles completed in a revolution. A two-pole

generator would complete one cycle per revolution and a fourpole

generator would complete two cycles per revolution. An

AC generator produces one cycle per revolution for each pair of

poles.

R1

One Revolution

48

Voltage and Current

Peak Value The sine wave illustrates how voltage and current in an AC

circuit rises and falls with time. The peak value of a sine wave

occurs twice each cycle, once at the positive maximum value

and once at the negative maximum value.

Peak Value

Peak Value

Time

+

0

-

Peak-to-Peak Value The value of the voltage or current between the peak positive

and peak negative values is called the peak-to-peak value.

Time

Peak-to-Peak

Value

+

0

-

Instantaneous Value The instantaneous value is the value at any one particular time.

It can be in the range of anywhere from zero to the peak value.

+

0

-

Time

Instantaneous Value

49

Calculating Instantaneous The voltage waveform produced as the armature rotates

Voltage through 360 degrees rotation is called a sine wave because

instantaneous voltage is related to the trigonometric function

called sine (sin θ = sine of the angle). The sine curve represents

a graph of the following equation:

e = Epeak x sin θ

Instantaneous voltage is equal to the peak voltage times the

sine of the angle of the generator armature. The sine value is

obtained from trigonometric tables. The following table reflects

a few angles and their sine value.

Angle Sin θ Angle Sin θ

30 Degrees

60 Degrees

90 Degrees

120 Degrees

150 Degrees

180 Degrees

210 Degrees

240 Degrees

270 Degrees

300 Degrees

330 Degrees

360 Degrees

0.5

0.5

0.866

0.866

1

0

-0.5

-0.5

-0.866

-0.866

-1

0

The following example illustrates instantaneous values at 90,

150, and 240 degrees. The peak voltage is equal to 100 volts.

By substituting the sine at the instantaneous angle value, the

instantaneous voltage can be calculated.

90° = +100 Volts

150° = +50 Volts

240° = -86.6 Volts

+

0

-

Any instantaneous value can be calculated. For example:

240°

e = 100 x -0.866

e = -86.6 volts

50

Effective Value of an Alternating voltage and current are constantly changing values.

AC Sine Wave A method of translating the varying values into an equivalent

constant value is needed. The effective value of voltage and

current is the common method of expressing the value of AC.

This is also known as the RMS (root-mean-square) value. If the

voltage in the average home is said to be 120 volts, this is the

RMS value. The effective value figures out to be 0.707 times

the peak value.

169.7 Vpeak x 0.707 = 120 Vrms

Peak Value

169.7 Volts

+

0

-

The effective value of AC is defined in terms of an equivalent

heating effect when compared to DC. One RMS ampere of

current flowing through a resistance will produce heat at the

same rate as a DC ampere.

For purpose of circuit design, the peak value may also be

needed. For example, insulation must be designed to withstand

the peak value, not just the effective value. It may be that

only the effective value is known. To calculate the peak value,

multiply the effective value by 1.41. For example, if the effective

value is 100 volts, the peak value is 141 volts.

Review 7

1. The graphic representation of AC voltage or current

values over a period of time is a ____________

____________ .

2. Each phase of three phase AC power is offset by

____________ degrees.

3. An AC generator produces ____________ cycle per

revolution for each pair of poles.

4. What is the instantaneous voltage at 240 degrees for a

peak voltage of 150 volts?

5. What is the effective voltage for a peak voltage of 150

volts?

51

Inductance

The circuits studied to this point have been resistive.

Resistance and voltage are not the only circuit properties that

effect current flow, however. Inductance is the property of an

electric circuit that opposes any change in electric current.

Resistance opposes current flow, inductance opposes change

in current flow. Inductance is designated by the letter “L”. . The

unit of measurement for inductance is the henry (h).

Current Flow and Current flow produces a magnetic field in a conductor. The

Field Strength amount of current determines the strength of the magnetic

field. As current flow increases, field strength increases, and as

current flow decreases, field strength decreases.

0 Degrees

No Current

30 Degrees

Increasing

Current

90 Degrees

Maximum

Current

Any change in current causes a corresponding change in the

magnetic field surrounding the conductor. Current is constant

in DC, except when the circuit is turned on and off, or when

there is a load change. Current is constantly changing in AC,

so inductance is a continual factor. A change in the magnetic

field surrounding the conductor induces a voltage in the

conductor. This self-induced voltage opposes the change in

current. This is known as counter emf. This opposition causes

a delay in the time it takes current to attain its new steady

value. If current increases, inductance tries to hold it down. If

current decreases, inductance tries to hold it up. Inductance is

somewhat like mechanical inertia which must be overcome to

get a mechanical object moving or to stop a mechanical object

from moving. A vehicle, for example, takes a few moments to

accelerate to a desired speed, or decelerate to a stop.

52

Inductors Inductance is usually indicated symbolically on an electrical

drawing by one of two ways. A curled line or a filled rectangle

can be used.

Inductors are coils of wire. They may be wrapped around a core.

The inductance of a coil is determined by the number of turns

in the coil, the spacing between the turns, the coil diameter,

the core material, the number of layers of windings, the type of

winding, and the shape of the coil. Examples of inductors are

transformers, chokes, and motors.

Simple Inductive Circuit In a resistive circuit, current change is considered

instantaneous. If an inductor is used, the current does not

change as quickly. In the following circuit, initially the switch

is open and there is no current flow. When the switch is

closed, current will rise rapidly at first, then more slowly as the

maximum value is approached. For the purpose of explanation,

a DC circuit is used.

+_

L1

R1

Inductive Time Constant The time required for the current to rise to its maximum value is

determined by the ratio of inductance (in henrys) to resistance

(in ohms). This ratio is called the time constant of the inductive

circuit. A time constant is the time (in seconds) required for the

circuit current to rise to 63.2% of its maximum value. When

the switch is closed in the previous circuit, current will begin to

flow. During the first time constant current rises to 63.2% of its

maximum value. During the second time constant, current rises

to 63.2% of the remaining 36.8%, or a total of 86.4%. It takes

about five time constants for current to reach its maximum

value.

53

100.0%

98.1%

94.9%

86.4%

63.2%

First Time

Constant

Second Time

Constant

Third Time

Constant

Fourth Time

Constant

Fifth Time

Constant

Similarly, when the switch is opened, it will take five time

constants for current to reach zero. It can be seen that

inductance is an important factor in AC circuits. If the frequency

is 60 hertz, current will rise and fall from its peak value to zero

120 times a second.

100.0%

36.8%

13.6%

5.1%

1.9%

0%

First Time

Constant

Second Time

Constant

Third Time

Constant

Fourth Time

Constant

Fifth Time

Constant

Calculating the Time The time constant is designated by the symbol “τ”. To

Constant of an determine the time constant of an inductive circuit use one of

Inductive Circuit the following formulas:

τ (in seconds) =

L (henrys)

R (ohms)

τ (in milliseconds) =

L (millihenrys)

R (ohms)

τ (in microseconds) =

L (microhenrys)

R (ohms)

54

In the following illustration, L1 is equal to 15 millihenrys and

R1 is equal to 5 Ω. When the switch is closed, it will take

3 milliseconds for current to rise from zero to 63.2% of its

maximum value.

+_

L1 15 mh

R1 5 Ω

15 mh

5 Ω

τ = 3 milliseconds

τ =

Formula for Series Inductors The same rules for calculating total resistance can be applied. In

the following circuit, an AC generator is used to supply electrical

power to four inductors. There will always be some amount of

resitance and inductance in any circuit. The electrical wire used

in the circuit and the inductors both have some resistance and

inductance. Total inductance is calculated using the following

formula:

AC Generator

R1 L1

Lt = L1 + L2 + L3

Lt = L1 + L2 + L3 + L4

Lt = 2 mh + 2 mh + 1 mh + 1 mh

Lt = 6 mh

L2 L3 L4

2 mh 2 mh 1 mh 1 mh

55

Formula for Parallel In the following circuit, an AC generator is used to supply

Inductors electrical power to three inductors. Total inductance is

calculated using the following formula:

R1

L1

5 mh

L2

10 mh

L3

20 mh

1

Lt

= 1

L1

+ 1

L2

+ 1

L3

1

Lt

= 1

5

+ 1

10

+ 1

20

1

Lt

= 7

20

Lt = 2.86 mh

56

Capacitance

Capacitance and Capacitors Capacitance is a measure of a circuit’s ability to store an elec

cal charge. A device manufactured to have a specific amount of

capacitance is called a capacitor. A capacitor is made up of a

pair of conductive plates separated by a thin layer of insulating

material. Another name for the insulating material is dielectric

material. When a voltage is applied to the plates, electrons are

forced onto one plate. That plate has an excess of electrons

while the other plate has a deficiency of electrons. The plate

with an excess of electrons is negatively charged. The plate

with a deficiency of electrons is positively charged.

Negative Plate

Dielectric Material

Positive Plate

Direct current cannot flow through the dielectric material

because it is an insulator; however it can be used to charge a

capacitor. Capacitors have a capacity to hold a specific quantity

of electrons. The capacitance of a capacitor depends on the area

of the plates, the distance between the plates, and the material

of the dielectric. The unit of measurement for capacitance is

farads (F). Capacitors usually are rated in μF (microfarads), or pF

(picofarads).

Capacitor Circuit Symbols Capacitance is usually indicated symbolically on an electrical

drawing by a combination of a straight line with a curved line, or

two straight lines.

57

Simple Capacitive Circuit In a resistive circuit, voltage change is considered

instantaneous. If a capacitor is used, the voltage across the

capacitor does not change as quickly. In the following circuit,

initially the switch is open and no voltage is applied to the

capacitor. When the switch is closed, voltage across the

capacitor will rise rapidly at first, then more slowly as the

maximum value is approached. For the purpose of explanation,

a DC circuit is used.

+

_

C1

R1

Capacitive Time Constant The time required for voltage to rise to its maximum value in a

circuit containing capacitance is determined by the product of

capacitance, in farads, times resistance, in ohms. This product is

the time constant of a capacitive circuit. The time constant gives

the time in seconds required for voltage across the capacitor to

reach 63.2% of its maximum value. When the switch is closed

in the previous circuit, voltage will be applied. During the first

time constant, voltage will rise to 63.2% of its maximum value.

During the second time constant, voltage will rise to 63.2% of

the remaining 36.8%, or a total of 86.4%. It takes about five

time constants for voltage across the capacitor to reach its

maximum value.

100.0%

98.1%

94.9%

86.4%

63.2%

First Time

Constant

Second Time

Constant

Third Time

Constant

Fourth Time

Constant

Fifth Time

Constant

58

Similarly, during this same time, it will take five time constants

for current through the resistor to reach zero.

100.0%

36.8%

13.6%

5.1%

1.9%

0%

First Time

Constant

Second Time

Constant

Third Time

Constant

Fourth Time

Constant

Fifth Time

Constant

Calculating the Time To determine the time constant of a capacitive circuit, use one

Constant of a of the following formulas:

Capacitive Circuit

τ (in seconds) = R (megohms) x C (microfarads)

τ (in microseconds) = R (megohms) x C (picofarads)

τ (in microseconds) = R (ohms) x C (microfarads)

In the following illustration, C1 is equal to 2 μF, and R1 is equal

to 10 Ω. When the switch is closed, it will take 20 microseconds

for voltage across the capacitor to rise from zero to 63.2%

of its maximum value. It will take five time constants, 100

microseconds for this voltage to rise to its maximum value.

+_

V

C1 2μF

R1 10 Ω

τ = RC

τ = 2μF x 10 Ω

τ = 20 microseconds

59

Formula for Connecting capacitors in series decreases total capacitance.

Series Capacitors The effect is like increasing the space between the plates. The

formula for series capacitors is similar to the formula for parallel

resistors. In the following circuit, an AC generator supplies

electrical power to three capacitors. Total capacotance is

calculated using the following formula:

1

Ct

= 1

C1

+ 1

C2

+ 1

C3

1

Ct

= 1

5

+

1

Ct

= 7

20

Ct = 2.86 μF

1

10

+ 1

20

R1 C1 C2 C3

5 μF 10 μF 20 μF

Formula for In the following circuit, an AC generator is used to supply

Parallel Capacitors electrical power to three capacitors. Total capacitance is

calculated using the following formula:

R1

C1 C2 C3

5 μF 10 μF 20 μF

Ct = C1 + C2 + C3

Ct = 5 μF + 10 μF + 20 μF

Ct = 35 μF

60

Review 8

1. The total inductance for this circuit is ___________ .

4 mh 2 mh 3 mh 1 mh

R1 L1 L2 L3 L4

2. The total inductance for this circuit is ____________ .

L1

R1

L2 L3

5 mh 10 mh 10 mh

3. The total capacitance for this circuit is ____________ .

R1 C1 C2 C3

5 μF 10 μF 10 μF

4. The total capacitance for this circuit is ____________ .

R1

C1 C2 C3

5 μF 10 μF 10 μF

61

Inductive and Capacitive Reactance

In a purely resistive AC circuit, opposition to current flow is

called resistance. In an AC circuit containing only inductance,

capacitance, or both, opposition to current flow is called

reactance. Total opposition to current flow in an AC circuit that

contains both reactance and resistance is called impedance,

designated by the symbol “Z”. Reactance and impedance are

expressed in ohms.

Inductive Reactance Inductance only affects current flow when the current is

changing. Inductance produces a self-induced voltage (counter

emf) that opposes changes in current. In an AC circuit, current

is changing constantly. Inductance in an AC circuit, therefore,

causes a continual opposition. This opposition to current flow is

called inductive reactance and is designated by the symbol XL.

Inductive reactance is dependent on the amount of inductance

and frequency. If frequency is low, current has more time

to reach a higher value before the polarity of the sine wave

reverses. If frequency is high, current has less time to reach

a higher value. In the following illustration, voltage remains

constant. Current rises to a higher value at a lower frequency

than a higher frequency.

+

0

_

+

0

_

Current Current

Low Frequency High Frequency

The formula for inductive reactance is:

XL = 2πfL

XL = 2 x 3.14 x frequency x inductance

62

In a 60 hertz, 10 volt circuit containing a 10 mh inductor, the

inductive reactance would be:

XL = 2πfL

XL = 2 x 3.14 x 60 x 0.10

XL = 3.768 Ω

Once inductive reactance is known, Ohm’s Law can be used to

calculate reactive current.

I = E

Z

I =

I = 2.65 Amps

10

3.768

Phase Relationship Current does not rise at the same time as the source voltage

between Current and in an inductive circuit. Current is delayed depending on the

Voltage in an amount of inductance. In a purely resistive circuit, current and

Inductive Circuit voltage rise and fall at the same time. They are said to be “in

phase.” In this circuit there is no inductance. Resistance and

impedance are the same.

+

0

_

Voltage

Current

In a purely inductive circuit, current lags behind voltage by 90

degrees. Current and voltage are said to be “out of phase”. In

this circuit, impedance and inductive reactance are the same.

+

0

_

90 Degrees

Voltage

Current

63

All inductive circuits have some amount of resistance. AC

current will lag somewhere between a purely resistive circuit,

and a purely inductive circuit. The exact amount of lag depends

on the ratio of resistance and inductive reactance. The more

resistive a circuit is, the closer it is to being in phase. The more

inductive a circuit is, the more out of phase it is. In the following

illustration, resistance and inductive reactance are equal.

Current lags voltage by 45 degrees.

+

0

_

45 Degrees

Voltage

Current

XL = 10 Ω

R = 10 Ω

Calculating Impedance in When working with a circuit containing elements of inductance,

an Inductive Circuit capacitance, and resistance, impedance must be calculated.

Because electrical concepts deal with trigonometric functions,

this is not a simple matter of subtraction and addition. The

following formula is used to calculate impedance in an inductive

circuit:

Z = R2 + XL2

In the circuit illustrated above, resistance and inductive

reactance are each 10 ohms. Impedance is 14.1421 ohms.

A simple application of Ohm’s Law can be used to find total

circuit current.

Z = 102 + 102

Z =

Z = 14.1421 Ω

200

Vectors Another way to represent this is with a vector. A vector is

a graphic represention of a quantity that has direction and

magnitude. A vector on a map might indicate that one city is

50 miles southwest from another. The magnitude is 50 miles

and the direction is southwest. Vectors are also used to show

electrical relationships. As mentioned earlier, impedance (Z) is

the total opposition to current flow in an AC circuit containing

reactance, inductance, and capacitance.

64

The following vector illustrates the relationship between

reactance and inductive reactance of a circuit containing equal

values of each. The angle between the vectors is the phase

angle represented by the symbol θ. When inductive reactance

is equal to resistance the resultant angle is 45 degrees. It is this

angle that determines how much current will lag voltage.

θ

XL = 10 Ω

R = 10 Ω

Z = 14.1421 Ω

Capacitive Reactance Capacitance also opposes AC current flow. Capacitive reactance

is designated by the symbol XC.The larger the capacitor, the

smaller the capacitive reactance. Current flow in a capacitive AC

circuit is also dependent on frequency. The following formula is

used to calculate capacitive reactance.

XC = 1

2πfC

Capacitive reactance is equal to 1 divided by 2 times pi, times

the frequency, times the capacitance. The capacitive reactance

for a 60 hertz circuit with a 10 microfarad capacitor is:

XC = 1

2πfC

XC = 1

2 x 3.14 x 60 x 0.000010

XC = 265.39 Ω

Once capacitive reactance is known, Ohm’s Law can be used to

calculate reactive current.

I =

I = 0.0376 Amps

E

Z

I = 10

265.39

65

Phase Relationship between The phase relationship between current and voltage are

Current and Voltage opposite to the phase relationship of an inductive circuit. In a

purely capacitive circuit, current leads voltage by 90 degrees.

+

0

_

90 Degrees

Voltage

Current

All capacitive circuits have some amount of resistance. AC

current will lead somewhere between a purely resistive circuit

and a purely capacitive circuit. The exact amount of lead

depends on the ratio of resistance and capacitive reactance. The

more resistive a circuit is, the closer it is to being in phase. The

more capacitive a circuit is, the more out of phase it is. In the

following illustration, resistance and capacitive reactance are

equal. Current leads voltage by 45 degrees.

+

0

_

45 Degrees

Voltage

Current

XC = 10 Ω

R = 10 Ω

Calculating Impedance in The following formula is used to calculate impedance in a

a Capacitive Circuit capacitive circuit:

Z = R2 + XC2

In the circuit illustrated above, resistance and capacitive re

tance are each 10 ohms. Impedance is 14.1421 ohms.

Z = 102 + 102

Z =

Z = 14.1421 Ω

200

66

The following vector illustrates the relationship between

resistance and capacitive reactance of a circuit containing equal

values of each. The angle between the vectors is the phase

angle represented by the symbol θ. When capacitive reactance

is equal to resistance the resultant angle is -45 degrees. It is

this angle that determines how much current will lead voltage.

θ

XC = 10 Ω

R = 10 Ω

Z = 14.1421 Ω

Review 9

1. In a circuit containing inductance, capacitance, or both,

opposition to current flow is called ____________ .

2. Total opposition to current flow in a circuit that contains

both reactance and resistance is called ____________ .

3. In a 50 hertz circuit, containing a 10 mh inductor, the

inductive reactance is ____________ ohms.

4. In a purely inductive circuit, ____________

a. current and voltage are in phase

b. current leads voltage by 90 degrees

c. current lags voltage by 90 degrees

5. In a purely capacitive circuit, ____________

a. current and voltage are in phase

b. current leads voltage by 90 degrees

c. current lags voltage by 90 degrees

6. In a 50 hertz circuit, containing a 10 microfarad

capacitor, the capacitive reactance is ____________

ohms.

7. In a circuit with 5 Ω resistance, and 10 Ω inductive

reactance, impedance is ____________ ohms.

8. In a circuit with 5 Ω resistance, and 4 Ω capacitive

reactance, impedance is ____________ ohms.

67

Series R-L-C Circuit

Circuits often contain elements of resistance, inductance, and

capacitance. In an inductive AC circuit, current lags voltage by

90 degrees. In a AC capacitive circuit, current leads voltage by

90 degrees. It can be seen that inductance and capacitance

are 180 degrees apart. Since they are 180 degrees apart, one

element will cancel out all or part of the other element.

R

XL

XC

An AC circuit is:

• Resistive if XL and XC are equal

• Inductive if XL is greater than XC

• Capacitive if XC is greater than XL

Calculating Total The following formula is used to calculate total impedance

Impedance in a Series of a circuit containing resistance, capacitance, and inductance:

R-L-C Circuit

Z = R2 + (XL - XC)2

In the case where inductive reactance is greater than capacitive

reactance, subtracting XC from XL results in a positive number.

The positive phase angle is an indicator that the net circuit

reactance is inductive, and current lags voltage.

In the case where capacitive reactance is greater than inductive

reactance, subtracting XC from XL results in a negative number.

The negative phase angle is an indicator that the net circuit

reactance is capacitive and current leads voltage. In either case,

the value squared will result in positive number.

68

Calculating Reactance and In the following 120 volt, 60 hertz circuit, resistance is 1000 Ω,

Impedance in a Series inductance is 5 mh, and capacitance is 2 μF. To calculate

R-L-C Circuit total impedance, first calculate the value of XL and XC, then

impedance can be calculated.

R = 1000 Ω L = 5 mh C = 2 μF

XL = 2πfL

XL = 6.28 x 60 x 0.005

XL = 1.884 Ω

XC = 1

2πfC

XC = 1

6.28 x 60 x 0.000002

XC = 1,327 Ω

Z = R2 + (XL - XC)2

Z = 10002 + (1.884 - 1,327)2

Z = 1,000,000 + ( - 1,325.116)2

Z = 1,000,000 + 1,755,932.41

Z =

Z = 1,660.1 Ω

2,755,932.41

Calculating Circuit Current Ohm’s Law can be applied to calculate total circuit current.

in a Series R-L-C Circuit

I = E

Z

I =

I = 0.072 Amps

120

1,660.1

69

Parallel R-L-C Circuit

Calculating Impedance in a Total impedance (Zt) can be calculated in a parallel R-L-C circuit

Parallel R-L-C Circuit if values of resistance and reactance are known. One method

of calculating impedance involves first calculating total current,

then using the following formula:

Zt =

Et

It

Total current is the vector sum of current flowing through the

resistance plus, the difference between inductive current and

capacitive current. This is expressed in the following formula:

It = IR2 + (IC - IL)2

In the following 120 volt, 60 hertz circuit, capacitive reactance

has been calculated to be 25 Ω and inductive reactance

50 Ω. Resistance is 1000 Ω. A simple application of Ohm’s Law

will find the branch currents. Remember, voltage is constant

throughout a parallel circuit.

R = 1000 Ω XL = 50 Ω XC = 25 Ω

IR = E

R

IR =

IR = 0.12 Amps

120

1000

IL = E

XL

IL =

IL = 2.4 Amps

120

50

IC = E

XC

IL =

IL = 4.8 Amps

120

25

70

Once the branch currents are known, total current can be

calculated.

It = IR2 + (IC - IL)2

It = 0.122 + (4.8 - 2.4)2

It = 0.0144 + 5.76

It =

It = 2.4 Amps

5.7744

Impedance is now found with an application of Ohm’s Law.

Zt =

Et

It

Zt =

Zt = 50 Ω

120

2.4

71

Power and Power Factor in an AC Circuit

Power consumed by a resistor is dissipated in heat and not

returned to the source. This is true power. True power is the rate

at which energy is used.

Current in an AC circuit rises to peak values and diminishes to

zero many times a second. The energy stored in the magnetic

field of an inductor, or plates of a capacitor, is returned to the

source when current changes direction.

Although reactive components do not consume energy, they do

increase the amount of energy that must be generated to do

the same amount of work. The rate at which this non-working

energy must be generated is called reactive power.

Power in an AC circuit is the vector sum of true power and

reactive power. This is called apparent power. True power is

equal to apparent power in a purely resistive circuit because

voltage and current are in phase. voltage and current are also in

phase in a circuit containing equal values of inductive reactance

and capacitive reactance. If voltage and current are 90 degrees

out of phase, as would be in a purely capacitive or purely

inductive circuit, the average value of true power is equal to

zero. There are high positive and negative peak values of power,

but when added together the result is zero.

True Power and The formula for apparent power is:

Apparent Power Formulas

P = EI

Apparent power is measured in volt-amps (VA).

True power is calculated from another trigonometric function,

the cosine of the phase angle (cos θ). The formula for true

power is:

P = EI cos θ

True power is measured in watts.

72

In a purely resistive circuit, current and voltage are in phase.

There is a zero degree angle displacement between current

and voltage. The cosine of zero is one. Multiplying a value by

one does not change the value. In a purely resistive circuit the

cosine of the angle is ignored.

In a purely reactive circuit, either inductive or capacitive, current

and voltage are 90 degrees out of phase. The cosine of 90

degrees is zero. Multiplying a value times zero results in a zero

product. No power is consumed in a purely reactive circuit.

Calculating Apparent Power In the following 120 volt circuit, current is equal to 84.9

in a simple R-L-C Circuit milliamps. Inductive reactance is 100 Ω and capacitive

reactance is 1100 Ω. The phase angle is -45 degrees. By

referring to a trigonometric table, the cosine of -45 degrees is

found to be .7071.

120 VAC

R = 1000 Ω XL = 100 Ω XC = 1100 Ω

The apparent power consumed by the circuit is:

P = EI

P = 120 x 0.0849

P = 10.2 VA

The true power consumed by the circuit is:

P = EI cos θ

P = 120 x 0.0849 x 0.7071

P = 7.2 Watts

Another formula for true power is:

P = I2R

P = 0.08492 x 1000

P = 7.2 Watts

73

Power Factor Power factor is the ratio of true power to apparent power in an

AC circuit. Power factor is expressed in the following formula:

PF = True Power

Apparent Power

Power factor can also be expressed using the formulas for true

power and apparent power. The value of EI cancels out because

it is the same in the numerator and denominator. Power factor

is the cosine of the angle.

PF =

PF = cos θ

EI cos θ

EI

In a purely resistive circuit, where current and voltage are in

phase, there is no angle of displacement between current and

voltage. The cosine of a zero degree angle is one. The power

factor is one. This means that all energy delivered by the source

is consumed by the circuit and dissipated in the form of heat.

In a purely reactive circuit, voltage and current are 90 degrees

apart. The cosine of a 90 degree angle is zero. The power factor

is zero. This means the circuit returns all energy it receives from

the source to the source.

In a circuit where reactance and resistance are equal, voltage

and current are displaced by 45 degrees. The cosine of a 45

degree angle is .7071. The power factor is .7071. This means the

circuit uses approximately 70% of the energy supplied by the

source and returns approximately 30%.

74

Review 10

1. An AC circuit is ____________ if inductive reactance and

capacitive reactance are equal.

2. A series AC circuit is ____________ if there is more

inductive reactance than capacitive reactance.

3. A series AC circuit is ____________ if there is more

capacitive reactance than inductive reactance.

4. In a 120 VAC, 60 hertz series circuit, with 1000 Ω

of resistance, 10 mh of inductance and 4 μF of

capacitance, impedance is ____________ Ω and current

is ____________ amps.

5. In the illustrated circuit,

120 VAC

60 Hz

R = 1000 Ω XL = 200 Ω XC = 1000 Ω

It is __________ amps, and impedance is __________ Ω.

6. True power is measured in ____________ .

7. A circuit with 0.2 amps flowing through 100 Ω of

resistance, is consuming ____________ watts.

75

Transformers

Mutual Induction Transformers are electromagnetic devices that transfer electrical

energy from one circuit to another by mutual induction. Mutual

induction is the coupling of inductances by their mutual

magnetic fields. In a single-phase transformer there are two

coils, a primary and a secondary coil. The following circuit

illustrates mutual induction. The AC generator provides electrical

power to the primary coil. The magnetic field produced by

the primary induces a voltage into the secondary coil, which

supplies power to a load.

Primary Coil Secondary Coil

Transformers are used to step a voltage up to a higher level,

or down to a lower level. Transformers are used extensively

in power distribution systems, allowing power companies

to transfer electrical energy many miles. Power generators

typically generate high voltages. This voltage varies, depending

on the generator, but a typical voltage might be 15 KV. The

voltage is stepped up through a transformer to higher levels

for transmission to substations. Typical voltages range from

115 KV to 765 KV. The electrical power is received at substation

transformers many miles away where it is stepped down.

Typical voltage might be 34 KV or 69 KV. From here, electrical

power is fed to a distribution substation. It can also be fed

directly to factory locations. If the power is fed to a factory,

transformers at the factory site reduce the voltage to usable

levels. The power fed to a distribution substation is reduced by

transformers at the substation for factory and home use.

76

Coefficient of Coupling Mutual inductance between two coils depends on their flux

linkage. Maximum coupling occurs when all the lines of flux

from the primary coil cut through the secondary winding.

The amount of coupling which takes place is referred to as

coefficient of coupling. To maximize coefficient of coupling, both

coils are often wound on an iron core which is used to provide

a path for the lines of flux. The following discussion of step-up

and step-down transformers applies to transers with an iron

core.

Lines of Flux

Confined to

Iron Core

Lines of Flux

that don’t Couple

Voltage, Current, and the There is a direct relationship between voltage, impedance,

Number of Turns in a Coil current, and the number of coil turns in a transformer. This

relationship can be used to find either primary or secondary

voltage, current, and the number of turns in each coil. It is the

number of turns which determine if a transformer is a step up

or step down transformer. The following “rules-of-thumb” apply

to transformers:

• If the primary coil has fewer turns than the secondary coil,

it is a step-up transformer.

• If the primary coil has more turns than the secondary coil,

it is a step-down transformer.

When the number of turns on the primary and seconday coils of

a transformer are equal, input voltage, impedance, and current

are equal to output voltage, impedance, and current.

77

Step-Up Transformer A step-up transformer is used when it is desirable to step

voltage up in value. The following circuit illustrates a stepup

transformer. The primary coil has fewer turns than the

secondary coil. The number of turns in a transformer is given as

a ratio. When the primary has fewer turns than the secondary,

voltage and impedance are stepped up. In the circuit illustrated,

voltage is stepped up from 120 VAC to 240 VAC. Because

impedance is also stepped up, current is stepped down from 10

amps to 5 amps.

1:2

Primary Coil

900 Turns

120 VAC Supply

10 Amps

Secondary Coil

1800 Turns

240 VAC

5 Amps

Step-Down Transformer A step-down transformer is used when it is desirable to

step voltage down in value. The following circuit illustrates a

step-down transformer. The primary coil has more turns than

the secondary coil. The step-down ratio is 2:1. voltage and

impedance are stepped down, current is stepped up.

2:1

Primary Coil

1800 Turns

Seconday Coil

900 Turns

240 VAC Supply

5 Amps

120 VAC Out

10 Amps

78

Single-Phase Transformer 120 or 240 VAC single-phase transformers are used to supply

lighting, receptacle, and small appliance loads. A transformer

with a 240 VAC secondary can be used to supply 240 VAC to

larger appliances such as stoves, air conditioners and heaters.

A 240 VAC secondary can be tapped in the center to provide

two sources of 120 VAC power.

Primary Primary

120 VAC 240 VAC

120 VAC 120 VAC

Secondary Ground Secondary

Formulas for Calculating the There are a number of useful formulas for calculating, voltage,

Number of Primary and current, and the number of turns between the primary and

Secondary Turns of a secondary of a transformer. These formulas can be used with

Transformer either step-up or step-down transformers. The following legend

applies to the transformer formulas:

ES = secondary voltage

EP = primary voltage

IS = secondary current

IP = primary current

NS = turns in the secondary coil

NP = turns in the primary coil

To find voltage:

ES =

EP x IP

IS

EP =

ES x IS

IP

To find current:

IS =

EP x IP

ES

IP =

ES x IS

EP

To find the number of coil turns:

NS =

ES x NP

EP

NP =

EP x NS

ES

79

Using the values for the step-down transformer in the example

of the previous page, the secondary voltage can be verified.

ES =

EP x IP

IS

ES =

240 Volts x 5 Amps

10 Amps

ES =

ES = 120 Volts

1200

10

Transformer Ratings Transformers are rated in kVA (kilovolt-amps). This rating is

used rather than watts because loads are not purely resistive.

Only resistive loads are measured in watts. The kVA rating

determines the current a transformer can deliver to its load

without overheating. Given volts and amps, kVA can be

calculated. Given kVA and volts, amps can be calculated.

kVA =

Volts x Amps

1000

Amps =

kVA x 1000

Volts

Using the illustrated step-down transformer, the kVA rating can

be calculated. The kVA rating of a transformer is the same for

both the primary and the secondary.

Primary kVA =

240 x 5

1000

Primary kVA = 1.2 kVA

Secondary kVA =

120 x 10

1000

Secondary kVA = 1.2 kVA

Transformer Losses Most of the electrical energy provided to the primary of a

transformer is transferred to the secondary. Some energy,

however, is lost in heat in the wiring or the core. Some losses in

the core can be reduced by building the core of a number of flat

sections called laminations.

80

Three-Phase Transformers

Delta Connections Three-phase transformers are used when three-phase power

is required for larger loads such as industrial motors. There

are two basic three-phase transformer connections, delta and

wye. Delta transformers are used where the distance from

the supply to the load is short. A delta is like three singlephase

transformers connected together. The secondary of a

delta transformer is illustrated below. For simplicity, only the

secondary of a three-phase transformer is shown. The voltages

shown on the illustration are secondary voltages available

to the load. Delta transformers are schematically drawn in a

triangle. The voltages across each winding of the delta triangle

represents one phase of a three phase system. The voltage is

always the same between any two wires. A single phase (L1 to

L2) can be used to supply single phase loads. All three phases

are used to supply three phase loads.

480 Volts 480 Volts

480 Volts

480 Volts

480 Volts

L1

L3

L2

L1 to L2 = 480 volts

L2 to L3 = 480 volts

L1 to L3 = 480 volts

81

Balanced Delta Current When current is the same in all three coils, it is said to be

balanced. In each phase, current has two paths to follow. For

example, current flowing from L1 to the connection point at

the top of the delta can flow down through one coil to L2, and

down through another coil to L3. When current is balanced, coil

current is 58% of the line current measured on each phase. If

the line current is 50 amps on each phase, coil current would be

29 amps.

29 Amps 29 Amps

29 Amps

50 Amps

50 Amps

50 Amps

L1

L3

L2

Unbalanced Delta Current When current is different in all three coils, it is unbalanced. The

following diagram depicts an unbalanced system.

Coil A

30 Amps

Coil B

20 Amps

Coil C

10 Amps

43.6 Amps L1

36 Amps L3

26.4 Amps L2

Though current is usually measured with an ammeter, line cur

rent of an unbalanced delta transformer can be calculated with

the following formulas:

IL1 = IA2 + IB2 + (IA x IB)

IL2 = IB2 + IC2 + (IB x IC)

IL3 = IA2 + IC2 + (IA x IC)

82

Wye Connections The wye connection is also known as a star connection. Three

transformers are connected to form a “Y” shape. The wye

transformer secondary, (shown below) has four leads, three

phase connectors, and one neutral. The voltage across any

phase (line-to-neutral) will always be less than the line-to-line

voltage. The line-to-line voltage is 1.732 times the line-to-neutral

voltage. In the circuit below, line-to-neutral voltage is 277 volts.

Line-to-line voltage will be 480 volts (277 x 1.732).

277 Volts

277 Volts

277 Volts

480 Volts

480 Volts

480 Volts

L1

L2

L3

N

Review 11

1. If the primary of a transformer has more turns than the

secondary, it is a ____________ transformer.

2. If the primary of a transformer has fewer turns than the

secondary, it is a ____________ transformer.

3. The secondary voltage of an iron-core transformer with

240 volts on the primary, 40 amps on the primary, and

20 amps on the secondary is ____________ volts.

4. A transformer with a 480 volt, 10 amp primary, and a

240 volt, 20 amp secondary will be rated for

____________ kVA.

5. A wye connected, three-phase transformer secondary,

with 240 volts line-to-line will have ____________ volts

line-to-neutral.

83

Review Answers

Review 1 1) electron (-), proton (+), neutron (neutral); 2) free electrons;

3) many; 4) a, c, e, g; 5) many, few.

Review 2 1) electrons; 2) negative; 3) positive; 4) repel, attract; 5) voltage;

6) b; 7) a.

Review 3 1)

I = E

R; 2) amps, volts, ohms; 3) .5 amps; 4) 45 Ω; 5) 2 amps;

6) 6 volts, 6 volts; 7) 20 volts, 80 volts.

Review 4 1) 5 Ω; 2) 5.45 Ω; 3) 3.33 Ω; 4) 12 volts; 5) 6 amps;

6) 2.4 amps, 1.6 amps.

Review 5 1) 12 Ω, 22 Ω; 2) 40 Ω, 13.33 Ω.

Review 6 1) power; 2) P = E x I; 3) 36 watts; 4) iron, north-south;

5)north, south; 6) left, thumb, lines of flux.

Review 7 1) sine wave; 2) 120 degrees; 3) one; 4) -129.9 volts;

5) 106.05 volts rms.

Review 8 1) 10 mh; 2) 2.5 mh; 3) 2.5 μF; 4) 25 μF.

Review 9 1) reactance; 2) impedance; 3) 3.14 Ω; 4) c; 5) b; 6) 318.5 Ω;

7) 11.18 Ω; 8) 6.4 Ω.

Review 10 1) resistive; 2) inductive; 3) capacitive; 4) 1198 Ω, .1 amp;

5) 84.9 milliamps,1414.2 Ω; 6) watts; 7) 4 watts;

Review 11 1) step-down; 2) step-up; 3) 480 volts; 4) 4.8 kVA;

5) 138.56 volts.

84

Final Exam

The final exam is intended to be a learning tool. The book

may be used during the exam. A tear-out answer sheet is

provided. After completing the test, mail the answer sheet in for

grading. A grade of 70% or better is passing. Upon successful

completion of the test a certificate will be issued.

Questions 1. A material that is a good insulator is

a. copper c. silver

b. aluminum d. rubber

2. A material with more protons than electrons has a

a. negative charge c. neutral charge

b. positive charge d. no charge

3. In a simple electric circuit with a 12 volt supply, and a

24 Ω resistor, current is

a. 2 amps c. 0.2 amps

b. 5 amps d. 0.5 amps

4. The total resistance in a series circuit containing three,

10 Ω, resistors is

a. 10 Ω c. 3.33 Ω

b. 30 Ω d. 100 Ω

5. In a 12 volt series circuit where R1=10 Ω, R2=20 Ω,

and R3=10 Ω, current flow through R2 is

a. 0.3 amps c. 0.25 amps

b. 0.5 amps d. 3.33 amps

85

6. In a circuit containing three 30 Ω resistors in parallel,

the total resistance is

a. 30 Ω c. 10 Ω

b. 90 Ω d. 0.1 Ω

7. The rate at which work is done is called

a. energy c. efficiency

b. power d. power factor

8. Power in a simple 12 volt, 4 amp series circuit is

a. 3 amps c. 48 amps

b. 3 watts d. 48 watts

9. The instantaneous voltage at 150 degrees of an AC sine

wave whose peak voltage is 480 volts is

a. 415.7 volts c. 240 volts

b. 480 volts d. 0 volts

10. The effective voltage of an AC sine wave whose peak

voltage is 480 volts is

a. 415.7 volts c. 480 volts

b. 339.4 volts d. 679 volts

11. The time constant of a series circuit with a 10 mh

inductor, and a 5 Ω resistor is

a. 2 milliseconds c. 2 microseconds

b. 2 seconds d. .5 seconds

12. The total inductance of a series circuit containing

three inductors with values of 10mh, 20 mh, and 40 mh

is

a. 5.7 pF c. 70 Ω

b. 5.7 mh d. 70 mh

13. The time constant for a series circuit with a 20 Ω

resistor and a 4 μF capacitor is

a. 80 microseconds c. 5 microseconds

b. 80 milliseconds d. 5 milliseconds

86

14. Total capacitance for a series circuit containing a 2 μF,

4 μF, and 8 μF capacitors is

a. 14 μF c. 1.14 μF

b. 0.875 μF d. 4 μF

15. Total opposition to current flow in an AC circuit that

contains both reactance and resistance is called

a. resistance c. impedance

b. reactance d. capacitance

16. In a 60 hertz circuit containing 5 millihenrys of

inductance, inductive reactance is

a. 1.884 Ω c. 0.0005 Ω

b. 1884 Ω d. 0.05 Ω

17. In a purely inductive circuit

a. current leads voltage by 90 degrees

b. current lags voltage by 90 degrees

c. current and voltage are in phase

d. current leads voltage by 30 degrees

18. In a series AC circuit with a 20 Ω resistor and a 10 Ω

inductive reactance, impedance is

a. 30 Ω c. 14.1 Ω

b. 10 Ω d. 22.4 Ω

19. A series AC circuit containing more capacitive reactance

than inductive reactance is

a. inductive c. capacitive

b. resistive d. in phase

20. An iron-core transformer with 120 volt, 10 amp primary,

and 5 amp secondary is a

a. step down transformer with a 60 volt secondary

b. step up transformer with a 240 volt secondary

c. step up transformer with a 480 volt secondary

d. step down transformer with a 30 volt secondary

87

88

quickSTEP Online Courses

quickSTEP online courses are available at

http://www.sea.siemens.com/step.

The quickSTEP training site is divided into three sections:

Courses, Downloads, and a Glossary. Online courses

include reviews, a final exam, the ability to print a certificate

of completion, and the opportunity to register in the Sales

& Distributor training database to maintain a record of your

accomplishments.

From this site the complete text of all STEP courses can be

downloaded in PDF format. These files contain the most recent

changes and updates to the STEP courses.

A unique feature of the quickSTEP site is our pictorial glossary.

The pictorial glossary can be accessed from anywhere within

a quickSTEP course. This enables the student to look up an

unfamiliar word without leaving the current work area.