Alternating Current Circuits
The difference between Direct Current (D.C.) and Alternating Current (A.C.) is that current in a
D.C. circuit does not vary and only flows in one direction. D.C. circuits are at their peak value (+)
or at their zero point (0) and do not vary in between. They could be compared to a light switch that
is either completely on, or off.
Direct Current (D.C.)
A.C. circuits flow in two directions. Alternating first from negative to positive, then positive to
negative. An A.C. circuit starts at a zero point (0), gradually reaches a maximum (+) peak value,
then drops gradually past the zero point (0) until reaching (-) peak, then gradually completes the
cycle by returning to zero. In the United States electricity is generated normally at 60 of these
cycles (hertz) per second.
Alternating Current (A.C.)
Because of its uniqueness an A.C. circuit voltage has to be measured differently. Peak voltage of a
typical A.C. circuit is approximately 170 volts. But, it’s only at that maximum point for a brief
moment, most of the time the voltage is somewhere between 0 and 170. The Code defines voltage
as the “greatest root-mean square difference of potential.” This is commonly referred to as the
“effective voltage” which is about 70.7% of the measured peak voltage, or in this case 170 x .707 =
120 volts. We commonly use “effective voltage” as a measurement of voltage on all circuits.
4-1
A.C. Resistance
When A.C. or D.C. current flows through a type of resistor, like a heating element, electrical energy
is converted to heat. In fact, all of the ohms law formulas can be used to express wattage, voltage
and amps in simple resistance circuits.
This works fine if all we want to do is turn on an incandescent light bulb or cook food on a broiler.
But, what if we want to ring a doorbell, use a motor, or a transformer. These devices require coils
which cause electrical energy to be converted into mechanical energy in the form of work. A
common term we use for a coil of wire is an inductor.
Inductors
An inductor is coil of wire in a circuit which first causes a build up then a delay in the release of
current in a circuit. Even though inductance is a physical characteristic of a conductor wound into
a coil it is often defined in terms of its effect on the flow of current.
Inductance is defined as that property of an electric circuit that tends to oppose any change
of current through a circuit.
As currents rise in a coil, the magnetic field around the conductor produces a counter voltage that
opposes the current flow causing it to rise slowly before being released. This type of opposition to
current flow causes current to lag behind voltage producing a type of resistance called inductivereactance (XL).
Although there are drawbacks to using inductors in a circuit they are, for the most part, very
useful devices. When a coil (inductor) is looped around a metal post it's called an electromagnet.
This can be useful in picking up large metal objects or ringing a doorbell. Transformers use mutual
induction to transfer electricity from one coil of wire to another coil of wire through one common
iron core at different voltages and ampacities. Motors and generators induce magnetic fields
through coils to produce mechanical motion. Inductance is measured in henrys (L).
Capacitors
Capacitors or Condensers (C) are used in an electrical circuit to store and then release a voltage, or
difference of potential. Some circuits require capacitors to accomplish work.
4-2
They are used in electric discharge lighting to attain a high voltage 60 cycle pulse which ignites a
gas in the fluorescent bulb. They are also used in air-conditioners as booster starters. A capacitor is
made up of two conductive plates with an insulative material in between. A potential builds in the
conductive plates while separated until a peak voltage is reached. Then, either the voltage
automatically breaks through the barrier or the charge is released through a mechanical switch. In
either case voltages much higher than normally available are attained for short periods of time.
Capacitor circuit voltage tends to lag behind its current causing a type of resistance called
capacitive-reactance (XC). You may have experienced reactance in a circuit when you get a sudden
annoying buzzing noise when you turn on a fluorescent light while listening to the radio.
Capacitance is measured in farads (C).
Impedance
The difference between a resistive circuit and an inductive, or capacitive, circuit is that the circuit
changes when a voltage is applied. In an inductive circuit the voltage leads and current lags
behind. In a capacitive circuit current leads and voltage lags behind.
This is best remembered in the saying "ELI the ICE man" where old ELI stands for...
Voltage (E) leads, in Inductance (L), the Current (I)
and what ELI does for a living, he's the ICE man...
Current (I) leads, in Capacitance (C), the Voltage (E)
Impedance (Z), on the other hand, is the total opposition to current flow of an AC circuit. This
includes the Inductive-Reactance (XL), Capacitive-Reactance (XC), and Resistance (R) of a circuit.
Impedance, in this case “Z” is normally measured in ohms and can be expressed in the
following formula...
For a circuit containing resistance, inductance, and capacitance...
Z =
√R
2
+ ( XL - XC )2
4-3
For a circuit containing resistance and inductance...
√R
Z =
2
+ XL 2
2
+ XC 2
For a circuit containing resistance and capacitance...
√R
Z =
For a circuit containing only resistance...
Z=R
For a circuit containing only inductance...
Z = XL
For a circuit containing only capacitance...
Z = XC
Formulas for voltage, current, and impedance...
E = I x Z
I = E
Z
Z= E
I
Example: Calculate the Impedance (Z) of a circuit with the following.....
!
!
!
R = 30
L = 20
C = 10
"
Z =
√ R2 + ( XL - XC )2
"
Z =
√ 302 + (20 + 10)2
"
Z =
√ 900 + (900)
= √1800 = 42.4 Ω
4-4
Phase Angle
In D.C. circuits resistance (R) and voltage (E) are in phase. In other words, they both reach their
peaks and zero points at the same time.
In A.C. circuits voltages and currents have both magnitude and direction. Inductive- reactance (XL)
and capacitive-reactance (XC) are measured in ohms but their directions are opposite. Inductivereactance (XL) lags resistance by 90 ̊ while capacitive-reactance (XC) leads resistance by 90 ̊.
Inductance produces opposition to the flow of current and also makes current lag behind voltage
in a circuit. In a purely inductive circuit (one without resistance) the current would lag the voltage
by 90 degrees (1/4 cycle) behind the voltage. The term that we commonly use for this is “out of
phase.” In actual circuits containing resistance and inductance the current will lag, or be out of
phase, between 0 and 90 degrees behind the voltage. The angle of lag of a current in an inductive
circuit can be calculated from this formula...
"
"
"
"
"
"
"
"
"
"
"
"
Tangent Of Angle Of Lag = XL
"
"
"
"
"
R
Capacitance produces opposition to voltage and tends to make it lag behind the current of a
circuit. In a purely capacitive circuit (one without resistance) the voltage would lag the current by
90 degrees (1/4 cycle) behind the current. The term that we commonly use for this is “out of
phase.” In actual circuits containing resistance and capacitance the voltage will lag, or be out of
phase, 0 to 90 degrees behind the current. The angle of lag of a current in a capacitive circuit can be
calculated from this formula...
"
"
"
"
"
"
"
"
"
"
"
"
Tangent Of Angle Of Lag = XC
"
"
"
"
"
R
4-5
Cosine Of The Angle
Cosines for different angles is the trigonomic equivalent of what’s called “Power Factor.” The
symbol Ø, the Greek letter theta, is often used to designate the angle of lag, or lead of a circuit.
Hence power factor is sometimes referred to a cosine Ø (cosine theta) meaning the cosine of the
angle Ø.
The phase angle of a circuit containing resistance, inductance, and capacitance can be found using
the following formula...
"
"
"
"
"
"
Cosine Of The Phase Angle = " R = Power Factor
"
"
"
"
"
"
"
Z
I2R Losses
Wasted internal work presents itself as heat and thus reduces the work performed by a discharge
light or motor. The loss due to this wasted work can be expressed by this formula...
W = I2 x R
Where (W) = the loss in watts, (I) = the current in amps on the conductor, and (R) = the resistance
in ohms. It requires much larger equipment and conductors to deliver the a certain amount of
power at a low power factor than at a high power factor (close to 1).
In industrial plants a low power factor is usually due to underloaded induction motors because the
power factor of motors is much less at partial loads than at full loads. This can be corrected by
installing smaller motors, capacitive discharge lighting, or by installing capacitor banks to offset
the inductive effects of motors.
Power Factor
Power in an A.C. circuit is, as we have learned, is voltage times current (W = E x I) when only
resistance is available. When inductance and capacitance is available we have to take it into
account. To do that we include Power Factor in the formula...
W = E x I x PF
Power factor is the ratio of watts, commonly called “true power” to the volt-amps (vA), commonly
called “apparent power” of an A.C. circuit. True power is what is actually consumed in a circuit.
Apparent power is what is available in the circuit. We usually express this difference as a
percentage, or decimal point. In other words, a power factor of .8 (or 80%). Thus, a power factor
of .8 means that the current and voltage is out of phase, meaning that only 80% of the wattage will
be available. Or, there will be a 20% loss in wattage in the circuit. The highest power factor possible
is 1, or 100%. Wattmeters are used to measure real power.
4-6
Here’s the formulas commonly used to find power factor, watts, and volt-amps
Power Factor =
Watts
Volt-Amps
Watts = Volt-Amps x Power Factor
Volt-Amps =
Watts
Power Factor
Types Of Loads (Power Factors)
Typical Power Factors
Minimum
Maximum
Incandescent Lights
1.0
1.0
Low-Voltage
Transformers
0.96
0.98
Welding Transformers
0.50
0.70
Sodium-Vapor Lights
0.80
0.85
Flourescent Lights
0.50
0.95
Mercury-Vapor Lights
0.50
0.95
Small Motor (under 1 hp)
0.55
0.75
Large Motor (1-10 hp)
0.75
0.86
3Ø Motor (5-20 hp)
0.80
0.89
3Ø Motor (20 -100 hp)
0.82
0.90
4-7
Here’s a simplified way of determining the Power Factor (P.F.) formulas above...
Just put your thumb on the value you are looking for.
Example #1: How much power is consumed in a circuit which operates at 115 volts, draws 8
amperes and has a power factor of 80% ?
Solution:"
Watts = Volt-Amps x Power Factor
"
"
"
Watts = 115 volts x 8 amps (W=E x I) x .8 (80%)
"
"
"
Watts = 736
Example #2: Measurements were taken in an A.C. circuit and the current flowing was 20 amps, 120
volts, and 1800 watts. What is the power factor of this circuit ?
Solution:" "
"
"
"
"
"
"
"
"
"
Power Factor =
"
"
"
"
Watts
Volt-Amps
"
"
"
"
"
"
"
"
"
"
"
"
Power Factor =
"
"
"
"
115v x 8a
1800w
"
"
"
"
"
"
Power Factor ="
4-8
.75 (74%)
Capacitance Problems
1.!The phenomenon whereby a circuit stores electrical energy is called ____.
! (a) !inductance !
! (b) capacitance!
(c)! resistance
(d)!susceptance
2.!A capacitor (condenser) opposes ____.
! (a)!change in voltage and current !
! (b)!change in voltage!
(c)! change in current
(d)!none of these
3.!A ____ stores electrical energy in as a spring stores mechanical energy.
! (a)!resistor!
! (b)!coil!
(c)! capacitor
(d)!inductor
4.!Capacitance is measured in ____.
! (a)!ohms !
! (b)!volts!
(c)! farads
(d)!henrys
5.!An electrical capacitor (condenser) is best defined as a ____.
!
!
!
!
(a)!coil of wire !
(b)!wrapping of layers of metal foil !
(c)! coil of wire with layers of metal foil
(d)!wrapping of many layers of metal foil insulated by waxed paper
Inductance Problems
1.!Inductance is measured in ____.
! (a)!farads !
! (b)!henrys!
(c)! coulombs
(d)!amperes
2.!Inductance in a circuit ____.
! (a)!delays the change in current!
! (b)!prevents current from changing!
(c)! causes power loss
(d)!causes current to lead voltage
3.!The current will lag the voltage when ____ is present in the circuit.
! (a)!capacitance!
! (b)!inductance!
(c)! reluctance
(d)!resistance
4-9
4.!Pure XL in a circuit will cause ____ degree lag of the ampere curve to the voltage curve in an
! A.C. circuit.
! (a)!90 ̊!
! (b)!120 ̊!
(c)! 180 ̊
(d)!240
Reactance Problems
1.!Inductive reactance is measured in ____.
! (a)!farads!
! (b)!henrys!
(c)! ohms
(d)!coulombs
2.!Reactance will cause the current in a circuit to vary only when ____.
! (a)!A.C. current flows!
! (b)!D.C. current flows!
(c)! there is no resistance in the circuit
(d)!there is resistance in the circuit
3.!If frequency is constant, the inductive reactance of a circuit will ____.
!
!
!
!
(a)!remain constant regardless of voltage or current change
(b)!vary with voltage
(c)! vary directly with current
(d)!not effect the impedance
4.!When the power factor in a given circuit is unity, the reactive power is ____.
! (a)!at maximum!
! (b)!1.1414!
(c)! zero
(d)!a negative quantity
Impedance Problems
1.!The total opposition to the flow of alternating circuit is called ____.
! (a)!resistance (#) !
! (b)!impedance (Z)!
(c)! inductance (L)
(d)!capacitance (C)
2.!Impedance is measured in ____.
! (a)!farads !
! (b)!henrys!
(c)! ohms
(d)!coulombs
3.!Impedance is present in the following type circuit ?
! (a)!resistance only!
! (b)!A.C. only!
(c)! D.C. only
(d)!both A.C. and D.C.
4-10
4.!Which of the following would cause the most power to be dissipated in the form of heat ?
! (a)!inductive reactance!
! (b)!capacitive reactance!
(c)! resonance
(d)!resistance
Power Factor Problems
1.!A wattmeter indicates ____.
! (a)!true power !
! (b)!apparent power (PF not at unity)!
(c)! power factor
(d)!volt-amps
2.!When current is in phase with the voltage the power factor would be at unity, or ____.
! (a)!0!
! (b)!.1!
(c)! 1
(d)!100
3.!The power factor of an incandescent light bulb would be ____.
! (a)!unity!
! (b)!0.7 leading!
(c)! 0.7 lagging
(d)!zero
4. A low power factor is commonly caused by all of the following except ?
! (a)!induction motors!
! (b)!synchronous motors!
(c)! fluorescent lights
(d)!none of these
5.!A transformer is more efficiently utilized when the load has a ____ power factor.
! (a)!low!
! (b) medium!
(c)! average
(d)!high
6.!If a power factor meter is not available the power factor of a load can be measured using
! a ____.
! (a)!voltmeter!
! (b)!ammeter!
(c)! wattmeter
(d)!all of these
4-11