AC Circuit Theory

AC Circuit
Theory
Prepared by Bill Hardy,TESCO
For North Carolina Electric Meter School
Tuesday, June 27, 2017 at 1:00 p.m.

A Little History
• 1800 Volta
– First electric battery
• 1830-31 Faraday and Henry
– Changing magnetic field can induce an electric
current. Build first very crude electric motors in lab.
• 1832 Pixii
– First crude generation of an AC current.
• 1856 Siemens
– First really practical electric motor
• 1860s Varley, Siemens and Wheatstone
– Each develop electric dynamos (DC Generators).
Slide 2

A Little History
• 1870s
– First electric railroad and street lights in Berlin (DC).
• 1880
– First electric elevator (DC).
• 1885-88 Thomson, Ferraris, Tesla
– Each develop AC electric induction motors.
– Tesla is granted a US patent for induction motor in
1888.
• 1890 Dolivo-Dobrovolsky
– First three phase generator, motor and transformer
Slide 3

A Little History
• Edison and Westinghouse
– Edison favored DC power distribution, Westinghouse
championed AC distribution.
– The first US commercial electric systems were
Edison’s DC systems.
• First AC system was in 1893 in Redlands, CA.
Developed by Almirian Decker it used 10,000
volt, three phase primary distribution.
• Siemens, Gauland and Steinmetz were other
pioneers.
Slide 4

War of the Currents
Thomas Edison George Westinghouse Nikola Tesla
Slide 5

AC Theory - History
• By 1900 AC power systems had won the
battle for power distribution.
– Transformers allowed more efficient
distribution of power over large areas.
– AC motors were cheaper and easier to build.
– AC electric generators were easier to build.
Slide 6

AC vs DC
• Direct Current (DC) – an electric current
that flows in one direction.(IEEE100)
• Alternating Current (AC) – an electric
current that reverses direction at regularly
recurring intervals of time. (IEEE100)
Slide 7

AC Circuits
• An AC circuit has three general characteristics
– Value
– Frequency
– Phase
• In the US, the household value is 120 Volts with
other common voltages being 208, 240, 277 and
480 Volts. The frequency is 60 Hertz (cycles per
second).
Slide 8

AC Theory – Sine Wave
)2sin(   ftVV pk
0
)2sin(2   ftVV rms
120rmsV
169pkV
Slide 9

AC Theory - Phase
Sine Wave
(15.000)
(10.000)
(5.000)
0.000
5.000
10.000
15.000
0 60 120 180 240 300 360 420 480 540 600 660 720
Degrees
Amplitude
)2sin(10 ftV  )302sin(10  ftI 
Here current LAGS voltage.
Slide 10

AC vs DC
• In DC theory we learned
– Ohm’s Law
• Voltage = Current x Resistance
• V = IR
– Power
• P = I2R = V2/R
• For AC we would like the same equations to apply.
– Specifically we want to be able to say that a DC
voltage of 10 Volts applied to a resistor of value R
produces the same power dissipation as an AC
voltage of 10 volts applied to the same resistor.
Slide 11

AC Theory – RMS
• For DC voltage to equal AC voltage we need
  dtftV
RR
Vdc
)2(sin
1 22
0
2

R
V
R
Vdc
2
2
0
2

DCVV 20 
Slide 12

AC Theory - RMS
)sin(68.169)sin(2120 ttV  
(200)
(150)
(100)
(50)
0
50
100
150
200
0 60 120 180 240 300 360 420 480 540 600 660 720
Amplitude
Degrees
Sine Wave
RMS Value
Slide 13

AC Theory – RMS
• So if we want to have the V in our equation for
an AC signal represent the same value as the its
DC counterpart we have
• By convention in AC theory we refer to VDC as
the RMS (Root Mean Squared) voltage.
• When we talk about AC values we always mean
the RMS value not the peak value unless we say
so specifically
)2sin(2)(   ftVtV DC
Slide 14

AC vs DC
ACDC
RR VrmsVdc
Idc Irms
V = IR
P = VI = I2R = V2/R
Slide 15

AC Theory – Resistive Load
Sine Wave
-200
-150
-100
-50
0
50
100
150
200
0 60 120 180 240 300 360 420 480 540 600 660 720
Degrees
Amplitude
AC
RVrms
Irms
Resistors are measured in Ohms. When an AC voltage is applied to a resistor, the
current is in degrees. A resistive load is considered a “linear” load because when
the voltage is sinusoidal the current is sinusoidal.
Slide 16

AC Theory – Inductive Load
Sine Wave
-200
-150
-100
-50
0
50
100
150
200
0 60 120 180 240 300 360 420 480 540 600 660 720
Degrees
Amplitude
AC
LVrms
Irms
Inductors are measured in Henrys. When an AC voltage is applied to an inductor,
the current is 90 degrees out of phase. We say the current “lags” the voltage. A
inductive load is considered a “linear” load because when the voltage is sinusoidal
the current is sinusoidal.
Slide 17

AC Theory – Capacitive Load
AC CVrms
IrmsSine Wave
-200
-150
-100
-50
0
50
100
150
200
0 60 120 180 240 300 360 420 480 540 600 660 720
Degrees
Amplitude
Capacitors are measured in Farads. When an AC voltage is applied to a capacitor,
the current is 90 degrees out of phase. We say the current “leads” the voltage. A
capacitive load is considered a “linear” load because when the voltage is
sinusoidal the current is sinusoidal.
Slide 18

AC Theory – Active Power
• Active Power is defined as P = VI
• Power is a rate, i.e. Energy per unit time.
• The Watt is the unit for Power
– 1 Watt = 1000 Joules/sec
• Since the voltage and current at every point in
time for an AC signal is different, we have to
distinguish between instantaneous power and
average power.
• Generally when we say “power” we mean
average power.
Slide 19

AC Theory – Energy
• Energy is power integrated over a period of time.
• The units of Energy are:
– Watt-Hour (abbreviated Wh)
– Kilowatt-Hour (abbreviated kWh)
• A Wh is the total energy consumed when a load
draws one Watt for one hour.
Slide 20

AC Theory – Instantaneous Power
Sine Wave
(200)
(150)
(100)
(50)
0
50
100
150
200
0 60 120 180 240 300 360 420 480 540 600 660 720
Degrees
Amplitude
(25000)
(20000)
(15000)
(10000)
(5000)
0
5000
10000
15000
20000
25000
WATTS
)2sin(2120 ftV  )2sin(296 ftI  )2(sin23040 2
ftP 
For a resistive load: ))2cos(1()(sin2 2
tVItVIvip  
P = 11520 Watts
Slide 21

Sine Wave
-200
-150
-100
-50
0
50
100
150
200
0 60 120 180 240 300 360 420 480 540 600 660 720
Degrees
Amplitude
-15000
-10000
-5000
0
5000
10000
15000
)2sin(2120 ftV  )902sin(296  ftI  )2sin(11520 ftP 
For an inductive load:
P = 0 Watts
)2sin()90sin()sin(2 tVIttVIvip  
Slide 22

Sine Wave
-200
-150
-100
-50
0
50
100
150
200
0 60 120 180 240 300 360 420 480 540 600 660 720
Degrees
Amplitude
-15000
-10000
-5000
0
5000
10000
15000
)2sin(2120 ftV  )902sin(296  ftI  )2sin(11520 ftP 
For an capacitive load:
P = 0 Watts
)2()90()(2 tVISintSintVISinvip  
Slide 23

AC Theory – Complex Circuits
• Impedance – The equivalent to the concept of resistance
for an AC circuit. It is also measured in Ohms.
Designated by the symbol X.
• In AC circuits non-resistive impedance affects both the
amplitude and phase of the current.
• A resistor R has an impedance which is frequency
independent. There is no phase shift.
• An inductor has an impedance which is proportional the
frequency, XL = 2πfL. The phase is shifted by 90
degrees lagging.
• A capacitor has an impedance which is inversely
proportional the frequency, XC = 1/2πfC. The phase is
shifted by 90 degrees leading.
Slide 24

AC Theory – Complex Circuits
AC
C
Vrms
Irms
R
L
22
)
1
(
C
LR
V
I

 

Amplitude (Current)












R
C
L )
1
(
arctan 


Phase (Current)
VC
V
VL
VC
VR
Slide 25

Sine Wave
(200)
(150)
(100)
(50)
0
50
100
150
200
0 60 120 180 240 300 360 420 480 540 600 660 720
Degrees
Amplitude
(25000)
(20000)
(15000)
(10000)
(5000)
0
5000
10000
15000
20000
25000
WATTS
)2sin(2120 ftV  )602sin(296  ftI 
)604cos(2304019953))604cos()60(cos(23040  ftftVIP 
Slide 26

Time Out for Trig
(Right Triangles)
c
a
)cos(
a
b
)tan(
a
c
b
90°
The Right Triangle:
The Pythagorean theory
c2 = a2 + b2
c
b
)sin(
Slide 27

AC Theory – Power Triangle
(Sinusoidal Waveforms)
If V = sin(ωt) and I = sin(ωt - θ) (and the load is linear)
then
Active Power = VIcos(θ) Watts
Reactive Power = VIsin(θ) VARs
Apparent Power = VI VA
Power Factor = Active/Apparent = cos(θ)
Watts
VARsSlide 28

Harmonics
Curse of the Modern World
• Every thing discussed so far was based on
“Linear” loads.
– For linear loads the current is always a simple
sine wave. Everything we have discussed is
true.
• For nearly a century after AC power was in
use ALL loads were linear.
• Today, many loads are NON-LINEAR.
Slide 29

Harmonic Load Waveforms
ANSI C12.20 now addresses harmonic waveforms as well
as sinusoidal.
-400
-300
-200
-100
0
100
200
300
400
-150
-100
-50
0
50
100
150
0 90 180 270 360
-200
-150
-100
-50
0
50
100
150
200
-150
-100
-50
0
50
100
150
0 90 180 270 360
-8
-6
-4
-2
0
2
4
6
8
-200
-150
-100
-50
0
50
100
150
200
0 90 180 270 360
-150
-100
-50
0
50
100
150
0 60 120 180 240 300 360
Test waveforms being proposed for ANSI C12.20.
-250
-200
-150
-100
-50
0
50
100
150
200
250
-150
-100
-50
0
50
100
150
0.00 100.00 200.00 300.00
-150
-100
-50
0
50
100
150
-150
-100
-50
0
50
100
150
0.00 100.00 200.00 300.00
Current
Voltage
Slide 30

AC Theory - Phasors
• An easier way to view AC data
Slide 31
CURRENTVOLTAGE
A
B
C
Va 0°
I a 0°
Vb 120°
I b 120°
Vc 240°
I c 240°
THE VECTOR
DIAGRAM

AC Theory - Phasors
• The length of the phasor is
proportional to the value of the
quantity
• The angle of the phasor (by
convention phase A is drawn
as horizontal) shows the phase
of the quantity relative to
phase A voltage.
• Here the current “lags” the
voltage by 30 degrees.
)302sin(25.2  ftI 
)02sin(2120  ftV 
Slide 32
CURRENTVOLTAGE
A
Va 0°
I a 30°
THE VECTOR
DIAGRAM

AC Theory - Phasors
Phasors are particularly useful in poly-phase situations
Slide 33
CURRENTVOLTAGE
A
B
C
Va 0°
I a 12°
Vb 120°
I b 126°
Vc 240°
I c 247°
THE VECTOR
DIAGRAM

New Energy Definitions
• At the moment there is no non-sinusoidal
definition for VA
• New ANSI Standard coming very soon
C12.31
Slide 34

New Definitions
RMS Voltage
Slide 35
Basic Definition
Time Domain
Frequency Domain
Waveform

New Definitions
RMS Current
Slide 36
Basic Definition
Time Domain
Frequency Domain
Waveform

New Definitions
Active Power
Slide 37
Basic Definition
Time Domain
Frequency Domain

New Definitions
Apparent Power
Slide 38
Basic Definition
Time Domain
Frequency Domain

Questions and Discussion
Slide 39
Bill Hardy, CTO
TESCO – The Eastern Specialty Company
Bristol, PA
865-279-1090 (cell)
215-785-2338 (office)
This presentation can also be found under Meter
Conferences and Schools on the TESCO website:
www.tesco-advent.com

AC Circuit Theory

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