1) Effective current in an AC circuit is 0.707 times the maximum current. Effective voltage is 0.707 times the maximum voltage. 2) Inductive reactance is directly proportional to frequency and inductance. Capacitive reactance is inversely proportional to frequency and capacitance. 3) Impedance is the total opposition to current flow in an AC circuit consisting of resistance and reactance. Power is consumed only by the resistive component of impedance and is proportional to the cosine of the phase angle.

2. Ronak S SutariyaRonak S Sutariya
Branch :Branch :ComputerComputer
Sub:Sub: Elements Of Electrical EngineeringElements Of Electrical Engineering
Enrollment No:Enrollment No: 151290107052151290107052
Topic :Topic : A.A. C CIRCUITSC CIRCUITS

3. θ
450
900
1350
1800
2700
3600
E
R = Emax
E = Emax sin θ
Rotating Vector DescriptionRotating Vector Description
The coordinate of the emf at any instant is the
value of Emax sin θ. Observe for incremental
angles in steps of 450
. Same is true for i.
The coordinate of the emf at any instant is the
value of Emax sin θ. Observe for incremental
angles in steps of 450
. Same is true for i.
θ
450
900
1350
1800
2700
3600
E
Radius = Emax
E = Emax sin θ

4. Effective AC CurrentEffective AC Current
iimaxmax
The average currentThe average current
in a cycle is zero—in a cycle is zero—
half + and half -.half + and half -.
But energy is expended,But energy is expended,
regardless of direction.regardless of direction.
So theSo the “root-mean-“root-mean-
square”square” value is useful.value is useful.
2
2 0.707
rms
I I
I = =
I = imax
TheThe rmsrms valuevalue IIrmsrms isis
sometimes called thesometimes called the
effectiveeffective currentcurrent IIeffeff::
The effective ac current:
ieff = 0.707 imax

5. AC DefinitionsAC Definitions
OneOne effective ampereeffective ampere is that ac current foris that ac current for
which the power is the same as for onewhich the power is the same as for one
ampere of dc current.ampere of dc current.
OneOne effective volteffective volt is that ac voltage thatis that ac voltage that
gives an effective ampere through agives an effective ampere through a
resistance of one ohm.resistance of one ohm.
Effective current: ieff = 0.707 imax
Effective current: ieff = 0.707 imax
Effective voltage: Veff = 0.707 Vmax
Effective voltage: Veff = 0.707 Vmax

6. Pure Resistance in AC CircuitsPure Resistance in AC Circuits
A
a.c. Source
R
V
Voltage and current are in phase, and Ohm’sVoltage and current are in phase, and Ohm’s
law applies for effective currents and voltages.law applies for effective currents and voltages.
Voltage and current are in phase, and Ohm’sVoltage and current are in phase, and Ohm’s
law applies for effective currents and voltages.law applies for effective currents and voltages.
Ohm’s law: Veff = ieffR
Vmax
iimaxmax
Voltage
Current

7. AC and InductorsAC and Inductors
Time, t
I
i
CurrentCurrent
RiseRise
τ
0.63I
Inductor
The voltageThe voltage VV peaks first, causing rapid rise inpeaks first, causing rapid rise in ii
current which then peaks as the emf goes to zero.current which then peaks as the emf goes to zero.
VoltageVoltage leadsleads ((peaks beforepeaks before) the current by 90) the current by 9000
..
Voltage and current are out of phaseVoltage and current are out of phase..
Time, t
I i
CurrentCurrent
DecayDecay
τ
0.37I
Inductor

8. A Pure Inductor in AC CircuitA Pure Inductor in AC Circuit
A
L
V
a.c.
Vmax
iimaxmax
Voltage
Current
The voltage peaks 90The voltage peaks 9000
before the current peaks.before the current peaks.
One builds as the other falls and vice versa.One builds as the other falls and vice versa.
The voltage peaks 90The voltage peaks 9000
before the current peaks.before the current peaks.
One builds as the other falls and vice versa.One builds as the other falls and vice versa.
TheThe reactancereactance may be defined as themay be defined as the nonresistivenonresistive
oppositionopposition to the flow of ac current.to the flow of ac current.

9. Inductive ReactanceInductive Reactance
A
L
V
a.c.
TheThe backback emfemf inducedinduced
by a changing currentby a changing current
provides opposition toprovides opposition to
current, calledcurrent, called inductiveinductive
reactance Xreactance XLL..
Such losses areSuch losses are temporarytemporary, however, since the, however, since the
currentcurrent changes directionchanges direction, periodically re-supplying, periodically re-supplying
energy so that no net power is lost in one cycle.energy so that no net power is lost in one cycle.
Inductive reactanceInductive reactance XXLL is a function of both theis a function of both the
inductanceinductance and theand the frequencyfrequency of the ac current.of the ac current.

10. Calculating Inductive ReactanceCalculating Inductive Reactance
A
L
V
a.c.
Inductive Reactance:
2 Unit is theLX fLπ= Ω
Ohm's law: L LV iX=
TheThe voltagevoltage readingreading VV in the above circuit at thein the above circuit at the
instant theinstant the acac current iscurrent is ii can be found from thecan be found from the
inductanceinductance inin HH and theand the frequencyfrequency inin HzHz..
(2 )LV i fLπ= Ohm’s law: VL = ieffXL

11. AC andAC and
CapacitanceCapacitance
Time, t
Qmax
q
Rise inRise in
ChargeCharge
Capacitor
τ
0.63 I
Time, t
I
i
CurrentCurrent
DecayDecay
Capacitor
τ
0.37 I
The voltageThe voltage VV peaks ¼ of a cycle after the currentpeaks ¼ of a cycle after the current
ii reaches its maximum. The voltagereaches its maximum. The voltage lagslags thethe
current.current. CurrentCurrent ii and V out of phaseand V out of phase..

12. A Pure Capacitor in ACA Pure Capacitor in AC
CircuitCircuit
Vmax
iimaxmax
Voltage
CurrentA V
a.c.
C
The voltage peaks 90The voltage peaks 9000
afterafter the current peaks.the current peaks.
One builds as the other falls and vice versa.One builds as the other falls and vice versa.
The voltage peaks 90The voltage peaks 9000
afterafter the current peaks.the current peaks.
One builds as the other falls and vice versa.One builds as the other falls and vice versa.
The diminishing currentThe diminishing current ii builds charge onbuilds charge on CC
which increases thewhich increases the back emfback emf ofof VVCC..
The diminishing currentThe diminishing current ii builds charge onbuilds charge on CC
which increases thewhich increases the back emfback emf ofof VVCC..

13. Capacitive ReactanceCapacitive Reactance
NoNo net powernet power is lost in a complete cycle, evenis lost in a complete cycle, even
though the capacitor does provide nonresistivethough the capacitor does provide nonresistive
opposition (opposition (reactancereactance) to the flow of ac current.) to the flow of ac current.
Capacitive reactanceCapacitive reactance XXCC is affected by both theis affected by both the
capacitancecapacitance and theand the frequencyfrequency of the ac current.of the ac current.
A V
a.c.
CEnergyEnergy gains andgains and
losses are alsolosses are also
temporarytemporary for capacitorsfor capacitors
due to the constantlydue to the constantly
changing ac current.changing ac current.

14. Calculating Inductive ReactanceCalculating Inductive Reactance
Capacitive Reactance:
1
Unit is the
2
CX
fCπ
= Ω
Ohm's law: VC CiX=
TheThe voltagevoltage readingreading VV in the above circuit at thein the above circuit at the
instant theinstant the acac current iscurrent is ii can be found from thecan be found from the
inductanceinductance inin FF and theand the frequencyfrequency inin HzHz..
2
L
i
V
fLπ
=
A V
a.c.
C
Ohm’s law: VC = ieffXC

15. Series LRC CircuitsSeries LRC Circuits
L
VR VC
CR
a.c.
VL
VT
A
Series ac circuit
Consider anConsider an inductorinductor LL,, aa capacitorcapacitor CC,, andand
aa resistorresistor RR all connected inall connected in seriesseries withwith anan
ac sourceac source. The instantaneous current and. The instantaneous current and
voltages can be measured with meters.voltages can be measured with meters.
Consider anConsider an inductorinductor LL,, aa capacitorcapacitor CC,, andand
aa resistorresistor RR all connected inall connected in seriesseries withwith anan
ac sourceac source. The instantaneous current and. The instantaneous current and
voltages can be measured with meters.voltages can be measured with meters.

16. Phase in a Series AC CircuitPhase in a Series AC Circuit
The voltageThe voltage leadsleads current in an inductor andcurrent in an inductor and lagslags
current in a capacitor.current in a capacitor. In phaseIn phase for resistancefor resistance RR..
θ
450
900
1350
1800
2700
3600
V V = Vmax sin θ
VR
VC
VL
RotatingRotating phasor diagramphasor diagram generates voltagegenerates voltage
waves for each elementwaves for each element RR,, LL, and, and CC showingshowing
phase relations. Currentphase relations. Current ii is alwaysis always in phasein phase withwith
VVR.R.

17. Impedance in an AC CircuitImpedance in an AC Circuit
φ
R
XL - XC
ZZ
ImpedanceImpedance 2 2
( )T L CV i R X X= + −
ImpedanceImpedance ZZ is defined:is defined:
2 2
( )L CZ R X X= + −
Ohm’s law for ac currentOhm’s law for ac current
and impedance:and impedance:
or T
T
V
V iZ i
Z
= =
The impedance is the combined opposition to ac
current consisting of both resistance and reactance.
The impedance is the combined opposition to ac
current consisting of both resistance and reactance.

18. Power in an AC CircuitPower in an AC Circuit
No power is consumed by inductance orNo power is consumed by inductance or
capacitance. Thus power is a function of thecapacitance. Thus power is a function of the
component of the impedance along resistance:component of the impedance along resistance:
No power is consumed by inductance orNo power is consumed by inductance or
capacitance. Thus power is a function of thecapacitance. Thus power is a function of the
component of the impedance along resistance:component of the impedance along resistance:
In terms of ac voltage:In terms of ac voltage:
P = iV cos φP = iV cos φ
In terms of the resistance R:In terms of the resistance R:
P = i2
RP = i2
R
φ
R
XL - XC
ZZ
ImpedanceImpedance
PP lost inlost in RR onlyonly
The fractionThe fraction CosCos φφ is known as theis known as the power factor.power factor.

19. The TransformerThe Transformer
AA transformertransformer is a device that uses inductionis a device that uses induction
and ac current to step voltages up or down.and ac current to step voltages up or down.
R
a.c.
Np Ns
Transformer
P PN
t
∆Φ
= −
∆
E S SN
t
∆Φ
= −
∆
EInduced
emf’s are:
Induced
emf’s are:
An ac source of emfAn ac source of emf
EEpp is connected tois connected to
primary coil withprimary coil with NNpp
turns. Secondary hasturns. Secondary has
NNss turns and emf ofturns and emf of EEss..
An ac source of emfAn ac source of emf
EEpp is connected tois connected to
primary coil withprimary coil with NNpp
turns. Secondary hasturns. Secondary has
NNss turns and emf ofturns and emf of EEss..

20. Transformers (Continued):Transformers (Continued):
R
a.c.
Np Ns
Transformer
P PN
t
∆Φ
= −
∆
E
S SN
t
∆Φ
= −
∆
E
Recognizing thatRecognizing that ∆φ∆φ//∆∆tt is the same in each coil,is the same in each coil,
we divide first relation by second and obtain:we divide first relation by second and obtain:
The transformer
equation:
The transformer
equation:
P P
S S
N
N
=
E
E

21. Transformer EfficiencyTransformer Efficiency
There is no power gain in stepping up the voltageThere is no power gain in stepping up the voltage
since voltage is increased by reducing current. Insince voltage is increased by reducing current. In
an ideal transformer with no internal losses:an ideal transformer with no internal losses:
or SP
P P S S
s P
i
i i
i
= =
E
E E
E
An idealAn ideal
transformer:transformer:
R
a.c.
Np Ns
Ideal Transformer
The above equation assumes no internal energyThe above equation assumes no internal energy
losses due to heat or flux changes.losses due to heat or flux changes. ActualActual
efficienciesefficiencies are usually betweenare usually between 90 and 100%.90 and 100%.
The above equation assumes no internal energyThe above equation assumes no internal energy
losses due to heat or flux changes.losses due to heat or flux changes. ActualActual
efficienciesefficiencies are usually betweenare usually between 90 and 100%.90 and 100%.

22. SummarySummary
Effective current: ieff = 0.707 imax
Effective current: ieff = 0.707 imax
Effective voltage: Veff = 0.707 Vmax
Effective voltage: Veff = 0.707 Vmax
Inductive Reactance:
2 Unit is theLX fLπ= Ω
Ohm's law: L LV iX=
Capacitive Reactance:
1
Unit is the
2
CX
fCπ
= Ω
Ohm's law: VC CiX=

23. Summary (Cont.)Summary (Cont.)
2 2
( )T R L CV V V V= + − tan L C
R
V V
V
φ
−
=
2 2
( )L CZ R X X= + −
or T
T
V
V iZ i
Z
= =
tan L CX X
R
φ
−
=
1
2
rf
LCπ
=

24. Summary (Cont.)Summary (Cont.)
In terms of ac voltage:In terms of ac voltage:
P = iV cos φP = iV cos φ
In terms of the resistance R:In terms of the resistance R:
P = i2
RP = i2
R
Power in AC Circuits:Power in AC Circuits:
P P
S S
N
N
=
E
E P P S Si i=E E
Transformers:Transformers:

25. THANK YOUTHANK YOU