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A.c circuits
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.
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A.c circuits
- 2. Ronak S SutariyaRonakS 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 CurrentEffectiveAC 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 OneOneeffective 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 inAC 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 InductorsACand 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 Inductorin 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. TheThebackback 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 ReactanceCalculatingInductive 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 Capacitorin 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 NoNonet 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 ReactanceCalculatingInductive 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 CircuitsSeriesLRC 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 aSeries 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 anAC 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 anAC 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 AAtransformertransformer 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. NpNs 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 Thereis 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.) 22 ( )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.) Interms 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:
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