Chapter 9: Sinusoids and Phasor

1 Chapter 9: Sinusoids and Relationships for circuit and s Laws in the Frequency Application: Summary1 How to determine v(t)and i(t)?2vs(t) = 10 VHow can we apply what we have learned before to determine i(t)and v(t)? Motivation (1)DC Motivation (2) Theoutputofthecircuitwillconsistoftwopar ts: atransientpartthatdiesoutastimeincreases ; ,thetransientpartdiesoutquickly,perhapsi na coupleofmilliseconds. ACcircuitsarethesubjectofthischapter. Inparticular, It'susefultoassociatea complexnumberwitha Usingphasorsandimpedances,weobtainanewre presentationofthelinearcircuit,calledthe frequency-domainrepresentation. Sinusoids (1) A sinusoidis a signal that has the form of the sine or cosine function.

2 A general expression for the sinusoid,whereVm= the amplitudeof the sinusoid = the angular frequency in radians/s t = the argument of the sinusoid= the phase4)sin()( +=tVtvm2period: T = A periodic functionis one that satisfies v(t)= v(t + nT), for all tand for all integers n. Sinusoids (2)5 HzTf1=f 2= Only two sinusoidal values with the same frequency can be compared by their amplitude and phase difference. If phase difference is zero, they are in phase; if phase difference is not zero, they are out of v1by or v1lags v2by Sinusoids (3)6 Example:Given a sinusoid, ,calculate its amplitude, phase, angular frequency, period, and :Amplitude = 5, phase = 60o, angular frequency = 4 rad/s, period = s, frequency = 2 (460 )t Example:Find the phase angle between and , does i1lead or lag i2?

3 25377sin(41oti+ =)40377cos(52oti =Solution:Since sin( t+90o) = cos ti1leads i2by 155o. Phasor (1) A phasoris a complex number that represents the amplitudeand phaseof a sinusoid. where Iis called a Phasor . Phasors may be used when the circuit is linear, the steady-stateresponse is sought, and all independent sources are sinusoidal and have the same frequency. It can be represented in one of the ( )Re{} j jtitI e e = mmjIeI == I =rz jrez=)sin(cos jrjyxz+=+=where 22r xy= +1tanyx =Mathematic operation of complex number: s identity8)()(212121yyjxxzz+++=+)()(21212 1yyjxxzz + = 212121 + =rrzz212121 =rrzz =rz112 =rz jrerjyxz = = = sincosjej = Phasor (2)9 Phasor (3) Transform a sinusoid from the time domain to the Phasor domain:(time domain) ( Phasor domain)10)cos()( +=tVtvmmV = V Amplitude and phase difference are two principal concerns in the study of voltage and current Sinusoids .)

4 Phasorwill be defined from the cosine function in all our proceeding study. If a voltage or current expression is in the form of a sine, it will be changed to a cosine by subtracting from the Phasor (4)The differences between v(t)and V: v(t)is instantaneous or time-domainrepresentationVis the frequency or Phasor -domain representation. v(t)is time dependent, Vis not. v(t)is always real with no complex term, Vis generally : Phasoranalysis applies only when frequency is constant; when it is applied to two or more sinusoid signals only if they have the same frequency. 11)(tvV = Vdtdvj V vdtj VLaplace transform[ ]0()( )()stLftFsfte dt == No initial set s = j Phasor (5)Example: Use Phasor approach, determine the current i(t)in a circuit described by the integro-differential : 12 += +)752cos(50384tdtdiidti121250; ; === = Phasor (6) We can derive the differential equations for the following circuit in order to solve for vo(t)in phase domain Vo.

5 154sin(340020350022ootvdtdvdtvd =++13 However, the derivation may sometimes be very tedious. Instead of first deriving the differential equation and then transforming it into phasorto solve for Vo,we can transform all the RLCcomponents into phasorfirst, then apply the KCLlaws and other theorems to set up a phasorequation involving Phasor (7) PhasorRelationships for circuit Elements (1)14 Resistor:Inductor:Capacitor:Vleads Iby 90 Ileads Vby 90 PhasorRelationships for circuit Elements (2)Example: If voltage v(t) = 6 cos(100t 30o) is applied to a 50 F capacitor, calculate the current, i(t), through the :i(t) = 30 cos(100t+ 60o) Impedance and Admittance (1)17 The impedance Zof a circuit is the ratio of the phasorvoltage Vto the phasorcurrent I,measured in ohms.)

6 Where R= Re(Z) is the resistance and X= Im(Z) is the reactance. Positive Xis for L(or lagging) and negative Xis for C(or leading). The admittance Yis the reciprocal of impedance, Unit: siemens(S)RjX= = +VZI1 GjB= = = + Impedance and Admittance (2)18 AfterweknowhowtoconvertRLCcomponentsfrom timetophasordomain,wecantransforma timedomaincircuitintoaphasor/frequencydo maincircuit. Hence, : Determine v(t)and i(t).5 cos(10 ) Vsvt=Answers:i(t)= cos(10t ) A; v(t)= cos(10t + ) Kirchhoff s Laws in the Freq. Domain(1)19 BothKVLandKCLareholdinthephasordomainorm orecommonlycalledfrequencydomain. Moreover,thevariablestobehandledarephaso rs, whicharecomplexnumbers. frequencycomplex Kirchhoff s Laws in the Freq.

7 Domain(2) Impedance Combinations (1)21 The following principles used for DC circuit analysis all apply to AC circuit . For transformation 22 Example: Determine the input impedance at = 10 rad/s. Answer:Zin = Example: Find the input impedance at = 50 rad/s. Impedance Combinations (2) Application: Phase-Shifters Series RC shift circuits:- Leading output -Lagging output Withthepervasiveuseofacelectricpowerinth ehomeandindustry,itisimportantforenginee rstoanalyzecircuitswithsinusoidalindepen dentsources. Thesteady-stateresponseofa linearcircuittoa sinusoidalinputis itselfa sinusoidhavingthesamefrequencyastheinput signal.

8 Circuitsthatcontaininductorsandcapacitor sarerepresentedbydifferentialequations. Whentheinputtothecircuitis sinusoidal,thephasorsandimpedancescanbeu sedtorepresentthecircuitinthefrequencydo main. Inthefrequencydomain,thecircuitis representedbyalgebraicequations. Thesteady-stateresponseofa linearcircuitwitha sinusoidalinputis , Summary (1) ,forexample, ,usingphasors. A circuitcontainsseveralsinusoidalsources, twocases: Whenallofthesinusoidalsourceshavethesame frequency,theresponsewillbea sinusoidwiththatfrequency,andtheproblemc anbesolvedinthesamewaythatitwouldbeifthe rewasonlyonesource. Whenthesinusoidalsourceshavedifferentfre quencies,superpositionis usedtobreakthetime-domaincircuitupintose veralcircuits,eachwithsinusoidalinputsal latthesamefrequency.

9 Eachoftheseparatecircuitsis Summary (2)