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Network Theory II

by: Sidney Stehr
Sidney Stehr
GPA 3.51


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This 4 page Class Notes was uploaded by Sidney Stehr on Sunday October 25, 2015. The Class Notes belongs to ECGR 2112 at University of North Carolina - Charlotte taught by Staff in Fall. Since its upload, it has received 36 views. For similar materials see /class/229005/ecgr-2112-university-of-north-carolina-charlotte in Electronics and Computer Technology at University of North Carolina - Charlotte.

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Date Created: 10/25/15
EEZOl Respanse 0f the basic elements to AC Phas0r Notaticn Let s go back and look at the voltage wrt time in a RLC series circuit 479 Take for now as a g1ven W 2 34851745 7 7 9A i248sin377t00A The voltage is also sinusoidal of the 0 same frequency and a magnitude of 0 1H l7OVpeak but unknown phase 6 7 OS 1W 3 7 7 t V First f1nd R XL and XC 100 F R 470 given XL 60L 3771H 3770 XC IwC 1377 100X10396F 26 52 Next find the maximum voltages across the elements V12 MR 248A472 1166V VL IMXL 248A3772 935V VC IMXC 248A2652 657V If we simply add these up we get 2758V Clearly this would violate Kirchhoff s voltage law as we know it To help understand why let s plot these voltages We can now see as we noted earlier that the various voltages all have differing phases with respect to each other The problem we face is how to add sine waves of differing phases and frequencies Here are three options to consider 1 Graphically Use the plotted waveforms and graphically add them to get the resultant waveform Looking at the plot above we can easily check the validity of this method by testing a few points For example if we look at values of the component waves at t0 12R 0V VL max 935 Vand VC min 65 7V If we add these we see the result in the sum waveform 278V Also if we look at the point in time when 12 W 0V the sum vR and we can see that the curves cross at that point We could continue this type of Denard Lynch Page 1 0f 4 Oct 2009 EE201 Response of the basic elements to AC Phasor Notation process until we plotted the entire result but this would obviously be a very laborious and time consuming endeavour although it would work for any phase and any frequency 2 Mathematically Consider two waveforms V1sina1t 1 and V2sina2t 2 We can use the identities sinoc3 sinacos cosasin and sinoc B sinacos cosasin where we can let a ocand 3 and if we impose the restriction that 001 002 we can with some manipulation reduce the sum to the following expression VJcosq 1 V2cos 2sinwt V1sin 1 V2sin 2sinat900 and recalling that sinat900 cosaJt V1cos 1 V2cos 2sinwt V1sin 1 V2sin 2coswt Note that we have 2 terms dependent only on Vand both not time varying that are 900 to each other This method could reduce the work considerable although it is simplest if we restrict its use to sine waves of the same frequency but differing phase However a minor extension of this math leads to a very popular and simple method for dealing with AC sinusoids of the same frequency phasors 3 Phasor notation Charles Proteus Steinmetz is credited with developing this phasor notation method of representing AC sinusoidal waveforms and it has reduced the complexity of working with sinusoids of differing phases to almost that of dealing with DC values Phasor notation is a transformation of a sinusoidal timevarying signal into a complex twodimensional domain where combination is relatively trivial Works for any sinusoidally varying quantities of the same frequency It can be visualized as a counter clockwise rotating vector in a twodimensional complex plane How it works General transformation of a sinusoidal signal in the time domain to the phasor domain Time vt Asinc0tqV A A A Phasor V Ap V polar form cosp sinp rectangular form J5 J3 J5 Note phasors are conventionally shown as RMS values but the concept works equally well for maximum i e peak values Reverse transformation from the phasor domain back into the time domain Phasor V BZQV polar form or V x jyVrectangular form Time vt JEBsincot 9V or vt 1 x2 yz sincottan 1XDV x DenardLynch Page 2 of4 Oct 2009 EEZOl Respanse 0f the basic elements to AC Phas0r N0tati0n Of course in the reverse case there is know way of determining 0 Without additional information The Phasor Diagram Phasors are just vectors in a 2D complex plane A Phasor Diagram can be used to illustrate their relationship and addition An example Phasor Diagram A x a x a x a x r k k k J Le VZSlIlqh V1sinl1 A couple more things to complete our toolbox for AC circuits De ne one more neW term Impedance Z Which represents the opposition to flow in the phasor domain It is also measured in Ohms 2 and is de ned as h It 19 as0rv0 age or Z K phasor current I Note since Z or X is just a ratio it can be calcalatea from either RMS 0r Max values Consider the impedance Z of the basic ideal elements again using current as a reference Resistors V is in phase With I therefore 40 ZR V R4009 40 Inductors V leads I eLi V4 0 ZL 9 2949009 40 Capacitors V lags I iCe V4 O ZC 9 XCz 9OOQ 40 Denara Lynch Page 3 0f 4 Oct 2009 EEZOl Respanse 0f the basic elements to AC Phas0r Notatian Note that reactanceX and resistanceR are scalars and n0t complex While impedance is a complex vector Which can be represented on a 2D plane Z does n0t represent a time varying quantity like a phasor does Impedance Z can also be represented and added on a 2D plane called an Impedance Diagram similar to but not a Phasor Diagram Impedance Diagram 139 ZL ZTot a 7 ZR ZC Finally our favourite laws These are virtually identical to those for the DC world except we use phasors Ohms LaW for AC Circuits VIZandZ Kirchhoff s Laws for AC circuits KVL The Z phasor voltages around a loop O KCL The Z phasor currents into a node O Denard Lynch Page 4 0f 4 Oct 2009


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