The resistance of a capacitor decreases The resistance of an inductor increases in .NET framework
The resistance of a capacitor decreases The resistance of an inductor increases Code 128B Encoder In Visual Studio .NET Using Barcode generation for VS .NET Control to generate, create Code128 image in .NET applications. Code 128A Reader In .NET Framework Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET applications. The Phasor Transform
Print Barcode In .NET Framework Using Barcode maker for .NET Control to generate, create bar code image in Visual Studio .NET applications. Bar Code Scanner In .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in VS .NET applications. A phasor is a complex representation of a phaseshifted sine wave If f (t) = A cos( t + ) Then the phasor is given by F= A (732) (731) Code 128C Drawer In Visual C#.NET Using Barcode creator for .NET framework Control to generate, create Code 128 Code Set A image in .NET applications. Code128 Printer In .NET Using Barcode drawer for ASP.NET Control to generate, create Code 128A image in ASP.NET applications. The Phasor Transform
Print Code 128C In VB.NET Using Barcode printer for .NET framework Control to generate, create Code 128A image in Visual Studio .NET applications. Bar Code Drawer In VS .NET Using Barcode generation for .NET framework Control to generate, create bar code image in VS .NET applications. To see how this works, we begin by considering Euler s identity (714) Since e j( t+ ) = cos( t + ) + j sin( t + ), we can take (731) to be the real part of this expression, that is f (t) = Re[Ae j( t+ ) ] (733) Code 128 Code Set B Encoder In .NET Framework Using Barcode generation for .NET Control to generate, create Code128 image in Visual Studio .NET applications. Printing GS1 RSS In Visual Studio .NET Using Barcode generation for Visual Studio .NET Control to generate, create GS1 DataBar Limited image in .NET applications. If a source is given as a sine wave, we can always rewrite it as a cosine wave because cos(x 90 ) = sin x (734) Creating Barcode In Visual Studio .NET Using Barcode printer for .NET framework Control to generate, create bar code image in Visual Studio .NET applications. Generating Rationalized Codabar In Visual Studio .NET Using Barcode creation for VS .NET Control to generate, create Ames code image in .NET applications. In a given circuit with a source f (t) = A cos( t + ), the frequency part will be the same for all components in the circuit Hence, we can do our analysis by focusing on the phase lead or lag for each voltage and current in the circuit This is done by writing each voltage and current by using what is called a phasor transform We denote phasor transforms with boldface letters For f (t) = A cos( t + ), the phasor transform is just F = Ae j (735) Barcode Maker In VB.NET Using Barcode maker for .NET framework Control to generate, create barcode image in VS .NET applications. Encoding EAN13 Supplement 5 In Java Using Barcode generation for BIRT reports Control to generate, create EAN13 Supplement 5 image in BIRT applications. The functions f (t) = A cos( t + ) and F = Ae j constitute a phasor transform pair We write this relationship as f (t) F EXAMPLE 72 If i(t) = 20 cos(12t + 30 ), what is the phasor transform SOLUTION First note that the current i(t) is the real part of i(t) = Re[20e j(12t+30 ) ] = Re[20 cos(12t + 30 ) + j20 sin(12t + 30 )] The phasor transform is I = 20e j30 Or we can write it in shorthand as I = 20 30 UPC  13 Generation In Java Using Barcode maker for Java Control to generate, create European Article Number 13 image in Java applications. Encode USS Code 39 In None Using Barcode generator for Office Word Control to generate, create Code 39 Full ASCII image in Microsoft Word applications. (736) Matrix 2D Barcode Encoder In Java Using Barcode creator for Java Control to generate, create Matrix 2D Barcode image in Java applications. Bar Code Creator In Visual Studio .NET Using Barcode creator for ASP.NET Control to generate, create barcode image in ASP.NET applications. Circuit Analysis Demysti ed
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Phasor transforms are very useful in electrical engineering because they allow us to convert differential equations into algebra In particular, the differentiation operation is converted into simple multiplication Again, let s start with f (t) = A cos( t + ), which allows us to write f (t) = Re [Ae j( t+ ) ] Then df = A sin( t + ) dt But notice that d Ae j( t+ ) = j Ae j( t+ ) dt Since f (t) = Re [Ae j( t+ ) ], it follows that df = j Re [Ae j( t+ ) ] dt For the phasor transform, we have the relation df j Ae j dt (738) (737) Now let s consider integration We integrate from the time 0 just before the circuit is excited to some time t, so de ne g(t) = t 0 Ae j( + ) d
Noting that at t = 0 , we take the function to be zero Letting u = j( + ) we have du = j d , d = 1 du j And we obtain g(t) = The Phasor Transform
1 j Ae ju du =
1 Ae j( + ) j
t 0 1 Ae j( t+ ) j
Hence, integration, which is the inverse operation to differentiation, results in division by j in the phasor domain Given a sinusoidal function f (t) with phasor transform F we have the phasor transform pair t 0 f ( ) d
1 F j
(739) Circuit Analysis Using Phasors
With differentiation and integration turned into simple arithmetic we can do steadystate analysis of sinusoidally excited circuits quite easily The current owing through a capacitor is given by i(t) = C dv dt When we work with phasors, this relation becomes I = j CV The voltage across a capacitor is related to the current via v(t) = 1 C (740) i( ) d
The phasor transform of this relation is V= 1 I j C (741) Now let s turn to the inductor The voltage across an inductor is related to the current through the time derivative v(t) = L di dt

