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# Circuit-slide-1 - Chapter I Circuits Complex numbers...

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1 Chapter I Circuits Complex numbers, Phasors, and Circuits Power in Circuits

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2 z a jb = + 1 j = - exp( ) cos( ) sin( ) cos( ) sin( ) z A j A jA a A b A φ φ φ φ φ = = + = = 1 tan b a φ - � � = � � � � 2 2 A a b = + real part Imaginary part or in polar (exponential) form Complex Numbers, Phasors and Circuits Complex numbers are defined by points or vectors in the complex plane, and can be represented in Cartesian coordinates where
3 ( 29 * * z a jb a jb = + = - ( 29 ( 29 * 2 * 2 2 2 . . . z z a jb a jb z z a b z A = + - = + = = ( 29 ( 29 * * * * exp( ) exp( ) exp 2 cos( ) sin( ) z A A j z A j j z A jA φ φ π φ φ φ = = - = - = - Every complex number has a complex conjugate so that In polar form we have

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4 ( 29 ( 29 ( 29 . .... n z a jb a jb a jb = + + + ( 29 exp exp( ) n n n z A j A jn φ φ = = ( 29 2 exp 2 exp n n n k z A j j k A j j n n φ π φ π = + = + The polar form is more useful in some cases. For instance, when raising a complex number to a power, the Cartesian form is cumbersome, and impractical for non-integer exponents. In polar form , instead, the result is immediate In the case of roots , one should remember to consider φ + 2k π as argument of the exponential, with k = integer, otherwise possible roots are skipped: The results corresponding to angles up to 2 π are solutions of the root operation.
5 exp 2 j j π = exp 2 j j π - = - exp( ) exp( ) cos( ) 2 jz jz z + - = exp( ) exp( ) sin( ) 2 jz jz z j - - = ( 29 exp cos( ) sin( ) jz z j z = In electromagnetic problems

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Circuit-slide-1 - Chapter I Circuits Complex numbers...

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