V-I Relationships for Resistor, Inductor, and Capacitor

6. The V-I Relationship for a Resistor Let the current through the resistor be a sinusoidal given as Is also sinusoidal with amplitude amplitude And phase The sinusoidal voltage and current in a resistor are in phase

7. The V-I Relationship for an Inductor Let the current through the resistor be a sinusoidal given as Now we rewrite the sin function as a cosine function The sinusoidal voltage and current in an inductor are out of phase by 90o The voltage lead the current by 90oor the current lagging the voltage by 90o

8. The V-I Relationship for a Capacitor Let the voltage across the capacitor be a sinusoidal given as Now we rewrite the sin function as a cosine function The sinusoidal voltage and current in an inductor are out of phase by 90o The voltage lag the current by 90oor the current leading the voltage by 90o

9. The Sinusoidal Response KVL This is first order differential equations which has the following solution We notice that the solution is also sinusoidal of the same frequency w However they differ in amplitude and phase

10. Complex Numbers Rectangular Representation

11. Complex Numbers (Polar form) Rectangular Representation

12. Euler’s Identity Euler’s identity relates the complex exponential function to the trigonometric function Adding Subtracting

13. Euler’s Identity The left side is complex function The right side is complex function The left side is real function The right side is real function

14. Complex Numbers (Polar form) Rectangular Representation Short notation

15. Real Numbers Rectangular Representation Polar Representation OR

16. Rectangular Representation Polar Representation OR

17. Imaginary Numbers Rectangular Representation Polar Representation OR

18. Rectangular Representation Polar Representation OR

19. Complex Conjugate Complex Conjugate is defined as

20. Complex Numbers (Addition)

21. Complex Numbers (Subtraction)

22. Complex Numbers (Multiplication) Multiplication in Rectangular Form Multiplication in Polar Form Multiplication in Polar Form is easier than in Rectangular form

23. Complex Numbers (Division) Divisionin Rectangular Form

24. Complex Numbers (Division) Divisionin Polar Form Division in Polar Form is easier than in Rectangular form

25. Complex Conjugate Identities ( can be proven) OR Other Complex Conjugate Identities ( can be proven)

26. Let the current through the indictor be a sinusoidal given as Let the current through the indictor be a sinusoidal given as From Linearity if then

27. The solution which was found earlier

28. The V-I Relationship for an Inductor Let the current through the resistor be a sinusoidal given as Now we rewrite the sin function as a cosine function The sinusoidal voltage and current in an inductor are out of phase by 90o The voltage lead the current by 90oor the current lagging the voltage by 90o

29. From Linearity if The solution which was found earlier

30. The solution is the real part of This will bring us to the PHASOR method in solving sinusoidal excitation of linear circuit

32. The real part is the solution Now if you pass a complex current Phasor You get a complex voltage Phasor

33. The phasor The phasor is a complex number that carries the amplitude and phase angle information of a sinusoidal function The phasor concept is rooted in Euler’s identity We can think of the cosine function as the real part of the complex exponential and the sine function as the imaginary part Because we are going to use the cosine function on analyzing the sinusoidal steady-state we can apply

34. Moving the coefficient Vm inside Phasor Transform Were the notation Is read “ the phasor transform of

35. The V-I Relationship for a Resistor Let the current through the resistor be a sinusoidal given as Is also sinusoidal with amplitude amplitude And phase The sinusoidal voltage and current in a resistor are in phase

36. Now let us see the pharos domain representation or pharos transform of the current and voltage and Which is Ohm’s law on the phasor ( or complex ) domain

37. The voltage and the current are in phase Imaginary Real

38. The V-I Relationship for an Inductor Let the current through the resistor be a sinusoidal given as The sinusoidal voltage and current in an inductor are out of phase by 90o The voltage lead the current by 90oor the current lagging the voltage by 90o You can express the voltage leading the current by T/4 or1/4fseconds wereTis the period and fis the frequency

39. Now we rewrite the sin function as a cosine function ( remember the phasor is defined in terms of a cosine function) The pharos representation or transform of the current and voltage But since Therefore and

40. and The voltage lead the current by 90oor the current lagging the voltage by 90o Imaginary Real

41. The V-I Relationship for a Capacitor Let the voltage across the capacitor be a sinusoidal given as The sinusoidal voltage and current in an inductor are out of phase by 90o The voltage lag the current by 90oor the current leading the voltage by 90o

42. The V-I Relationship for a Capacitor The pharos representation or transform of the voltage andcurrent and

43. and The voltage lag the current by 90oor the current lead the voltage by 90o Imaginary Real

44. Phasor ( Complex or Frequency) Domain Time-Domain

45. Impedance and Reactance The relation between the voltage and current on the phasor domain (complex or frequency) for the three elements R, L, and C we have When we compare the relation between the voltage and current , we note that they are all of form: Which the state that the phasor voltage is some complex constant ( Z ) times the phasor current This resemble ( شبه) Ohm’s law were the complex constant ( Z ) is called “Impedance” (أعاقه ) Recall on Ohm’s law previously defined , the proportionality content R was real and called “Resistant” (مقاومه ) Solving for ( Z ) we have The Impedance of a resistor is In all cases the impedance is measured in Ohm’s W The Impedance of an indictor is The Impedance of a capacitor is

46. Impedance The Impedance of a resistor is In all cases the impedance is measured in Ohm’s W The Impedance of an indictor is The Impedance of a capacitor is The imaginary part of the impedance is called “reactance” The reactance of a resistor is We note the “reactance” is associated with energy storage elements like the inductor and capacitor The reactance of an inductor is The reactance of a capacitor is Note that the impedance in general (exception is the resistor) is a function of frequency At w = 0 (DC), we have the following short open

47. 9.5 Kirchhoff’s Laws in the Frequency Domain ( Phasor or Complex Domain) Consider the following circuit PhasorTransformation KVL Using Euler Identity we have Which can be written as Factoring Can not be zero Phasor KVL on the phasor domain So in general

48. Kirchhoff’s Current Law A similar derivation applies to a set of sinusoidal current summing at a node Phasor Transformation KCL KCL on the phasor domain

49. Example 9.6 for the circuit shown below the source voltage is sinusoidal (a) Construct the frequency-domain (phasor, complex) equivalent circuit ? (b) Calculte the steady state current i(t) ? The source voltage pahsor transformation or equivalent The Impedance of the indictor is The Impedance of the capacitor is

50. To Calculate the phasor current I