A sinusoidal current source (independent or dependent) produces a current That varies sinusoidally with time

We define the Root Mean Square value of v(t) or rms as

The Root Mean Square value of Expand using trigonometric identity

KVL Let assume the solution is Now substituting the solution i (t) into the above differential equation we have Now we seek to findAs follows On the circuit shown. The source is sinusoidal function and we seek the response in this case is the current i(t) 9.2 The Sinusoidal Response

Solution assumption Equating the terms on the left with the terms on the right, we have Solving

However as the circuit become more complicated ( i.e, more connections of R,L and C ), the order of the differential equation will increase and the solution using the previous method will not be practical (2) Substitute the solution assumed in (1) into the differential equation We can summarized the previous method as follows: Therefore another technique will be developed as will be shown next

In this chapter, we develop a technique for calculating the response directly without solving the differential equation Solution for i (t) should be a sinusoidal of frequency 3 We notice that only the amplitude and phase change Example

Time DomainComplex Domain Deferential EquationComplex Algebraic Equation We are going to use complex analysis in the complex domain to do all the algebraic operations ( +, , X, ÷ ) Therefore we will review complex arithmetic

9.3 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 Euler’s identity relates the complex exponential function to the trigonometric function 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

Phasor Transform Were the notation Is read “ the phasor transform of We can move the coefficient V m inside The quantity is a complex number define to be the phasor that carries the amplitude and phase angle of a given sinusoidal function

Summation Property of Phasor (can be shown)

Since Next we derive y using phsor method

The V  I Relationship for a Resistor Let the current through the resistor be a sinusoidal given as Is also sinusoidal with amplitudeAnd phase The sinusoidal voltage and current in a resistor are in phase

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

The voltage and the current are in phase Real Imaginary

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 90 o The voltage lead the current by 90 o or the current lagging the voltage by 90 o You can express the voltage leading the current by T/4 or 1/4f seconds were T is the period and f is the frequency

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

The voltage lead the current by 90 o or the current lagging the voltage by 90 o Real Imaginary

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 90 o The voltage lag the current by 90 o or the current leading the voltage by 90 o

The V  I Relationship for a Capacitor The pharos representation or transform of the voltage and current and

The voltage lag the current by 90 o or the current lead the voltage by 90 o Real Imaginary

Time-Domain Phasor ( Complex or Frequency) Domain

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 The Impedance of an indictor is The Impedance of a capacitor is In all cases the impedance is measured in Ohm’s 

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

Time Domain Phasor (Complex) Domain

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

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

9.6 Series, Parallel, Simplifications and Ohm’s law in the phosor domain We seek an equivalent impedance between a and b

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

To Calculate the phasor current I

and Ohm’s law in the phosor domain

Example 9.7 Combining Impedances in series and in Parallel (a) Construct the frequency-domain (phasor, complex) equivalent circuit ? (b) Find the steady state expressions for v,i 1, i 2, and i 3 ? ? (a)

Delta-to Wye Transformations  to Y Y to 

Example 9.8

9.7 Source Transformations and Thevenin-Norton Equivalent Circuits Source Transformations Thevenin-Norton Equivalent Circuits

Example 9.9

Example 9.10 Source Transformation Sincethen Next we find the Thevenin Impedance KVL

Thevenin Impedance

9.8 The Node-Voltage Method Example 9.11 KCL at node 1 KCL at node 2Since (1) (2) Two Equations and Two Unknown, solving

To Check the work

9.9 The Mesh-Current Method Example 9.12 KVL at mesh 1 KVL at mesh 2Since Two Equations and Two Unknown, solving (1) (2)

9.12 The Phasor Diagram