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

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 Now we rewrite the sin function as a cosine function

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 Now we rewrite the sin function as a cosine function

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  However they differ inamplitudeandphase

Complex Numbers Rectangular Representation

Complex Numbers (Polar form) Rectangular Representation

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

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

Complex Numbers (Polar form) Rectangular Representation Short notation

Rectangular Representation OR Polar Representation Real Numbers

Rectangular Representation OR Polar Representation

Rectangular Representation Imaginary Numbers Polar Representation OR

Rectangular Representation OR Polar Representation

Complex Conjugate Complex Conjugate is defined as

Complex Numbers (Addition)

Complex Numbers (Subtraction)

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

Complex Numbers (Division) Division in Rectangular Form

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

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

Let the current through the indictor be a sinusoidal given as Let the current through the indictor be a sinusoidal given as From Linearity ifthen

The solution which was found earlier

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 Now we rewrite the sin function as a cosine function

From Linearity if The solution which was found earlier

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

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

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

Phasor Transform Were the notation Is read “ the phasor transform of Moving the coefficient V m inside

The V  I Relationship for a Resistor Let the current through the resistor be a sinusoidal given as Is also sinusoidal with amplitude 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 and

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

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

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 indictor 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

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)

Ex 6.4:Determine the voltage v(t) in the circuit Impedance of capacitor is

A single-node pair circuit Hence time-domain voltage becomes

Ex 6.5 Determine the current i(t) and voltage v(t) Single loop phasor circuit The current By voltage division The time-domain

Ex 6.6 Determine the current i(t) The phasor circuit is Combine resistor and inductor

Use current division to obtain capacitor current Hence time-domain current is:

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

Example 9.9

Ex 6.7 Determine i(t) using source transformation Phasor circuit Transformed source Voltage of source: Hence the current In time-domain

Phasor circuit Ex 6.9 Find voltage v(t) by reducing the phasor circuit at terminals a and b to a Thevenin equivalent Remove the load Voltage Divsion

parallel

The Thevenin impedance can be modeled as 1.19 resistor in series with a capacitor with value or