The V-I Relationship for a Resistor

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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
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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 90o or the
current lagging the voltage by 90o
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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 90o or the
current leading the voltage by 90o
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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
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Complex Numbers
Rectangular Representation
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Complex Numbers (Polar form)
Rectangular Representation
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Eulers Identity
Eulers identity relates the complex exponential
function to the trigonometric function
Adding
Subtracting
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Eulers 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
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Complex Numbers (Polar form)
Rectangular Representation
Short notation
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Real Numbers
Rectangular Representation
Polar Representation
OR
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Rectangular Representation
Polar Representation
OR
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Imaginary Numbers
Rectangular Representation
Polar Representation
OR
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Rectangular Representation
Polar Representation
OR
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Complex Conjugate
Complex Conjugate is defined as
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Complex Numbers (Addition)
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Complex Numbers (Subtraction)
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Complex Numbers (Multiplication)
Multiplication in Rectangular Form
Multiplication in Polar Form
Multiplication in Polar Form is easier than in
Rectangular form
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Complex Numbers (Division)
Division in Rectangular Form
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Complex Numbers (Division)
Division in Polar Form
Division in Polar Form is easier than in
Rectangular form
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Complex Conjugate Identities ( can be proven)
OR
Other Complex Conjugate Identities ( can be
proven)
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Let the current through the resistor be a
sinusoidal given as
Let the current through the resistor be a
sinusoidal given as
From Linearity if
then
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The solution which was found earlier
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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 90o or the
current lagging the voltage by 90o
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From Linearity if
The solution which was found earlier
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The solution is the real part of
This will bring us to the PHASOR method in
solving sinusoidal excitation of linear circuit
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The real part is the solution
Now if you pass a complex current
Phasor
You get a complex voltage
Phasor
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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 Eulers 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
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Moving the coefficient Vm inside
Phasor Transform
Were the notation
Is read the phasor transform of
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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
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Now let us see the pharos domain representation
or pharos transform of the current and voltage
and
Which is Ohms law on the phasor ( or complex )
domain
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The voltage and the current are in phase
Imaginary
Real
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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 90o or the
current lagging the voltage by 90o
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
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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
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and
The voltage lead the current by 90o or the
current lagging the voltage by 90o
Imaginary
Real
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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 90o or the
current leading the voltage by 90o
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The V-I Relationship for a Capacitor
The pharos representation or transform of the
voltage and current
and
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and
The voltage lag the current by 90o or the
current lead the voltage by 90o
Imaginary
Real
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Phasor ( Complex or Frequency) Domain
Time-Domain
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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 ( ??? ) Ohms law were the complex
constant ( Z ) is called Impedance (????? )
Recall on Ohms law previously defined , the
proportionality content R was real and called
Resistant (?????? )
Solving for ( Z ) we have
The Impedance of a resistor is
In all cases the impedance is measured in Ohms
W
The Impedance of an indictor is
The Impedance of a capacitor is
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Impedance
The Impedance of a resistor is
In all cases the impedance is measured in Ohms
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
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9.5 Kirchhoffs Laws in the Frequency Domain (
Phasor or Complex Domain)
Consider the following circuit
Phasor Transformation
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
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Kirchhoffs Current Law
A similar derivation applies to a set of
sinusoidal current summing at a node
Phasor Transformation
KCL
KCL on the phasor domain
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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
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To Calculate the phasor current I
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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,i1,
i2, and i3 ? ?
(a)
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• Ex 6.4Determine the voltage v(t) in the circuit

Replace source with desired voltage
v(t) with
Impedance of capacitor is
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A single-node pair circuit
Hence time-domain voltage becomes
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Ex 6.5 Determine the current i(t) and voltage v(t)
Single loop phasor circuit
The current
By voltage division
The time-domain
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Ex 6.6 Determine the current i(t)
The phasor circuit is
Combine resistor and inductor
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Use current division to obtain capacitor current
Hence time-domain current is
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9.7 Source Transformations and Thevenin-Norton
Equivalent Circuits
Source Transformations
Thevenin-Norton Equivalent Circuits
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Example 9.9
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Ex 6.7 Determine i(t) using source transformation
Phasor circuit
Transformed source
Voltage of source
Hence the current
In time-domain
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Ex 6.9 Find voltage v(t) by reducing the phasor
circuit at terminals a and b to a Thevenin
equivalent
Phasor circuit
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The Thevenin impedance can be modeled as 1.19
resistor in series with a capacitor with value
or