Sinusoidal Signals & Phasors Sinusoidal Signals & Phasors Dr. Mohamed Refky Amin Electronics and...
Transcript of Sinusoidal Signals & Phasors Sinusoidal Signals & Phasors Dr. Mohamed Refky Amin Electronics and...
Sinusoidal Signals & Phasors
Dr. Mohamed Refky Amin
Electronics and Electrical Communications Engineering Department (EECE)
Cairo University
http://scholar.cu.edu.eg/refky/
OUTLINE
• Previously on ELCN102
• AC Circuits
• Sinusoidal Signals
• Phasor Representation
Dr. Mohamed Refky 2
Previously on ELCN102
Dr. Mohamed Refky
CapacitorsWhen a voltage source is connected to a capacitor, an electric
field is generated in the dielectric and charges are accumulated on
the plates.
𝑄 = 𝐶 × 𝑉
𝐶 =𝑄
𝑉
The amount of charge (𝑄) that a capacitor can store per volt
across the plates, is its capacitance (𝐶).
Coulomb Farad
Volt
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Previously on ELCN102
Dr. Mohamed Refky
Series and Parallel Combinations
Series Capacitors
1
𝐶𝑒𝑞=
1
𝐶1+
1
𝐶2+⋯+
1
𝐶𝑁
𝑄 = 𝐶 × 𝑉
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Previously on ELCN102
Dr. Mohamed Refky
Series and Parallel Combinations
Parallel Capacitors 𝑄 = 𝐶 × 𝑉
𝐶𝑒𝑞 = 𝐶1 + 𝐶2 +⋯+ 𝐶𝑁
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Previously on ELCN102
Dr. Mohamed Refky
InductorsWhen the current flowing through an
inductor changes, the magnetic field induces
a voltage in the conductor, according to
Faraday’s law of electromagnetic induction,
to resist this change in the current.
𝑣𝐿 𝑡 = 𝐿𝑑𝑖𝐿 𝑡
𝑑𝑡
𝐿 is the inductance in Henri (𝐻)
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Previously on ELCN102
Dr. Mohamed Refky
Series and Parallel Combinations
Series Inductors
𝐿𝑒𝑞 = 𝐿1 + 𝐿2 +⋯+ 𝐿𝑁
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Previously on ELCN102
Dr. Mohamed Refky
Series and Parallel Combinations
Parallel Inductors
1
𝐿𝑒𝑞=
1
𝐿1+
1
𝐿2+⋯+
1
𝐿𝑁
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Previously on ELCN102
Dr. Mohamed Refky
Transient AnalysisThe transient response of the circuit is the response when the input
is change suddenly or a switches status change.
𝑣 𝑡 =
𝑣1 𝑡 , 𝑡0 < 𝑡 < 𝑡1𝑣2 𝑡 , 𝑡1 < 𝑡 < 𝑡2
⋮𝑣𝑛 𝑡 , 𝑡𝑛−1 < 𝑡 < 𝑡𝑛
𝑣 𝑡 the same
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Previously on ELCN102
Dr. Mohamed Refky
Steady State AnalysisThe steady state response of the circuit is the response when the
status of the circuit does not change for long time.
𝑣 𝑡 the same
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Previously on ELCN102
Dr. Mohamed Refky
Time Domain Analysis 1st Order Systems
𝑉𝑖 and 𝑉𝑓 are the initial and final capacitor voltages, respectively.
𝜏 = 𝑅𝑒𝑞𝐶, 𝑅𝑒𝑞 is the resistance seen between the capacitor nodes
while all sources are switched off.
𝑣𝑐 𝑡 = 𝑉𝑓 − 𝑉𝑓 − 𝑉𝑖 𝑒−𝑡𝜏
RC Circuits
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Previously on ELCN102
Dr. Mohamed Refky
Time Domain Analysis 1st Order Systems
𝐼𝑖 and 𝐼𝑓 are the initial and final inductor current, respectively.
𝜏 = 𝐿/𝑅𝑒𝑞, 𝑅𝑒𝑞 is the resistance seen between the inductor nodes
while all sources are switched off.
𝑖𝐿 𝑡 = 𝐼𝑓 − 𝐼𝑓 − 𝐼𝑖 𝑒−𝑡𝜏
RL Circuits
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Previously on ELCN102
Dr. Mohamed Refky
AC CircuitsAn AC circuit is a combination of active elements (Voltage and
current sources) and passive elements (resistors, capacitors and
coils).
Unlike resistance, capacitors and coils can store energy and do
not dissipate it. Thus, capacitors and coils are called storage
elements.13
Previously on ELCN102
Dr. Mohamed Refky
AC CircuitsAn AC circuit is a combination of active elements (Voltage and
current sources) and passive elements (resistors, capacitors and
coils).
The sources are usually AC sinusoidal voltage or current sources
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Sinusoidal Signals
Dr. Mohamed Refky
DefinitionA sinusoid is a signal that has the form of the sine or cosine
function.
𝑉𝐴𝐶 = 𝑉𝑚 sin 𝜔𝑡 𝜔𝑇 = 2𝜋 → 𝜔 =2𝜋
𝑇
amplitude
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Sinusoidal Signals
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DefinitionA sinusoid is a signal that has the form of the sine or cosine
function.
• 𝑉𝑚 is the amplitude of the sinusoid
• 𝜔 is the angular frequency in rad/s
• 𝜔𝑡 is the argument of the sinusoid
• 𝑓 =𝜔
2𝜋is the sinusoid frequency
• 𝑇 =1
𝑓is the sinusoid period
𝑉𝐴𝐶 = 𝑉𝑚 sin 𝜔𝑡 𝜔𝑇 = 2𝜋 → 𝜔 =2𝜋
𝑇
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Sinusoidal Signals
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Why Do We Study Sinusoidal Signals?We study the sinusoid because:
• A sinusoidal signal is easy to generate and transmit. It is the
form of voltage generated throughout the world and supplied
to homes, factories, laboratories.
• Through Fourier analysis, any practical periodic signal can be
represented by a sum of sinusoids.
• A sinusoid is easy to handle mathematically. The derivative
and integral of a sinusoid are themselves sinusoids.
• For a linear time invariant system (LTI), a sinusoid is an eigen
function to the system.
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Sinusoidal Signals
Dr. Mohamed Refky
Eigen function of an LTI system
If 𝑥 𝑡 is an Eigen function to an LTI system, the response of the
system to the input 𝑥 𝑡 is
𝑦 𝑡 = 𝛼𝑥 𝑡
𝛼 is generally complex number causing a change in both the
magnitude and phase.
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Sinusoidal Signals
Dr. Mohamed Refky
Phase shift
𝑉𝐴𝐶 = 𝑉𝑚 sin 𝜔𝑡𝑉𝐴𝐶 = 𝑉𝑚 sin 𝜔𝑡 + 𝜙𝑉𝐴𝐶 = 𝑉𝑚 sin 𝜔𝑡 − 𝜙
The phase shift is positive if the signal is shifted to the left and isnegative if the signal is shifted to the right.
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Sinusoidal Signals
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Complex NumbersA complex number can be written in two forms: rectangular
form and polar form.
The rectangular form consists of a
real part and an imaginary part.
𝑍 = 𝑥 + 𝑗𝑦
The polar form consists of a
magnitude and phase.
𝑍 = 𝑟𝑒𝑗𝜃 = 𝑟∠𝜃
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Sinusoidal Signals
Dr. Mohamed Refky
Complex NumbersA complex number can be written in two forms: rectangular
form and polar form.
To convert from rectangular form
to polar form
𝑟 = 𝑥2 + 𝑦2,
To convert from polar form to
rectangular form
𝑥 = 𝑟 cos 𝜃 , 𝑦 = 𝑟 sin 𝜃
𝜃 = tan−1𝑦
𝑥
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Sinusoidal Signals
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Complex NumbersA complex number can be written in two forms: rectangular
form and polar form.
Complex numbers are added/subtracted easily in rectangular
form
𝑍1 = 𝑥1 + 𝑗𝑦1, 𝑍2 = 𝑥2 + 𝑗𝑦2
Then
𝑍1 + 𝑍2 = 𝑥1 + 𝑥2 + 𝑗 𝑦1 + 𝑦2
𝑍1 − 𝑍2 = 𝑥1 − 𝑥2 + 𝑗 𝑦1 − 𝑦2
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Sinusoidal Signals
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Complex NumbersA complex number can be written in two forms: rectangular
form and polar form.
Complex numbers are multiplied/divided easily in polar form
𝑍1 = 𝑟1𝑒𝑗𝜃1 = 𝑟1∠𝜃1, 𝑍2 = 𝑟2𝑒
𝑗𝜃2 = 𝑟2∠𝜃2
Then
𝑍1 × 𝑍2 = 𝑟1 × 𝑟2 𝑒𝑗 𝜃1+𝜃2 = 𝑟1 × 𝑟2 ∠ 𝜃1 + 𝜃2
𝑍1/𝑍2 = 𝑟1/𝑟2 𝑒𝑗 𝜃1−𝜃2 = 𝑟1/𝑟2 ∠ 𝜃1 − 𝜃2
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Phasor Representation
Dr. Mohamed Refky
Definition
The locus of 𝑒𝑗𝜃 is a circle with radius 1.
𝑒𝑗𝜃
𝜃 = 0𝑜𝜃 = 30𝑜𝜃 = 60𝑜𝜃 = 135𝑜𝜃 = 225𝑜𝜃 = 315𝑜 𝜃 = −45𝑜
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Phasor Representation
Dr. Mohamed Refky
Definition
sin 𝜃
𝑒𝑗𝜃 = cos 𝜃 + 𝑗 sin 𝜃
cos 𝜃 = 𝑅𝑒 𝑒𝑗𝜃
sin 𝜃 = 𝐼𝑚 𝑒𝑗𝜃
Euler’s identity
cos 𝜃
𝑒𝑗𝜃
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Phasor Representation
Dr. Mohamed Refky
DefinitionThe sinusoid function 𝑣 𝑡 = 𝑉𝑚 cos 𝜔𝑡 + 𝜙 can be written as
𝑣 𝑡 = 𝑅𝑒 𝑉𝑚𝑒𝑗 𝜔𝑡+𝜙
= 𝑅𝑒 𝑉𝑚𝑒𝑗 𝜙 𝑒𝑗 𝜔𝑡
= 𝑅𝑒 𝑉𝑒𝑗 𝜔𝑡
𝑉 = 𝑉𝑚𝑒𝑗 𝜙 = 𝑉𝑚∠𝜙
𝑉𝑚𝑒𝑗 𝜙 is the phasor representation of 𝑉𝑚 cos 𝜔𝑡 + 𝜙
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Phasor Representation
Dr. Mohamed Refky
DefinitionA sinusoid
𝑣 𝑡 = 𝑉𝑚 cos 𝜔𝑡 + 𝜙
= 𝑅𝑒 ( 𝑉𝑒𝑗𝜔𝑡)
can be represented by the projection,
on the horizontal axis, of a phasor
rotating with a constant angular
velocity 𝜔.
𝑉 = 𝑉𝑚∠𝜙
𝑉𝑚 is the circle radius
∠𝜙 is the initial phasor position27
Phasor Representation
Dr. Mohamed Refky
DefinitionA sinusoid
𝑣 𝑡 = 𝑉𝑚 sin 𝜔𝑡 + 𝜙
= 𝐼𝑚 ( 𝑉𝑒𝑗𝜔𝑡)
can be represented by the projection, on the vertical axis, of a
phasor rotating with a constant angular velocity 𝜔.
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Phasor Representation
Dr. Mohamed Refky
Phasors
The cosine function leads the sine function by 90𝑜
cos 𝜔𝑡 = sin 𝜔𝑡 + 90𝑜
sin 𝜔𝑡 = cos 𝜔𝑡 − 90𝑜
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Phasor Representation
Dr. Mohamed Refky
Graphical approachThe cosine function leads the sinefunction by 90𝑜
cos 𝜔𝑡 = sin 𝜔𝑡 + 90𝑜
sin 𝜔𝑡 = cos 𝜔𝑡 − 90𝑜
Graphical approach is very handyin representing the addition oftwo sinusoids of the samefrequency
𝑉 = 𝛼 cos 𝜔𝑡 + 𝛽 sin 𝜔𝑡
= 𝛾 cos 𝜔𝑡 − 𝜃 𝛾 = 𝛼2 + 𝛽2, 𝜃 = tan−1𝛽
𝛼30
Phasor Representation
Dr. Mohamed Refky
Sinusoid-Phasors transformation
Phasor domain is also known as the frequency domain.
Time-domain representation Phasor representation
𝑉𝑚 cos 𝜔𝑡 + 𝜙 𝑉𝑚∠𝜙
𝑉𝑚 sin 𝜔𝑡 + 𝜙 𝑉𝑚∠𝜙 − 90𝑜
𝐼𝑚 cos 𝜔𝑡 + 𝜃 𝐼𝑚∠𝜃
𝐼𝑚 sin 𝜔𝑡 + 𝜃 𝐼𝑚∠𝜃 − 90𝑜
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Phasor Representation
Dr. Mohamed Refky
Example (1)For the sinusoid 5sin(4𝜋𝑡 + 60𝑜) calculate its amplitude, phase,
angular frequency, frequency, and period.
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Phasor Representation
Dr. Mohamed Refky
Example (2)Transform these sinusoids to phasors representation:
𝑣 = 6cos(50𝑡 − 40𝑜)
𝑖 = −4 sin(50 𝑡 + 50𝑜)
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Phasor Representation
Dr. Mohamed Refky
Example (3)Transform these phasors representation to sinusoids:
𝑉 = 8𝑒−𝑗20𝑜
𝑖 = 3 + 𝑗4
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Phasor Representation
Dr. Mohamed Refky
Example (4)Calculate the phase angle (phase difference) between:
𝑣1 = −10cos(𝜔𝑡 + 50𝑜) 𝑎𝑛𝑑 𝑣2 = 12 sin(𝜔𝑡 − 10𝑜)
State which sinusoid is leading.
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Phasor Representation
Dr. Mohamed Refky
Phasor Relationships for Circuit Elements
If the current in a resistor 𝑅 is given by:
𝑖𝑅 𝑡 = 𝐼𝑚cos(𝜔𝑡)
The resistor voltage will be given by
𝑣𝑅 𝑡 = 𝑅 × 𝑖𝑅 𝑡 = 𝑅𝐼𝑚 cos 𝜔𝑡 = 𝑉𝑚cos(𝜔𝑡)
𝐼 = 𝐼𝑚∠0𝑜 𝑉 = 𝑉𝑚∠0
𝑜 = 𝑅𝐼𝑚∠0𝑜
Resistor
For a resistor, the voltage and current are in phase
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Phasor Representation
Dr. Mohamed Refky
Phasor Relationships for Circuit Elements
If the current in an inductor 𝐿 is given by:
𝑖𝐿 𝑡 = 𝐼𝐿cos(𝜔𝑡)
The inductor voltage will be given by
𝑣𝐿 𝑡 = 𝐿𝑑𝑖𝐿 𝑡
𝑑𝑡= −𝜔𝐿𝐼𝐿 sin 𝜔𝑡 = −𝑉𝐿 sin(𝜔𝑡)
𝐼 = 𝐼𝑚∠0𝑜 𝑉 = 𝑉𝐿∠90
𝑜 = 𝜔𝐿𝐼𝐿∠90𝑜
Inductor
For an inductor, the current lags the voltage by 90𝑜
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Phasor Representation
Dr. Mohamed Refky
Phasor Relationships for Circuit Elements
If the voltage in a capacitor 𝐶 is given by:
𝑣𝐶 𝑡 = 𝑉𝐶cos(𝜔𝑡)
The capacitor current will be given by
𝑖𝐶 𝑡 = 𝐶𝑑𝑣𝐶 𝑡
𝑑𝑡= −𝜔𝐶𝑉𝐶 sin 𝜔𝑡 = −𝐼𝐶 sin(𝜔𝑡)
𝑉 = 𝑉𝐶∠0𝑜 𝐼 = 𝐼𝐶∠90
𝑜 = 𝜔𝐶𝑉𝐶∠90𝑜
Capacitor
For an capacitor, the current leads the voltage by 90𝑜
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Phasor Representation
Dr. Mohamed Refky
Phasor Relationships for Circuit Elements
𝐼 = 𝐼𝑚∠0𝑜 𝑉 = 𝑅𝐼𝑚∠0
𝑜
𝑉 = 𝑉𝑚∠0𝑜 𝐼 = 𝐶𝜔𝑉𝑚∠90
𝑜𝐼 = 𝐼𝑚∠0𝑜 𝑉 = 𝐿𝜔𝐼𝑚∠90
𝑜
inductor capacitor
resistor
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Phasor Representation
Dr. Mohamed Refky
Phasor Relationships for Circuit Elements
The impedance 𝑍 of a circuit is the ratio of the phasor voltage 𝑉to the phasor current 𝐼, measured in Ω.
Resistor Inductor Capacitor
𝑣𝑅 𝑡 = 𝑅𝑖𝑅 𝑡 𝑣𝐿 𝑡 = 𝐿𝑑𝑖𝐿 𝑡
𝑑𝑡𝑖𝐶 𝑡 = 𝐶
𝑑𝑣𝐶 𝑡
𝑑𝑡
𝑉𝑅 = 𝑅 × 𝐼𝑅𝑉𝐿 = 𝜔𝐿𝐼𝐿∠90
𝑜
= 𝑗𝐿𝜔 × 𝐼𝐿
𝐼𝐶 = 𝜔𝐶𝑉𝐶∠90𝑜
= 𝑗𝜔𝐶 × 𝑉𝐶
𝑍𝑅 = 𝑅 𝑍𝐿 = 𝑗𝜔L 𝑍𝐶 =1
𝑗𝜔𝐶= −
𝑗
𝜔𝐶
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Phasor Representation
Dr. Mohamed Refky
Phasor Relationships for Circuit Elements
The admittance 𝑌 of a circuit is the ratio of the phasor current 𝐼to the phasor voltage 𝑉, measured in Ω−1.
Resistor Inductor Capacitor
𝑣𝑅 𝑡 = 𝑅𝑖𝑅 𝑡 𝑣𝐿 𝑡 = 𝐿𝑑𝑖𝐿 𝑡
𝑑𝑡𝑖𝐶 𝑡 = 𝐶
𝑑𝑣𝐶 𝑡
𝑑𝑡
𝑉𝑅 = 𝑅 × 𝐼𝑅𝑉𝐿 = 𝜔𝐿𝐼𝐿∠90
𝑜
= 𝑗𝐿𝜔 × 𝐼𝐿
𝐼𝐶 = 𝜔𝐶𝑉𝐶∠90𝑜
= 𝑗𝜔𝐶 × 𝑉𝐶
𝑌𝑅 =1
𝑅𝑌𝐿 =
1
𝑗𝜔L= −
𝑗
𝜔L𝑌𝐶 = 𝑗𝜔𝐶
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Phasor Representation
Dr. Mohamed Refky
Impedance and Admittance
The impedance 𝑍 of a circuit is the ratio of the phasor voltage 𝑉to the phasor current 𝐼, measured in Ω.
𝑍 = 𝑅 + 𝑗𝑋
𝑅 is the resistance & 𝑋 is the reactance
𝑍 is inductive if 𝑋 is +𝑣𝑒.
𝑍 is capacitive if 𝑋 is −𝑣𝑒.
𝑍, 𝑅, and 𝑋 are in units of Ω
Impedance
𝑍𝐿 = 𝑗𝜔L
𝑍𝐶 =1
𝑗𝜔𝐶= −
𝑗
𝜔𝐶
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Phasor Representation
Dr. Mohamed Refky
Impedance and Admittance
The admittance 𝑌 of a circuit is the ratio of the phasor current 𝐼 to
the phasor voltage 𝑉, measured in Ω−1.
𝑌 = 𝐺 + 𝑗𝐵
𝐺 is the conductance & 𝐵 is the susceptance.
𝑌 is inductive if 𝐵 is −𝑣𝑒.
𝑌 is capacitive if 𝐵 is +𝑣𝑒.
𝑌, 𝐺, and 𝐵 are in units of Ω−1
Admittance
𝑌𝐿 =1
𝑗𝜔L= −
𝑗
𝜔L
𝑌𝐶 = 𝑗𝜔𝐶
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Phasor Representation
Dr. Mohamed Refky
Impedance Combination
𝑉𝑒𝑞 = 𝑉1 + 𝑉2 +⋯+ 𝑉𝑁
𝐼 × 𝑍𝑒𝑞 = 𝐼 × 𝑍1 + 𝐼 × 𝑍2 +⋯+ 𝐼 × 𝑍𝑁
𝑍𝑒𝑞 = 𝑍1 + 𝑍2 +⋯+ 𝑍𝑁
Series Combination
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Phasor Representation
Dr. Mohamed Refky
Impedance Combination
𝐼𝑒𝑞 = 𝐼1 + 𝐼2 +⋯+ 𝐼𝑁
𝑉
𝑍𝑒𝑞=
𝑉
𝑍1+𝑉
𝑍2+⋯+
𝑉
𝑍𝑁
Parallel Combination
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Phasor Representation
Dr. Mohamed Refky
Impedance Combination
1
𝑍𝑒𝑞=
1
𝑍1+
1
𝑍2+⋯+
1
𝑍𝑁
Parallel Combination
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Phasor Representation
Dr. Mohamed Refky
Admittance Combination
𝑉𝑒𝑞 = 𝑉1 + 𝑉2 +⋯+ 𝑉𝑁
𝐼
𝑌𝑒𝑞=
𝐼
𝑌1+
𝐼
𝑌2+⋯+
𝐼
𝑌𝑁
Series Combination
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Phasor Representation
Dr. Mohamed Refky
Admittance Combination
1
𝑌𝑒𝑞=
1
𝑌1+1
𝑌2+⋯+
1
𝑌𝑁
Series Combination
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Phasor Representation
Dr. Mohamed Refky
Admittance Combination
𝐼𝑒𝑞 = 𝐼1 + 𝐼2 +⋯+ 𝐼𝑁
𝑉 × 𝑌𝑒𝑞 = 𝑉 × 𝑌1 + 𝑉 × 𝑌2 +⋯+ 𝑉 × 𝑌𝑁
𝑌𝑒𝑞 = 𝑌1 + 𝑌2 +⋯+ 𝑌𝑁
Parallel Combination
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Phasor Representation
Dr. Mohamed Refky
Voltage DividerWhen impedances are connected in series, the total voltage
across these impedances is divided between them with a ratio that
depends on the values of theses impedance.
𝑉𝑎𝑐 = 𝐼 × 𝑍1 + 𝐼 × 𝑍2
= 𝐼 𝑍1 + 𝑍2
𝐼 =𝑉𝑎𝑐
𝑍1 + 𝑍2
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Phasor Representation
Dr. Mohamed Refky
Voltage DividerWhen impedances are connected in series, the total voltage
across these impedances is divided between them with a ratio that
depends on the values of theses impedance.
𝑉𝑍1 = 𝐼 × 𝑍1 = 𝑉𝑎𝑐𝑍1
𝑍1 + 𝑍2= 𝑉𝑎𝑐
𝑍1𝑍𝑒𝑞
𝑉𝑍2 = 𝐼 × 𝑍2 = 𝑉𝑎𝑐𝑍2
𝑍1 + 𝑍2= 𝑉𝑎𝑐
𝑍2𝑍𝑒𝑞
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Phasor Representation
Dr. Mohamed Refky
Current DividerWhen impedances are connected in parallel, the total current is
divide between these impedances with a ratio that depends on the
values of theses impedances.
𝐼 = 𝐼1 + 𝐼2 =𝑉
𝑍1+𝑉
𝑍2
= 𝑉𝑍1 + 𝑍2𝑍1𝑍2
𝑉 = 𝐼𝑍1𝑍2
𝑍1 + 𝑍2
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Phasor Representation
Dr. Mohamed Refky
Current DividerWhen impedances are connected in parallel, the total current is
divide between these impedances with a ratio that depends on the
values of theses impedances.
𝐼1 =𝑉
𝑍1= 𝐼
𝑍2𝑍1 + 𝑍2
= 𝐼𝑍𝑒𝑞𝑍1
𝐼2 =𝐼
𝑍2= 𝐼
𝑍1𝑍1 + 𝑍2
= 𝐼𝑍𝑒𝑞𝑍2
𝑍𝑒𝑞 =𝑍1𝑍2
𝑍1 + 𝑍253
Phasor Representation
Dr. Mohamed Refky
Star-Delta Transformation
𝑍𝐴𝐵 = 𝑍𝐴 + 𝑍𝐵 +𝑍𝐴𝑍𝐵𝑍𝐶
𝑍𝐴𝐶 = 𝑍𝐴 + 𝑍𝐶 +𝑍𝐴𝑍𝐶𝑍𝐵
𝑍𝐵𝐶 = 𝑍𝐵 + 𝑍𝐶 +𝑍𝐵𝑍𝐶𝑍𝐴
𝑍𝐴 =𝑍𝐴𝐵𝑍𝐴𝐶
𝑍𝐴𝐶 + 𝑍𝐵𝐶 + 𝑍𝐴𝐵𝑍𝐶 =
𝑍𝐵𝐶𝑍𝐴𝐶𝑍𝐴𝐶 + 𝑍𝐵𝐶 + 𝑍𝐴𝐵
𝑍𝐵 =𝑍𝐴𝐵𝑍𝐵𝐶
𝑍𝐴𝐶 + 𝑍𝐵𝐶 + 𝑍𝐴𝐵
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Phasor Representation
Dr. Mohamed Refky
Example (5)Find the equivalent impedance of the shown circuit. Assume 𝜔= 50 𝑟𝑎𝑑/𝑠.
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Phasor Representation
Dr. Mohamed Refky
Example (6)Find the current 𝐼 for the circuit shown
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Phasor Representation
Dr. Mohamed Refky
Example (7)Find the current 𝐼 for the circuit shown
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Phasor Representation
Dr. Mohamed Refky
Example (8)For the circuit shown,
𝑅 = 5𝑘Ω, 𝐶 = 0.1𝜇𝐹 and 𝑣𝑎𝑐 𝑡 = 10 cos(4000𝑡)find the circuit current 𝑖 𝑡 and the capacitor voltage 𝑣𝑐 𝑡 .
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Phasor Representation
Dr. Mohamed Refky
Example (9)For the circuit shown,
𝑅 = 4Ω, 𝐿 = 0.2𝐻 and 𝑣𝑎𝑐 𝑡 = 5 𝑠𝑖𝑛(10𝑡)find the circuit current and the inductor voltage.
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