sinusoidal signals & phasors sinusoidal signals & phasors dr. mohamed refky amin electronics...

Sinusoidal Signals & Phasors

Dr. Mohamed Refky Amin

Electronics and Electrical Communications Engineering Department (EECE)

Cairo University

[email protected]

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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Dr. Mohamed Refky

Series and Parallel Combinations

Series Capacitors

1

𝐶𝑒𝑞=

1

𝐶1+

1

𝐶2+⋯+

1

𝐶𝑁

𝑄 = 𝐶 × 𝑉

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Dr. Mohamed Refky


Parallel Capacitors 𝑄 = 𝐶 × 𝑉

𝐶𝑒𝑞 = 𝐶1 + 𝐶2 +⋯+ 𝐶𝑁

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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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Dr. Mohamed Refky


Series Inductors

𝐿𝑒𝑞 = 𝐿1 + 𝐿2 +⋯+ 𝐿𝑁

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Dr. Mohamed Refky


Parallel Inductors

1

𝐿𝑒𝑞=

1

𝐿1+

1

𝐿2+⋯+

1

𝐿𝑁

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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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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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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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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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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


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

Dr. Mohamed Refky

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

Dr. Mohamed Refky

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

Dr. Mohamed Refky

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



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

Dr. Mohamed Refky



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

Dr. Mohamed Refky



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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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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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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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


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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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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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


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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Dr. Mohamed Refky

Example (1)For the sinusoid 5sin(4𝜋𝑡 + 60𝑜) calculate its amplitude, phase,

angular frequency, frequency, and period.

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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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Dr. Mohamed Refky

Example (3)Transform these phasors representation to sinusoids:

𝑉 = 8𝑒−𝑗20𝑜

𝑖 = 3 + 𝑗4

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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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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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Dr. Mohamed Refky


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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Dr. Mohamed Refky


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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Dr. Mohamed Refky


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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Dr. Mohamed Refky


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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Dr. Mohamed Refky


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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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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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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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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Dr. Mohamed Refky


𝐼𝑒𝑞 = 𝐼1 + 𝐼2 +⋯+ 𝐼𝑁

𝑉

𝑍𝑒𝑞=

𝑉

𝑍1+𝑉

𝑍2+⋯+

𝑉

𝑍𝑁

Parallel Combination

45


Dr. Mohamed Refky


1

𝑍𝑒𝑞=

1

𝑍1+

1

𝑍2+⋯+

1

𝑍𝑁


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Dr. Mohamed Refky

Admittance Combination

𝑉𝑒𝑞 = 𝑉1 + 𝑉2 +⋯+ 𝑉𝑁

𝐼

𝑌𝑒𝑞=

𝐼

𝑌1+

𝐼

𝑌2+⋯+

𝐼

𝑌𝑁

Series Combination

47


Dr. Mohamed Refky


1

𝑌𝑒𝑞=

1

𝑌1+1

𝑌2+⋯+

1

𝑌𝑁

Series Combination

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Dr. Mohamed Refky


𝐼𝑒𝑞 = 𝐼1 + 𝐼2 +⋯+ 𝐼𝑁

𝑉 × 𝑌𝑒𝑞 = 𝑉 × 𝑌1 + 𝑉 × 𝑌2 +⋯+ 𝑉 × 𝑌𝑁

𝑌𝑒𝑞 = 𝑌1 + 𝑌2 +⋯+ 𝑌𝑁


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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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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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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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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


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

54


Dr. Mohamed Refky

Example (5)Find the equivalent impedance of the shown circuit. Assume 𝜔= 50 𝑟𝑎𝑑/𝑠.

55


Dr. Mohamed Refky

Example (6)Find the current 𝐼 for the circuit shown

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Dr. Mohamed Refky

Example (7)Find the current 𝐼 for the circuit shown

57


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 .

58


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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sinusoidal signals & phasors sinusoidal signals & phasors dr. mohamed refky amin electronics...

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