lect 04: ac circuits

Electrical Circuits

Part (2) : AC Circuits

Lecture 4

باسم ممدوح الحلوانى . د

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Power in ac Circuits Power in ac Circuits Chapter (17) Chapter (17)

The Relationship between P, Q, and S The Relationship between P, Q, and S

Real, reactive, and apparent power are related by a very simple relationship through the power triangle.

Consider the series circuit of Figure (a) Let the current through the circuit be I = I ∠ 0o as shown in (b)

• VR is in phase with I, while VL leads it by 90o . • Kirchhoff’s voltage law applies for ac voltages in phasor form.

Thus as indicated in figure (c):

The voltage triangle of (c) may be redrawn as :

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Apparent Power Apparent Power

When a load has voltage V across it and current I through it , the power that appears to flow to it is :

When a load has voltage V across it and current I through it , the power that appears to flow to it is :

Since VI appears to represent power, it is called apparent power.

if the load contains both resistance and reactance, this product represents neither real power nor reactive power.

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The Relationship between P, Q, and S The Relationship between P, Q, and S

From the geometry of this triangle:

If the circuit is capacitive instead of inductive:

Real and Reactive Power Equations Real and Reactive Power Equations

V and I are the magnitudes of the rms values ϴ is the angle between them ϴ = P is always positive, Q is positive for inductive circuits and negative

for capacitive circuits.

Complex Power representation:

Magnitude:

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Complex Power Relations: Complex Power Relations:

Magnitude: is the product of the rms values of voltage and current.

P = 𝑽𝒓𝒎𝒔 𝑰𝒓𝒎𝒔 Cos(θv − θi) = 𝟏

𝟐 𝑽𝒎 𝑰𝒎 Cos(θv − θi) = S Cos(θv − θi)

Q = 𝑽𝒓𝒎𝒔 𝑰𝒓𝒎𝒔 Sin(θv − θi) = 𝟏

𝟐 𝑽𝒎 𝑰𝒎 Sin(θv − θi) = S Sin(θv − θi)

I* is the conjugate of current

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The Power Factor: The Power Factor:

the power factor cannot exceed 1.0 (or 100% if expressed in percent).

For a purely resistive circuit: θ = 0, pf = cos(0) = 1 • All the load’s apparent power is real power • This case is referred to as unity power factor.

For a Purely reactive circuit: θv − θi = ± 90◦, pf = 0. • In this case the average power is zero.

In between these two extreme cases, pf is said to be leading or lagging. • Leading power factor means that current leads voltage, which implies a

capacitive load. • Lagging power factor means that current lags voltage, implying an

inductive load.

A load’s power factor shows how much of its apparent power is actually real power: A load’s power factor shows how much of its apparent power is actually real power:

is the ratio of the average power to the apparent

The angle θ = θv − θi is called the power factor angle, which is equal to the angle of the load impedance

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The complex power may be expressed in terms of the load impedance:

Since:

Therefore

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Why Equipment Is Rated in VA instead of watts. Why Equipment Is Rated in VA instead of watts.

Assume that the generator is rated at : 600 V, 120 kVA.

Assume that the generator is rated at : 600 V, 120 kVA.

I = 120 kVA / 600 V = 200 A I = 120 kVA / 600 V = 200 A

In Figure (a), the generator is supplying a purely resistive load with 120 kW.

Since S = P for a purely resistive load, S = 120 kVA and the generator is supplying its rated current.

In Figure (b), the generator is supplying a load with P= 120 kW as before, but Q=160 kVAR.

Its apparent power is therefore S = 200 kVA,

In Figure (b), the generator is supplying a load with P= 120 kW as before, but Q=160 kVAR.

Its apparent power is therefore S = 200 kVA,

I = 200 kVA / 600 V = 333.3 A I = 200 kVA / 600 V = 333.3 A

Current capability:

Supplied current:

So, the current-carrying capability can be greatly exceeded (even though its power rating is not)

which means overloading and possible damage.

So, the current-carrying capability can be greatly exceeded (even though its power rating is not)

which means overloading and possible damage.

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Ch (17) : ac Series-Parallel Circuits Ch (17) : ac Series-Parallel Circuits

The rules and laws which were developed for dc circuits will apply equally well for ac circuits.

The rules and laws which were developed for dc circuits will apply equally well for ac circuits.

The major difference between solving dc and ac circuits is that analysis of ac circuits requires using vector algebra.

The major difference between solving dc and ac circuits is that analysis of ac circuits requires using vector algebra.

Ohm’s law, The voltage divider rule, Kirchhoff’s voltage law, Kirchhoff’s current law, and The current divider rule.

you should be able to add and subtract any number of vector quantities.

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ac Series Circuits ac Series Circuits

When working with ac circuits we no longer work with only resistance but with impedances

When working with ac circuits we no longer work with only resistance but with impedances

Use rectangular form to add real parts and imaginary parts

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ac Series Circuits ac Series Circuits

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Kirchhoff’s Voltage Law and the Voltage Divider Rule


When a voltage is applied to impedances in series, Ohm’s law may be used to determine the voltage across any impedance as:

When a voltage is applied to impedances in series, Ohm’s law may be used to determine the voltage across any impedance as:

The current in the circuit is: The current in the circuit is:

By substitution, we arrive at the voltage divider : By substitution, we arrive at the voltage divider :

It is very similar to the equation for the voltage divider rule in dc circuits. The fundamental differences in solving ac circuits are that : It is very similar to the equation for the voltage divider rule in dc circuits. The fundamental differences in solving ac circuits are that :

We use impedances rather than resistances and The voltages found are phasors.

we generally use the polar form rather than the rectangular form for division and multiplications we generally use the polar form rather than the rectangular form for division and multiplications

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Remember : The summation is generally done more easily in rectangular form than in the polar form.

Remember : The summation is generally done more easily in rectangular form than in the polar form.

Kirchhoff’s voltage law for ac circuits may be stated as: Kirchhoff’s voltage law for ac circuits may be stated as:

The phasor sum of voltage drops and voltage rises around a closed loop is equal to zero.

The phasor sum of voltage drops and voltage rises around a closed loop is equal to zero.

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Kirchhoff’s Voltage Law and the Voltage Divider Rule Kirchhoff’s Voltage Law and the Voltage Divider Rule

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ac Parallel Circuits ac Parallel Circuits

The admittance Y of any impedance is defined as a vector quantity which is the reciprocal of the impedance Z.

The admittance Y of any impedance is defined as a vector quantity which is the reciprocal of the impedance Z.

the unit of admittance is the siemens (S).

Purely reactive component :

Purely resistive component :

admittance of a resistor R is called conductance

admittance of a reactance X is called susceptance

Admittance diagram Admittance diagram

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ac Parallel Circuits ac Parallel Circuits

The total admittance is the vector sum of the admittances of the network. The total admittance is the vector sum of the admittances of the network.

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The Current Divider Rule The Current Divider Rule

For two branches in parallel For two branches in parallel

Kirchhoff’s Current Law (KCL): Kirchhoff’s Current Law (KCL):

The summation of current phasors entering and leaving a node is equal to zero. The summation of current phasors entering and leaving a node is equal to zero.

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Series-Parallel Circuits Series-Parallel Circuits

lect 04: ac circuits

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