Chapter 15 – Series & Parallel ac Circuits

Lecture 19

by Moeen Ghiyas

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Chapter 15 – Series & Parallel ac Circuits

(Series ac Circuits)

Impedance and Phasors Diagram

Series Configuration

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Resistive Elements - For the purely resistive circuit,

Time domain equations: v = Vm sin ωt and i = Im sin ωt

In phasor form:

Where V = 0.707Vm and where I = 0.707Im

Applying Ohm’s law and using phasor algebra, we have

Since i and v are in phase, thus, θR = 0°, if phase is to be same.

Thus, we define a new term, ZR as impedance of a resistive

element (which impedes flow of current)19/04/23 4

Inductive Reactance - For the inductive circuit,

Time domain equations: v = Vm sin ωt and i = Im sin ωt

In phasor form:

Where V = 0.707Vm and where I = 0.707Im

Applying Ohm’s law and using phasor algebra, we have

Since i lags v by 90°, thus, θL = 90°, for condition to be true.

Thus, we define term, ZL as impedance of an inductive element

(which impedes flow of current) 19/04/23 5

Capacitive Reactance - For a capacitive circuit,

Time domain equations: v = Vm sin ωt and i = Im sin ωt

In phasor form:

Where V = 0.707Vm and where I = 0.707Im

Applying Ohm’s law and using phasor algebra, we have

Since i leads v by 90°, thus, θC = –90°, for condition to be true.

Thus, we define term, ZC as impedance of a capacitive element

(which impedes flow of current) 19/04/23 6

However, it is important to realize that ZR is not a phasor,

even though the format is very similar to

the phasor notations for sinusoidal currents and voltages.

The term phasor is basically reserved for quantities that vary

with time, whereas R and its associated angle of 0° are

fixed, i.e. non-varying quantities.

Similarly ZL and ZC are also not phasor quantities

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Example – Find the current i for the circuit of fig. Sketch the

waveforms of v and i.

Solution:

In phasor form

From ohm’s law

Converting to time domain

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Sketch of waveform and Phasor Diagram

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Example – Find the voltage v for the circuit of fig. Sketch the

waveforms of v and i.

Solution:

In phasor form

From ohm’s law

Converting to time domain

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Sketch of waveform and Phasor Diagram

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Example – Find the voltage v for the circuit of fig. Sketch the

waveforms of v and i.

Solution:

In phasor form

From ohm’s law

Converting to time domain

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Sketch of waveform and Phasor Diagram

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Impedance Diagram - For any network,

Resistance is plotted on the positive real axis,

Inductive reactance on the positive imaginary axis, and

Capacitive reactance on the negative imaginary axis.

Impedance diagram reflects the individual and total impedance

levels of ac network.

Impedance Diagram

The magnitude of total impedance of a network defines the

resulting current level (through Ohm’s law)

For any configuration (series, parallel, series-parallel, etc.), the

angle associated with the total impedance is the angle by

which the applied voltage leads the source current.

Thus angle of impedance reveals whether the network is primarily

inductive or capacitive or simply resistive.

For inductive networks θT will be positive, whereas for capacitive

networks θT will be negative, and θT will be zero for resistive cct.

Overall properties of series ac circuits are the same as those

for dc circuits

For instance, the total impedance of a system is the sum of

the individual impedances:

EXAMPLE - Determine the input impedance to the series

network of fig. Draw the impedance diagram.

Solution:

EXAMPLE - Determine the input impedance to the series

network of fig. Draw the impedance diagram.

Solution:

Current is same in ac series circuits just like it is in dc circuits.

Ohm’s law applicability is same.

KVL applies in similar manner.

The power to the circuit can be determined by

where θT is the phase angle between E and I.19/04/23 19

Impedance Relation with Power Factor

We know that

Reference to figs and equations

θT is not only the impedance angle of ZT but also

θT is the phase angle between the input voltage

and current for a series ac circuit.

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Phasor Diagram

Impedance Diagram

Note: θT of ZT is with reference to voltage unlike FP . Also current I

is in phase with VR, lags the VL by 90°, and leads the VC by 90°.

R-L-C Example

Step 1 – Convert Available information to Phasor Notation

R-L-C Example

. Step 2 – Find ZT and

make impedance

diagram

R-L-C Example

Step 3 – Find I or E

R-L-C Example

Step 4 – Find phasor voltages across each element

R-L-C Example

I =

VR =

VL =

VC =

. Step 5 – Make phasor diagram

and

. apply KVL (for verification or if

req)Note: Current I in phase with VR,

lags the VL by 90°, and leads the VC

by 90°

R-L-C Example

Step 6 – Convert phasor values to time domain

R-L-C Example

Step 7 – Plot all

the voltages and

the current of the

circuit

R-L-C Example

Step 8 – Calculation of total power in watts delivered to the circuit

or

or

R-L-C Example

Step 9 – The power factor of the circuit is

or

(Series ac Circuits)

Impedance and Phasors Diagram

Series Configuration

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