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# ELECTRIC CIRCUIT ANALYSIS - I - PowerPoint PPT Presentation

ELECTRIC CIRCUIT ANALYSIS - I. Chapter 15 – Series & Parallel ac Circuits Lecture 19 by Moeen Ghiyas. TODAY’S lesson. Chapter 15 – Series & Parallel ac Circuits. Today’s Lesson Contents. (Series ac Circuits) Impedance and Phasors Diagram Series Configuration.

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### ELECTRIC CIRCUIT ANALYSIS - I

Chapter 15 – Series & Parallel ac Circuits

Lecture 19

by MoeenGhiyas

Chapter 15 – Series & Parallel ac Circuits

• (Series ac Circuits)

• Impedance and Phasors Diagram

• Series Configuration

• 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)

• 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)

• 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)

• 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

• 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

• Sketch of waveform and Phasor Diagram

• 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

• Sketch of waveform and Phasor Diagram

• 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

• Sketch of waveform and Phasor Diagram

• 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.

• 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.

Impedance Diagram

Phasor 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