ELECTRIC CIRCUIT ANALYSIS I Chapter 15 Series Parallel

ELECTRIC CIRCUIT ANALYSIS - I Chapter 15 – Series & Parallel ac Circuits Lecture 19 by Moeen Ghiyas 19/01/2022 1

Chapter 15 – Series & Parallel ac Circuits TODAY’S lesson

Today’s Lesson Contents � (Series ac Circuits) � Impedance and Phasors Diagram � Series Configuration 19/01/2022 3

IMPEDANCE AND THE PHASOR DIAGRAM � 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. 707 Vm and where I = 0. 707 Im � 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) 19/01/2022 4

IMPEDANCE AND THE PHASOR DIAGRAM � Inductive Reactance - For the inductive circuit, � Time domain equations: v = Vm sin ωt and i = Im sin ωt � In phasor form: � Where V = 0. 707 Vm and where I = 0. 707 Im � 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) 19/01/2022 5

IMPEDANCE AND THE PHASOR DIAGRAM � Capacitive Reactance - For a capacitive circuit, � Time domain equations: v = Vm sin ωt and i = Im sin ωt � In phasor form: � Where V = 0. 707 Vm and where I = 0. 707 Im � 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) 19/01/2022 6

IMPEDANCE AND THE PHASOR DIAGRAM �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 19/01/2022 7

IMPEDANCE AND THE PHASOR DIAGRAM � 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 19/01/2022 8

IMPEDANCE AND THE PHASOR DIAGRAM � Sketch of waveform and Phasor Diagram 19/01/2022 9

IMPEDANCE AND THE 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 19/01/2022 10

IMPEDANCE AND THE PHASOR DIAGRAM � Sketch of waveform and Phasor Diagram 19/01/2022 11

IMPEDANCE AND THE 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 19/01/2022 12

IMPEDANCE AND THE PHASOR DIAGRAM � Sketch of waveform and Phasor Diagram 19/01/2022 13

IMPEDANCE AND THE 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 AND THE PHASOR DIAGRAM � 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.

SERIES CONFIGURATION � 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:

SERIES CONFIGURATION � EXAMPLE - Determine the input impedance to the series network of fig. Draw the impedance diagram. � Solution:

SERIES CONFIGURATION � EXAMPLE - Determine the input impedance to the series network of fig. Draw the impedance diagram. � Solution:

SERIES CONFIGURATION � 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. 19/01/2022 19

SERIES CONFIGURATION � Impedance Relation with Power Factor � We know that Impedance Diagram Phasor Diagram � 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. 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°. 19/01/2022 20

SERIES CONFIGURATION � R-L-C Example � Step 1 – Convert Available information to Phasor Notation

SERIES CONFIGURATION � R-L-C Example �. Step 2 – Find ZT and make impedance diagram

SERIES CONFIGURATION � R-L-C Example � Step 3 – Find I or E

SERIES CONFIGURATION � R-L-C Example � Step 4 – Find phasor voltages across each element

SERIES CONFIGURATION � 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°

SERIES CONFIGURATION � R-L-C Example � Step 6 – Convert phasor values to time domain

SERIES CONFIGURATION � R-L-C Example � Step 7 – Plot all the voltages and the current of the circuit

SERIES CONFIGURATION � R-L-C Example � Step 8 – Calculation of total power in watts delivered to the circuit � or

SERIES CONFIGURATION � R-L-C Example � Step 9 – The power factor of the circuit is � or

Summary / Conclusion � (Series ac Circuits) � Impedance and Phasors Diagram � Series Configuration

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