ELECTRIC CIRCUIT ANALYSIS - I

1. ELECTRIC CIRCUIT ANALYSIS - I Chapter 15 – Series & Parallel ac Circuits Lecture 19 by MoeenGhiyas

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

3. Today’s Lesson Contents • (Series ac Circuits) • Impedance and Phasors Diagram • Series Configuration

4. 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 = 0.707Vm 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)

5. 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 = 0.707Vm 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)

6. 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 = 0.707Vm 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)

7. 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

8. 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

9. IMPEDANCE AND THE PHASOR DIAGRAM • Sketch of waveform and Phasor Diagram

10. 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

11. IMPEDANCE AND THE PHASOR DIAGRAM • Sketch of waveform and Phasor Diagram

12. 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

13. IMPEDANCE AND THE PHASOR DIAGRAM • Sketch of waveform and Phasor Diagram

14. 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.

15. 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.

16. 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:

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

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

19. 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.

20. SERIES CONFIGURATION • 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°.

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

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

23. SERIES CONFIGURATION • R-L-C Example • Step 3 – Find I or E

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

25. 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°

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

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

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

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

30. Summary / Conclusion • (Series ac Circuits) • Impedance and Phasors Diagram • Series Configuration