1. Chapter 15 – Series & Parallel ac Circuits Lecture 19 by Moeen Ghiyas 11/10/2015 1

2. Chapter 15 – Series & Parallel ac Circuits

3. (Series ac Circuits) Impedance and Phasors Diagram Series Configuration 11/10/2015 3

4. Resistive Elements- For the purely resistive circuit, Time domain equations:v = V m sin ωt and i = I m sin ωt In phasor form: WhereV = 0.707V m andwhere I = 0.707I m Applying Ohm’s law and using phasor algebra, we have Since i and v are in phase, thus, θ R = 0°, if phase is to be same. Thus, we define a new term, Z R as impedance of a resistive element (which impedes flow of current) 11/10/2015 4

5. Inductive Reactance- For the inductive circuit, Time domain equations:v = V m sin ωt and i = I m sin ωt In phasor form: WhereV = 0.707V m andwhere I = 0.707I m Applying Ohm’s law and using phasor algebra, we have Since i lags v by 90°, thus, θ L = 90°, for condition to be true. Thus, we define term, Z L as impedance of an inductive element (which impedes flow of current) 11/10/2015 5

6. Capacitive Reactance- For a capacitive circuit, Time domain equations:v = V m sin ωt and i = I m sin ωt In phasor form: WhereV = 0.707V m andwhere I = 0.707I m Applying Ohm’s law and using phasor algebra, we have Since i leads v by 90°, thus, θ C = –90°, for condition to be true. Thus, we define term, Z C as impedance of a capacitive element (which impedes flow of current) 11/10/2015 6

7. However, it is important to realize that Z R 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 Z L and Z C are also not phasor quantities 11/10/2015 7

8. 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 11/10/2015 8

9. Sketch of waveformandPhasor Diagram 11/10/2015 9

10. 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 11/10/2015 10

11. Sketch of waveformandPhasor Diagram 11/10/2015 11

12. 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 11/10/2015 12

13. Sketch of waveformandPhasor Diagram 11/10/2015 13

14. 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 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 θ T will be positive, whereas for capacitive networks θ T will be negative, and θ T will be zero for resistive cct.

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

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

19. 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 θ T is the phase angle between E and I. 11/10/2015 19

20. Impedance Relation with Power Factor We know that Reference to figs and equations θ T is not only the impedance angle of Z T but also θ T is the phase angle between the input voltage and current for a series ac circuit. 11/10/2015 20 Phasor Diagram Impedance Diagram Note: θ T of Z T is with reference to voltage unlike F P. Also current I is in phase with V R, lags the V L by 90°, and leads the V C by 90°.

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

22. R-L-C Example.Step 2 – Find Z T and make impedance diagram

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

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

25. R-L-C Example I = V R = V L = V C =.Step 5 – Make phasor diagram and. apply KVL (for verification or if req) Note: Current I in phase with V R, lags the V L by 90°, and leads the V C by 90°

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

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

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

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

30. (Series ac Circuits) Impedance and Phasors Diagram Series Configuration

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