1 ELECTRICAL TECHNOLOGY ET 201  Define series impedances and analyze series AC circuits using circuit techniques.

Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current (review) FIG Reviewing the frequency response of the basic elements.

3 (CHAPTER 15) SERIES AC CIRCUITS

Series Impedances The 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: [Ω][Ω]

5 Example 15.7 Draw the impedance diagram and find the total impedance Series Impedances Solution

6 Example 15.8 Draw the impedance diagram and find the total impedance Series Impedances Solution

Series AC Circuit In a series AC configuration having two impedances, the current I is the same through each element (as it was for the series DC circuit) The current is determined by Ohm’s Law:

8 Kirchhoff’s Voltage Law can be applied in the same manner as it is employed for a DC circuit. The power to the circuit can be determined by: Where E, I : effective values (E rms, I rms ) θ T : phase angle between E and I 15.3 Series Configuration

Power Factor For a purely resistive load; Hence; For purely inductive or purely capacitive load; Hence;

Power Factor Power factor can be lagging or leading. –Defined by the current through the load. Lagging power factor: –Current lags voltage –Inductive circuit Leading power factor: –Current leads voltage –Capacitive circuit

11 R-L 1. Phasor Notation 15.3 Series Configuration Series R-L circuit Apply phasor notation

12 R-L 2. Z T Impedance diagram: 15.3 Series Configuration

13 R-L 3. I 15.3 Series Configuration

14 R-L 4. V R and V L Ohm’s Law: 15.3 Series Configuration

15 R-L Kirchhoff’s voltage law: Or; In rectangular form, 15.3 Series Configuration

16 R-L Phasor diagram: I is in phase with the V R and lags the V L by 90 o. I lags E by o Series Configuration

17 R-L Power: The total power delivered to the circuit is Where E, I : effective values; θ T : phase angle between E and I Or; 15.3 Series Configuration

18 R-L Power factor: 15.3 Series Configuration

19 R-C 1. Phasor Notation 15.3 Series Configuration Series R-C circuit Apply phasor notation

20 R-C 2. Z T Impedance diagram: 15.3 Series Configuration

21 R-C 3. E 15.3 Series Configuration

22 R-C 4. V R and V C Ohm’s Law: 15.3 Series Configuration

23 R-C Kirchhoff’s voltage law: Or; 15.3 Series Configuration

24 R-C Phasor diagram: I is in phase with the V R and leads the V C by 90 o. I leads E by o Series Configuration

25 R-C Time domain: 15.3 Series Configuration

26 R-C Power: The total power delivered to the circuit is Or; 15.3 Series Configuration

27 R-C Power factor: Or; 15.3 Series Configuration

28 R-L-C 1. Phasor Notation TIME DOMAIN PHASOR DOMAIN 15.3 Series Configuration

29 R-L-C Impedance diagram: 2. Z T 15.3 Series Configuration

30 R-L-C 3. I 15.3 Series Configuration

31 R-L-C 4. V R, V L and V C Ohm’s Law: 15.3 Series Configuration

32 R-L-C Kirchhoff’s voltage law: Or; 15.3 Series Configuration

33 R-L-C Phasor diagram: I is in phase with the V R, lags the V L by 90 o, leads the V C by 90 o I lags E by o Series Configuration

34 R-L-C Time domain: 15.3 Series Configuration

35 R-L-C Power: The total power delivered to the circuit is Or; Power factor: 15.3 Series Configuration

36 The basic format for the VDR in AC circuits is exactly the same as that for the DC circuits. Where V x : voltage across one or more elements in a series that have total impedance Z x E : total voltage appearing across the series circuit. Z T : total impedance of the series circuit Voltage Divider Rule

37 Example 15.11(a) Calculate I, V R, V L and V C in phasor form Series Configuration

38 Example 15.11(a) - Solution Combined the R’s, L’s and C’s Series Configuration e

39 Example 15.11(a) – Solution (cont’d) Find the reactances. 1. Transform the circuit into phasor domain Series Configuration E

40 Example 15.11(a) – Solution (cont’d) 2. Determine the total impedance. 3. Calculate I Series Configuration E

41 Example 15.11(a) – Solution (cont’d) 4. Calculate V R, V L and V C 15.3 Series Configuration E

Series Configuration Example 15.11(b) Calculate the total power factor. Solution Angle between E and I is

43 Example 15.11(c) Calculate the average power delivered to the circuit. Solution 15.3 Series Configuration

44 Example 15.11(d) Draw the phasor diagram. Solution 15.3 Series Configuration

45 Example 15.11(e) Obtain the phasor sum of V R, V L and V C and show that it equals the input voltage E. Solution 15.3 Series Configuration

46 Example 15.11(f) Find V R and V C using voltage divider rule. Solution 15.3 Series Configuration E

Summaries of Series AC Circuits For a series AC circuits with reactive elements: The total impedance will be frequency dependent. The impedance of any one element can be greater than the total impedance of the network. The inductive and capacitive reactances are always in direct opposition on an impedance diagram. Depending on the frequency applied, the same circuit can be either predominantly inductive or predominantly capacitive.

Summaries of Series AC Circuits (continued…) At lower frequencies, the capacitive elements will usually have the most impact on the total impedance. At high frequencies, the inductive elements will usually have the most impact on the total impedance. The magnitude of the voltage across any one element can be greater than the applied voltage.

Summaries of Series AC Circuits (continued…) The magnitude of the voltage across an element as compared to the other elements of the circuit is directly related to the magnitude of its impedance; that is, the larger the impedance of an element, the larger the magnitude of the voltage across the element. The voltages across an inductor or capacitor are always in direct opposition on a phasor diagram.

Summaries of Series AC Circuits (continued…) The current is always in phase with the voltage across the resistive elements, lags the voltage across all the inductive elements by 90°, and leads the voltage across the capacitive elements by 90°. The larger the resistive element of a circuit compared to the net reactive impedance, the closer the power factor is to unity.