# Series Parallel Analysis of AC Circuits

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1 HAE 9 eries arallel Analysis of A ircuits hapter Outline 9. A eries ircuits 9.2 A arallel ircuits 9.3 A eries arallel ircuits 9.4 Analysis of Multiple-ource A ircuits Using uperposition 9. A EIE IUI In D analysis of series-parallel circuits, what type of numbers was utilized? If the signal source is A instead of D and the circuit contains inductors and/or capacitors, what type of numbers would you epect to be utilized? Why? Good news: the circuit analysis techniques are the same. What is the phase shift of the voltage with respect to the current for an inductor? Why? What is the phase shift of the voltage with respect to the current for a capacitor? Why? What is the phase shift of the voltage with respect to the current for a resistor? Why? What would you epect the phase shift to be for A signals in series, parallel, and series parallel circuits with resistances, inductors, and capacitors? Why? ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age trangeway, etersen, Gassert, and Lokken

2 How is series parallel circuit analysis with A sources approached? eview the circuit laws, concepts, and results for D circuit analysis by stating in words each relation that follows:. I = I 2 = I 3 =. = I (9.) 2. Vn Vrise Vrise2 Vrise3 Vdrop Vdrop2 Vdrop3 n 0 (9.2) How are the signs for voltage rises and voltage drops assigned (Figure 9. below)? Is the key to the sign of the voltage rise (or drop) the current direction or the direction that KVL is applied? n n 4. (9.3) V V (9.4) Which of these laws and concepts are applicable to A series circuits? Answer the following questions: Does constant current throughout a series circuit change for any signal? Does KVL change for any signal? Why or why not? Are 3 and 4 above valid? Why or why not? If the same D circuit analysis techniques are valid for A, and resistances are used in D circuit analysis, then what must be used in A circuit analysis? Why? tate in words the circuit laws, concepts, and results for A series circuits: I I I I (9.5) V n V rise V rise2 V rise3 V drop V drop2 V drop3 n 0 (9.6) ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 2 trangeway, etersen, Gassert, and Lokken

3 How is a voltage polarity assigned with A signals considering the polarity is alternating? n n Z Z Z Z Z Z (9.7) 4. V Z V Z (9.8) Eample 9.. (Eplain each step.) For the circuit shown in Figure 9.2 below, determine (a) Z, (b) I, (c) V and V L by the VD, and (d) V by Ohm s law. (e) Verify KVL. : v (t) = 0 sin(827t) V = 35 = 20 F L = 00 mh : a. Z b. I trategy: Determine Z, Z L, and Z Z Z Z Z L c. V and V L by VD d. V by Ohm s law e. Verify KVL Label a circuit diagram with voltage polarities and current directions. V I Z Z Z V V, V V Z Z L L V I Z V V V V 0 L olution: Z 35 ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 3 trangeway, etersen, Gassert, and Lokken

4 Z j j j (827)(20 ) Z jl j(827)(0.) j82.7 L a. Z Z Z Z 35 j60.46 j j j 22.2 L ote: All voltages and currents are epressed as peak values in this eample. ee Figure 9.3 below. b. V 0 0 I A Z 35 j22.24 c. V Z 35 V V Z 35 j22.24 Determine V L in a similar manner. Answer: V L V d. V I Z ( )( j60.46) V e. (efer to Figure 9.3) V V V V 0 erform KVL. L Is KVL satisfied? ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 4 trangeway, etersen, Gassert, and Lokken

5 9.2 A AALLEL IUI tate the important laws, concepts, and results that were found for D parallel circuits in words:. V = V = V 2 = V 3 = = V (9.9) 2. n 2 3 n I I I I I 0 (9.0) How is the sign of the current assigned to if it is entering the node? Leaving the node? 3. 2 in general (9.) for two parallel resistors (9.2) 4. I I in general (9.3) I I, I I for two parallel resistors (9.4) 5. I G V (9.5) 6. G = G + G G (9.6) Which of these laws and concepts are applicable to A series circuits? Answer the following questions: Does constant voltage across a parallel circuit change for any signal? Does KL change for any signal? Why or why not? Are 3-6 above valid? Why or why not? If the same D circuit analysis techniques are valid for A, and resistances are used in D circuit analysis, then what must be used in A circuit analysis? Why? ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 5 trangeway, etersen, Gassert, and Lokken

6 tate in words the circuit laws, concepts, and results for A parallel circuits: V V V V V (9.7) n 2 3 n I I I I I 0 (9.8) 3. Z Z Z Z 2 (general or special case?) (9.9) Z Z Z 2 ZZ 2 Z Z 2 (general or special case?) (9.20) 4. IZ Z I (general or special case?) (9.2) I I Z, I I Z 2 2 Z Z 2 Z Z 2 (general or special case?) (9.22) Eample 9.2. (Eplain each step.) For the circuit shown in Figure 9.4 below, determine (a) Z, (b) V, (c) I by the D, and (d) I by Ohm s law. (e) Verify KL. : = 0 X = 8 I 820 A : a. Z b. V c. I by the D d. I by Ohm s law e. Verify KL : Determine Z and Z Z Z Z Z Z Label a circuit diagram with voltage polarities and current directions. V I Z I I Z IZ or I Z Z Z (which equation is easier? note the previous step) ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 6 trangeway, etersen, Gassert, and Lokken

7 I V Z Apply KL to the top node. olution: Z 0 Z jx j8 a. Z ZZ (0)( j8) Z Z 0 j8 b. ee Figure 9.5 below for voltage polarity and current direction assignments. V I Z (8 20 )( ) V c. I IZ Z (8 20 )( ) A d. I V A Z j8 e. KL at node: I I I 0 erform KL: Is KL satisfied? ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 7 trangeway, etersen, Gassert, and Lokken

8 What is the inverse of resistance (G = /)? What is the inverse of reactance? B X (9.23) Identify: B L X L B (9.24) X If susceptance is the inverse of reactance, what is the inverse of impedance? Y Z (9.25) Is Y a comple number? Why or why not? How does admittance ( Y ) relate to conductance (G) and susceptance (B)? Eplain each step: (9.26) Z jx Why is the plus sign used with inductive reactance? Why is the minus sign used with capacitive reactance? (9.27) Y G jb onductance is the part of admittance and susceptance is the part of admittance. I I Y ( I V) Z V V (9.28) Why is the sign minus used with inductive susceptance? Why is the plus sign used with capacitive susceptance? What is the unit of admittance? abbreviation? 5. I Y V Z () (9.29) 6. 2 n n Y Y Y Y Y (9.30) Eample (Eplain each step.) For the circuit shown previously in Figure 9.4 (repeated below), determine (a) Y from G and B, and (b) Y from Z. Given: = 0 X = 8 Z j (from Eample 9.2.) ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 8 trangeway, etersen, Gassert, and Lokken

9 Desired: a. Y from G and B b. Y from Z trategy: a. G, B X, Y G jb b. Y Z olution: a. G 0. 0 B X 8 Y G jb 0. j j b. Y Z j j Why do the results from parts (a) and (b) match? Observation about the impedance admittance relationship: start with Z of the circuit in Figure 9.4 (repeated above). Eplain each step: Z j jx onsider: G (9.3) B (9.32) X ompare these results with those of Eample 9.2.2: G (9.33) B (9.34) X 8 hese answers do not match! Why not? Which approach is correct? ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 9 trangeway, etersen, Gassert, and Lokken

10 Eample (fill in) Determine the (a) total equivalent admittance and (b) parallel equivalent circuit for the series circuit shown in Figure 9.6a (below). : = 0 X = 3 (inductive) : a. Y b. and X : a. Y, epress in rectangular form Z olution: (fill in) b. G, and X B p Answers: a. Y j j b. 26.9, X 20.7 Eample (fill in) Determine the resistance and reactance values of the series equivalent circuit for the parallel circuit shown in Figure 9.7a (below). Given: Desired: trategy: olution: Answer: = ; X = ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 0 trangeway, etersen, Gassert, and Lokken

11 Eample (fill in) Determine the component values for the parallel equivalent circuit of the series circuit shown in Figure 9.8a (below) if = 20 krads/s. Given: Desired: trategy: olution: Eample Answer: 2.5, 2.00 F (fill in) For the circuit in Figure 9.9 (below), determine (a) the total equivalent admittance, and (b) the values for the components of the series equivalent circuit. he frequency is khz. Given: Desired: trategy: a. Y Y Y Y L G jb jb L j j L b. Z jx, X Y olution: Answers: a. Y ( j3.00) m (8.33 j3.0) m b. Z 05.4 j F Why are there only two reactive components in the series equivalent circuit? ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age trangeway, etersen, Gassert, and Lokken

12 Generalization of eries arallel onversions to Handbook Equations (Eplain each step.) ase : eries-to-arallel onversion see Fig. 9.0: Z jx, Y G jb (9.35) Y ( jx ) (9.36) Y ( jx ) jx ( jx ) ( jx ) X 2 2 (9.37) X Y j X X (9.38) G, 2 2 X 2 2 X (9.39) jx jb, 2 2 X X 2 2 X (9.40) X ase 2: arallel-to-eries onversion see Fig. 9.: Z ( jx ), Z jx (9.4) ( jx ) Z ( jx ) (9.42) Z jx jx jx jx jx ( jx ) ( jx ) X 2 ( ) ( ) ( ) 2 2 (9.43) X j X X X Z (9.44) j X X X X X (9.45) X 2 jx j 2 2 X, X X X (9.46) ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 2 trangeway, etersen, Gassert, and Lokken

13 9.3 A eries arallel ircuits KVL, total equivalent impedance, the VD, and the fact that the current is the same are generally applied to what type of circuit? KL, total equivalent impedance and admittance, the D, and the fact that the voltage is the same are generally applied to what type of circuit? he following general guidelines for D series parallel circuit analysis are etended to A:. he general approach to analyzing an A series parallel circuit is to identify the groups of that are in series and those that are in parallel and to apply the appropriate series and parallel circuit laws to the impedances in that group. 2. here are usually multiple approaches to analyzing an A series parallel circuit. think about your strategy formulate an efficient approach 3. A single-source series parallel circuit is fundamentally a series circuit or fundamentally a parallel circuit. he source is the key. If a single impedance is in series with the source on either side (or both sides) of the source, then the circuit is fundamentally a circuit. If the current path from the source divides on both sides of the source, then the circuit is fundamentally a circuit. 4. A group of series impedances, usually a leg in a parallel circuit, is called a. 5. In the VD and the D, the total voltage or current becomes the total voltage or the current for that group of impedances, not necessarily the entire circuit. Eample 9.3. (Eplain each step.) Determine the total equivalent admittance of the circuit in Figure 9.2 (below). Given: circuit in Figure 9.2 Desired: trategy: Y, Z j8 7, Y 2 j4 Y Z olution: Y 0.5 j j4 (identify group ) Z j j j0.25 Y Z 8.6 j j ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 3 trangeway, etersen, Gassert, and Lokken

14 Eample (Eplain each step.) Determine (a) I and (b) V o for the circuit shown in Figure 9.3 (below). Given: circuit in Figure 9.3 Desired: a. I b. V o trategy: Z [5 ( j3)], Z j6 Z 4 (identify Z ) V I Z V o I Z V o I Z olution: Z (5)( j3) j3 Z I j V A Z V o I Z ( )( ) V I V o A A Z 3 90 Eample edraw the circuit in Figure 9.4 (below), clearly showing series and parallel groups of components. olution: (ry to redraw the circuit first, then check it with the answer in Figure 9.5 on the net page.) ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 4 trangeway, etersen, Gassert, and Lokken

15 Eample (fill in steps) Determine (a) I and (b) V ab for the circuit shown in Figure 9.6 (below). Given: circuit in Figure 9.6 Desired: a. I b. V ab trategy: edraw and label the circuit (net to Fig. 9.6) olution: (use separate paper) Answers: a. I A b. V V ab ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 5 trangeway, etersen, Gassert, and Lokken

16 9.4 AALYI OF MULILE-OUE A IUI UIG UEOIIO onsider A series parallel circuits that contain several sources as well as several impedances. What condition on the sources must hold true if superposition with phasors is to be used? Why? (Hint: reactances) In superposition the circuit is analyzed one source at a time. he other sources are. How does one deactivate an A voltage source? Why does it guarantee 0 V? How is an A current source deactivated? Why does it guarantee 0 A? Write the general superposition procedure his last step is actually the superposition step: the phasor voltages and currents are superposed. ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 6 trangeway, etersen, Gassert, and Lokken

17 Eample 9.4. Determine the current I a and i a (t) in Figure 9.8 (below). eak values for sources are shown. Given: circuit in Figure 9.8 Desired: trategy: I a and i a (t) I a uses the peak value of i a (t). superposition olution: Deactivate the 5 V source; redraw the circuit (see Figure 9.9 below). Eplain each step: substrategy: Z j3 5 j6 (4 j2) 0 30 I Z D I a Z 5 j3 j6 4 j2 5 j I A ( )(6 90 ) I a A ( j6) (4 j2) Fill in the rest of the solution steps on separate paper: Deactivate the 0 V source; redraw the circuit; determine the substrategy; determine I and I. a a Answers: I A, I A, i a (t) =.93 sin(t ) A a a ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 7 trangeway, etersen, Gassert, and Lokken

18 A sources he properties of ideal A sources are An ideal A voltage source has a constant and can deliver any An ideal A current source delivers a constant at any Eplain a practical A voltage source model shown in Figure 9.2b (below). How was the following equation obtained? V V V V I Z (9.47) t Zint t int What does the previous equation predict about the terminal voltage of a practical A source? Eplain a practical A current source model shown in Figure 9.22b (below). How was the following equation obtained? Vt I t I I Z I int Z int (9.48) What does the previous equation predict about the terminal current of a practical A source? Which quantity was not calculated in this chapter that was determined in previous chapters (conspicuous by its absence)? ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 8 trangeway, etersen, Gassert, and Lokken

19 Learning Objectives Discussion: an you perform each learning objective for this chapter? (Eamine each one.) As a result of successfully completing this chapter, you should be able to:. Utilize phasors in A series parallel circuit voltage and current calculations. 2. Describe the fundamental properties of series and parallel circuits and subcircuits. 3. alculate all voltages and currents in single-source A series, parallel, and series parallel circuits. 4. erform series parallel circuit conversions. 5. Describe why and how superposition is used in multiple-source A series, parallel, and series parallel circuits. 6. alculate all voltages and currents in multiple-source A series, parallel, and series parallel circuits. 7. Eplain what an A current source is and how to analyze A series parallel circuits that contain a current source. ontemporary Electric ircuits, 2 nd ed., rentice-hall, 2008 lass otes h. 9 age 9 trangeway, etersen, Gassert, and Lokken

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