Series-Parallel Circuits
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1 Chapter 6 Series-Parallel Circuits Topics Covered in Chapter 6 6-1: Finding R T for Series-Parallel Resistances 6-2: Resistance Strings in Parallel 6-3: Resistance Banks in Series 6-4: Resistance Banks and Strings in Series-Parallel 2007 The McGraw-Hill Companies, Inc. All rights reserved.
2 Topics Covered in Chapter 6 6-5: Analyzing Series-Parallel Circuits with Random Unknowns 6-6: The Wheatstone Bridge 6-7: Troubleshooting: Opens and Shorts in Series- Parallel Circuits McGraw-Hill 2007 The McGraw-Hill Companies, Inc. All rights reserved.
3 6-1: Finding R for T Series-Parallel Resistances Overview of Series-Parallel Circuits A series-parallel circuit, or combination circuit, combines both series and parallel connections. Most electronic circuits fall into this category. Series-parallel circuits are typically used when different voltage and current values are required from the same voltage source. Series components form a series string. Parallel components form a parallel bank.
4 6-1: Finding R for T Series-Parallel Resistances Overview of Series-Parallel Circuits V There are three branches in this circuit; sections 1 and 2 are series strings.
5 6-1: Finding R for T Series-Parallel Resistances Overview of Series-Parallel Circuits 1 V 3 2 There are three series sections in this circuit; sections 1 and 2 are parallel banks.
6 6-1: Finding R for T Series-Parallel Resistances To find R T for a series-parallel circuit, add the series resistances and combine the parallel resistances. In this diagram, R 1 and R 2 are in series, and R 3 and R 4 are in parallel. However, R 2 is not in series with the parallel resistances: Resistances in series have the same current, but the current in R 2 is equal to the sum of the branch currents I 3 and I 4. Fig. 6-1b: Schematic diagram of a seriesparallel circuit. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
7 For Fig. 6-1b, 6-1: Finding R for T Series-Parallel Resistances The series resistances are: 0.5kΩ + 0.5kΩ = 1kΩ The parallel resistances are: 1kΩ / 2 = 0.5kΩ The series and parallel values are then added for the value of R T : 1kΩ + 0.5kΩ = 1.5 kω
8 6-2: Resistance Strings in Parallel In this figure, branch 1 has two resistances in series; branch 2 has only one resistance. Ohm s Law can be applied to each branch, using the same rules for the series and parallel components that were discussed in Chapters 4 and 5. Fig. 6-3a: Series string in parallel with another branch (schematic diagram). Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
9 6-2: Resistance Strings in Parallel Series Circuit Current is the same in all components. V across each series R is I R. V T = V 1 + V 2 + V etc. Parallel Circuit Voltage is the same across all branches. I in each branch R is V/R. I T = I 1 + I 2 + I etc.
10 6-2: Resistance Strings in Parallel V I is the same in this section. V is the same across each parallel branch.
11 6-2: Resistance Strings in Parallel The current in each branch equals the voltage applied across the branch divided by the branch R T. The total line current equals the sum of the branch currents for all parallel strings. The R T for the entire circuit equals the applied voltage divided by the total line current. For any resistance in a series string, the IR voltage drop across that resistance equals the string s current multiplied by the resistance. The sum of the voltage drops in the series string equals the voltage across the entire string.
12 6-3: Resistance Banks in Series In this figure, R 2 and R 3 are parallel resistances in a bank. The parallel bank is in series with R 1. There may be more than two parallel resistances in a bank, and any number of banks in series. Ohm s Law is applied to the series and parallel components as seen previously. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 6-4a: Parallel bank of R 2 and R 3 in series with R 1 (Original circuit).
13 6-3: Resistance Banks in Series To find the total resistance of this type of circuit, combine the parallel resistances in each bank and add the series resistances. V R = I R= 24V 4A 6Ω = 10 Ω (of R 2 + R 3 ) 2 branches + 1Ω (R 1 ) 6Ω = 24V 4A 6Ω = 5Ω + 1Ω
14 6-4: Resistance Banks and Strings in Series-Parallel To solve series-parallel (combination) circuits, it is important to know which components are in series with one another and which components are in parallel. Series components must be in one current path without any branch points. To find particular values for this type of circuit, Reduce and combine the components using the rules for individual series and parallel circuits. Reduce the circuit to its simplest possible form. Then solve for the needed values using Ohm s Law.
15 6-4: Resistance Banks and Strings in Example: Series-Parallel Find all currents and voltages in Fig Step 1: Find R T. Step 2: Calculate main line current as I T = V T / R T Fig. 6-5: Reducing a series-parallel circuit to an equivalent series circuit to find the RT. (a) Actual circuit. (b) R 3 and R 4 in parallel combined for the equivalent R T. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
16 6-4: Resistance Banks and Strings in Series-Parallel Fig. 6-5, cont. (c) R T and R 6 in series added for R 13. (d) R 13 and R 5 in parallel combined for R 18. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
17 6-4: Resistance Banks and Strings in Series-Parallel Fig. 6-5e: The R 18, R 1, and R 2 in series are added for the total resistance of 50Ω for R T. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
18 6-5: Analyzing Series-Parallel Circuits with Random Unknowns In solving such circuits, apply the same principles as before: Reduce the circuit to its simplest possible form. Apply Ohm s Law.
19 6-5: Analyzing Series-Parallel Circuits Example: with Random Unknowns In Fig. 6-6, we can find branch currents I 1 and I 2-3, and I T, and voltage drops V 1, V 2, and V 3, without knowing the value of R T. Fig. 6-6: Finding all the currents and voltages by calculating the branch currents first. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
20 6-5: Analyzing Series-Parallel Circuits with Random Unknowns Find I 1, I 2-3, and I T. I 1 = V R I 1 = 90V 30Ω (parallel branches have the same voltage) I 1 = 3A
21 6-5: Analyzing Series-Parallel Circuits with Random Unknowns I 2-3 = V R I T = I 1 + I 2-3 I 2-3 = I 2-3 = 90V 45Ω 90V 20Ω + 25Ω I T = 3A + 2A I T = 5A I 2-3 = 2A
22 6-5: Analyzing Series-Parallel Circuits with Random Unknowns Find voltage drops V 1, V 2, and V 3 : Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
23 6-5: Analyzing Series-Parallel Circuits with Random Unknowns V 1 = V A (parallel branches have the same voltage) V 1 = 90V or V 1 = I 1 R 1 V 2 = I 2-3 R 2 V 3 = I 2-3 R 3 V 1 = 3A 30Ω V 2 = 2A(20 Ω) V 3 = 2A(25 Ω) V 1 = 90V V 2 = 40V V 3 = 50V Note: V 2 + V 3 = V A 40V + 50V = 90V
24 6-5: Analyzing Series-Parallel Circuits with Random Unknowns R T R T R T = = = V A I T 90A 5A 18Ω
25 6-6: 6: The Wheatstone Bridge A Wheatstone bridge is a circuit that is used to determine the value of an unknown resistance. The unknown resistor (R X ) is in the same branch as the standard resistor (R S ). Fig. 6-10: Wheatstone bridge. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
26 6-6: 6: The Wheatstone Bridge Resistors R 1 and R 2 form the ratio arm; they have very tight resistance tolerances. The galvanometer (M 1 ), a sensitive current meter, is connected between the output terminals C and D. When R 1 / R 2 = R 3 / R 4, the bridge is balanced. When the bridge is balanced, the current in M 1 is zero.
27 6-6: 6: The Wheatstone Bridge Using a Wheatstone Bridge to Measure an Unknown Resistance R S is adjusted for zero current in M 1.. When the current in M 1 = 0A, the voltage division between R X and R S is equal to that between R 1 and R 2. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
28 6-6: 6: The Wheatstone Bridge Note: When the Wheatstone bridge is balanced, it can be analyzed as two series strings in parallel. Note the following relationship: R X = R 1 R S R 2 = R X R S R 1 R 2
29 6-7: Troubleshooting: Opens and Shorts in Series-Parallel Circuits In series-parallel circuits, an open or short in one part of the circuit changes the values in the entire circuit. When troubleshooting series-parallel circuits, combine the techniques used when troubleshooting individual series and parallel circuits.
30 6-7: Troubleshooting: Opens and Shorts in Series-Parallel Circuits Effect of a Short in a Series-Parallel Circuit The total current and total power increase.. Fig. 6-13: Effect of a short circuit with series-parallel connections. (a) Normal circuit with S 1 open. (b) Circuit with short between points A and B when S 1 is closed; now R 2 and R 3 are shortcircuited. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
31 6-7: Troubleshooting: Opens and Shorts in Series-Parallel Circuits Effect of a Short in a Series-Parallel Circuit The total current increases from 2A with S 1 open to 10A with S 1 closed. With S 1 closed, R 2 and R 3 are shorted out. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
32 6-7: Troubleshooting: Opens and Shorts in Series-Parallel Circuits Effect of an Open in a Series-Parallel Circuit Fig. 6-14: Effect of an open path in a seriesparallel circuit. (a) Normal circuit with S 2 closed. (b) Series circuit with R 1 and R 2 when S 2 is open. Now R 3 in the open path has no current and zero IR voltage drop. With S 2 open, R 3 is effectively removed from the circuit. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
33 6-7: Troubleshooting: Opens and Shorts in Series-Parallel Circuits Effect of an Open in a Series-Parallel Circuit With S 2 open the voltage across points C and D equals the voltage across R 2,which is 89V. The voltage across R 3 is zero. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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