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Chapter6Series-Parallel CircuitsTopics Covered in Chapter 66-1: Finding RT for Series-Parallel Resistances6-2: Resistance Strings in Parallel6-3: Resistance Banks in Series6-4: Resistance Banks and Strings in Series-Parallel 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Topics Covered in Chapter 6 6-5: Analyzing Series-Parallel Circuits with RandomUnknowns 6-6: The Wheatstone Bridge 6-7: Troubleshooting: Opens and Shorts in SeriesParallel CircuitsMcGraw-Hill 2007 The McGraw-Hill Companies, Inc. All rights reserved.

6-1: Finding RT forSeries-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 differentvoltage and current values are required from the samevoltage source. Series components form a series string. Parallel components form a parallel bank.

6-1: Finding RT forSeries-Parallel Resistances Overview of Series-Parallel Circuits13V2There are three branches in thiscircuit; sections 1 and 2 are series strings.

6-1: Finding RT forSeries-Parallel Resistances Overview of Series-Parallel Circuits13V2There are three series sections in thiscircuit; sections 1 and 2 are parallel banks.

6-1: Finding RT forSeries-Parallel Resistances To find RT for a series-parallelcircuit, add the seriesresistances and combine theparallel resistances. In this diagram, R1 and R2 arein series, and R3 and R4 are inparallel. However, R2 is not inseries with the parallelresistances: Resistances inseries have the same current,but the current in R2 is equalto the sum of the branchcurrents I3 and I4.Fig. 6-1b: Schematic diagram of a seriesparallel circuit.Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-1: Finding RT forSeries-Parallel Resistances For Fig. 6-1b, 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 thevalue of RT:1kΩ 0.5kΩ 1.5 kΩ

6-2: Resistance Strings in Parallel In this figure, branch 1has two resistances inseries; branch 2 hasonly one resistance. Ohm’s Law can beapplied to each branch,using the same rules forthe series and parallelcomponents that werediscussed in Chapters 4and 5.Fig. 6-3a: Series string in parallel withanother branch (schematic diagram).Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-2: Resistance Strings in Parallel Series Circuit Current is the same in allcomponents. V across each series R isI R. VT V1 V2 V3 . etc. Parallel Circuit Voltage is the sameacross all branches. I in each branch R is V/R. IT I1 I2 I3 . etc.

6-2: Resistance Strings in ParallelVI is the samein thissection.V is the same across each parallel branch.

6-2: Resistance Strings in Parallel The current in each branch equals the voltage appliedacross the branch divided by the branch RT. The total line current equals the sum of the branchcurrents for all parallel strings. The RT for the entire circuit equals the applied voltagedivided by the total line current. For any resistance in a series string, the IR voltage dropacross that resistance equals the string’s currentmultiplied by the resistance. The sum of the voltage drops in the series string equalsthe voltage across the entire string.

6-3: Resistance Banks in Series In this figure, R2 and R3are parallel resistances ina bank. The parallel bankis in series with R1. There may be more thantwo parallel resistances ina bank, and any numberof banks in series. Ohm’s Law is applied tothe series and parallelcomponents as seenpreviously.Fig. 6-4a: Parallel bank of R2 and R3 inseries with R1 (Original circuit).Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-3: Resistance Banks in Series To find the total resistance of this type of circuit,combine the parallel resistances in each bank and addthe series resistances.VR I24VR 4A24V6Ω 4A10 Ω (of R2 R3)6Ω 1Ω (R1)2 branches6Ω 5Ω 1Ω

6-4: Resistance Banks and Strings inSeries-Parallel To solve series-parallel (combination) circuits, it isimportant to know which components are in series withone another and which components are in parallel. Series components must be in one current path withoutany branch points. To find particular values for this type of circuit, Reduce and combine the components using the rulesfor individual series and parallel circuits. Reduce the circuit to its simplest possible form. Then solve for the needed values using Ohm’s Law.

6-4: Resistance Banks and Strings inSeries-Parallel Example: Find all currents and voltages in Fig. 6-5. Step 1: Find RT. Step 2: Calculate main line current as IT VT / RTFig. 6-5: Reducing a series-parallel circuit to an equivalent series circuit to find the RT. (a)Actual circuit. (b) R3 and R4 in parallel combined for the equivalent RT.Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-4: Resistance Banks and Strings inSeries-ParallelFig. 6-5, cont. (c) RT and R6 in series added for R13. (d) R13 and R5 in parallel combined for R18.Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-4: Resistance Banks and Strings inSeries-ParallelFig. 6-5e: The R18, R1, and R2 in series are added for the total resistance of 50Ω for RT.Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-5: Analyzing Series-Parallel Circuitswith Random Unknowns In solving such circuits, apply the same principles asbefore: Reduce the circuit to its simplest possible form. Apply Ohm’s Law.

6-5: Analyzing Series-Parallel Circuitswith Random Unknowns Example: In Fig. 6-6, we canfind branchcurrents I1 and I2-3,and IT, andvoltage drops V1,V2, and V3, withoutknowing the valueof RT.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.

6-5: Analyzing Series-Parallel Circuitswith Random Unknowns Find I1, I2-3, and IT.VI1 R90V (parallel branches have the same voltage)I1 30ΩI1 3A

6-5: Analyzing Series-Parallel Circuitswith Random UnknownsVI2-3 R90VI2-3 20Ω 25ΩIT I1 I2-3IT 3A 2AIT 5A90VI2-3 45ΩI2-3 2A

6-5: Analyzing Series-Parallel Circuitswith Random Unknowns Find voltage drops V1, V2,and V3:Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-5: Analyzing Series-Parallel Circuitswith Random UnknownsV1 VA (parallel branches have the same voltage)V1 90VorV1 I1R1V2 I2-3R2V3 I2-3R3V1 3A 30Ω V2 2A(20 Ω)V3 2A(25 Ω)V1 90VV2 40VV3 50VNote: V2 V3 VA40V 50V 90V

6-5: Analyzing Series-Parallel Circuitswith Random UnknownsRT RT RT VAIT90A5A18Ω

6-6: The Wheatstone Bridge A Wheatstone bridge is a circuit that is used todetermine the value of an unknown resistance. The unknown resistor (RX) is in the same branch as thestandard resistor (RS).Fig. 6-10: Wheatstone bridge.Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-6: The Wheatstone Bridge Resistors R1 and R2 form the ratio arm; they have verytight resistance tolerances. The galvanometer (M1), a sensitive current meter, isconnected between the output terminals C and D. When R1 / R2 R3 / R4, the bridge is balanced. When the bridge is balanced, the current in M1 is zero.

6-6: The Wheatstone Bridge Using a Wheatstone Bridge to Measure an UnknownResistance RS is adjusted for zero current in M1. When the current in M1 0A, the voltage divisionbetween RX and RS is equal to that between R1 and R2.Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-6: The Wheatstone BridgeNote: When the Wheatstone bridge is balanced, it can beanalyzed as two series strings in parallel. Note thefollowing relationship:R1RX RSR2RXR1 RS R2

6-7: Troubleshooting: Opens andShorts in Series-Parallel Circuits In series-parallel circuits, an open or short in one part ofthe circuit changes the values in the entire circuit. When troubleshooting series-parallel circuits, combinethe techniques used when troubleshooting individualseries and parallel circuits.

6-7: Troubleshooting: Opens andShorts 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 S1open. (b) Circuit with short between points A and B when S1 is closed; now R2 and R3 are shortcircuited.Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-7: Troubleshooting: Opens andShorts in Series-Parallel Circuits Effect of a Short in a Series-Parallel CircuitThe total current increases from 2Awith S1 open to 10A with S1 closed.Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.With S1 closed, R2 and R3are shorted out.

6-7: Troubleshooting: Opens andShorts in Series-Parallel Circuits Effect of an Open in a Series-Parallel CircuitFig. 6-14: Effect of an open path in a seriesparallel circuit. (a) Normal circuit with S2closed. (b) Series circuit with R1 and R2 whenS2 is open. Now R3 in the open path has nocurrent and zero IR voltage drop.With S2 open, R3 is effectively removed fromthe circuit.Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6-7: Troubleshooting: Opens andShorts in Series-Parallel Circuits Effect of an Open in a Series-Parallel CircuitWith S2 open the voltage across points C and D equals thevoltage across R2,which is 89V. The voltage across R3 iszero.Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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 .

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