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Lecture 2: Resistive Circuits Nilsson 2.5, 3.1-3.4, 3.7 ENG17 : Circuits I Spring 2015 1 April 2, 2015

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Overview Dependent Sources Resistors in Series Resistors in Parallel Voltage Divider Current Divider Voltage / Current Division Δ-to-Y Equivalent Circuits 2

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Finding v o 3 Strategy: If we find i o, we know v o If we find i Δ, we know the dependent source The current in 500V source is i Δ Use KCL and KVL to do find unknown currents

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Finding v o 4 KVL: KCL: only 1 loop option Ohm’s Law: can only use node b or c

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Overview Dependent Sources Resistors in Series Resistors in Parallel Voltage Divider Current Divider Voltage / Current Division Δ-to-Y Equivalent Circuits 5

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Series Connected Elements Resistors in series carry the same current –KCL proves this Resistors in series can be summed –KVL proves this 6

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Overview Dependent Sources Resistors in Series Resistors in Parallel Voltage Divider Current Divider Voltage / Current Division Δ-to-Y Equivalent Circuits 7

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Parallel Connected Elements Resistors in parallel have same voltage across their terminals –KVL proves this Resistors in parallel can be summed inversely –KCL proves this –Conductances are summed 8

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Overview Dependent Sources Resistors in Series Resistors in Parallel Voltage Divider Current Divider Voltage / Current Division Δ-to-Y Equivalent Circuits 9

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Voltage Divider One voltage supply, multiple voltage levels 10

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Load R L = load = 1+ circuit elements that draw power from circuit –May alter voltage across parallel element –But not if R L >> R 2 11

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Example 12

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Example Wheatstone Bridge –2 voltage dividers in parallel 13

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Overview Dependent Sources Resistors in Series Resistors in Parallel Voltage Divider Current Divider Voltage / Current Division Δ-to-Y Equivalent Circuits 14

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Current Divider 15 One current supply, multiple current levels

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Example 16

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Example 17 Similar analysis

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Overview Dependent Sources Resistors in Series Resistors in Parallel Voltage Divider Current Divider Voltage / Current Division Δ-to-Y Equivalent Circuits 18

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Voltage Division 19 Generalization of voltage divider circuit

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Current Division 20 Generalization of current divider circuit

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Example 21

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Overview Dependent Sources Resistors in Series Resistors in Parallel Voltage Divider Current Divider Voltage / Current Division Δ-to-Y Equivalent Circuits 22

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Unable to get R eq For example, Wheatstone bridge circuit (below) cannot be simplified with standard parallel/series rules Must use circuit transformation 23

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Defining Delta/Pi and Y/T 24 Δ Y T π

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Delta-to-Wye Circuit Transformation Equivalent resistance between nodes Note: resistor values are NOT THE SAME 25

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Apply to Wheatstone Bridge 26

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Recap Dependent Sources Resistors in Series Resistors in Parallel Voltage Divider Current Divider Voltage / Current Division Δ-to-Y Equivalent Circuits 27

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