Lecture 2: Resistive Circuits Nilsson 2.5, , 3.7 ENG17 : Circuits I Spring April 2, 2015

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

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

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

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

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

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

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

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

Voltage Divider One voltage supply, multiple voltage levels 10

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

Example 12

Example Wheatstone Bridge –2 voltage dividers in parallel 13

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

Current Divider 15 One current supply, multiple current levels

Example 16

Example 17 Similar analysis

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

Voltage Division 19 Generalization of voltage divider circuit

Current Division 20 Generalization of current divider circuit

Example 21

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

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

Defining Delta/Pi and Y/T 24 Δ Y T π

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

Apply to Wheatstone Bridge 26

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