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Chapter 3 EGR 271 – Circuit Theory I 1 Reading Assignment: Chapter 3 in Electric Circuits, 9th Edition by Nilsson Chapter 3: Analysis of simple resistive circuits Series Resistance Two or more resistors in series can be replaced by a single equivalent resistance, Req, as shown below. I R1 R2 R3 I ... + V _ Use KVL to show that: Req = R1 + R2 + + R N RN + V _ Req Chapter 3 EGR 271 – Circuit Theory I 2 Example: Determine the current I in the circuit shown below. I 3 5 8V 6V 15V 2 7 20V 4 Chapter 3 EGR 271 – Circuit Theory I 3 Parallel Resistance Two or more resistors in parallel can be replaced by a single equivalent resistance, Req, as shown below. I I ... + V _ R1 R2 R3 RN + V _ ... 1 R Use KCL to show that: eq 1 1 1 R1 R2 RN Req Chapter 3 EGR 271 – Circuit Theory I 4 Example: Find the equivalent resistance for the circuit shown below. 48 36 72 Example: Find the voltage V in the circuit shown below. + V _ 3A 120 60 7A 80 240 8A Chapter 3 EGR 271 – Circuit Theory I 5 Special Case - 2 Resistors in Parallel An easy formula can be used to calculate the equivalent of two resistors in parallel. R1 R2 Show that the general formula for parallel resistance can be simplified as follows: R eq 1 1 1 1 R1 R2 RN (For 2 or more resistors) R eq R1 R 2 R1 R 2 product sum (For 2 resistors only) Chapter 3 EGR 271 – Circuit Theory I 6 Example: Find the equivalent resistance in each case using the formula for two resistors only. 30 70 80 Equal Value Resistors • Show the result of two equal value resistors in parallel. • Show the result of N equal value resistors in parallel. 30 60 Chapter 3 EGR 271 – Circuit Theory I 7 Series/Parallel Combination of Resistors Circuits can sometimes be reduced through repeated series or parallel combinations of resistors. Example: Find the current I in the circuit shown below. I 100V 40 180 960 160 1.2k 240 Chapter 3 EGR 271 – Circuit Theory I Two useful tools for analyzing resistive circuits • Voltage Division (or the Voltage Divider Law) • Current Division (or the Current Divider Law) Voltage Division • Applies to series circuits only. • Voltage divides proportionally between series R’s with the largest R getting the most voltage. • Show that Ri Vi VS R eq 8 Chapter 3 EGR 271 – Circuit Theory I 9 Example: Use voltage division to find the voltage V1 in the circuit shown 40 140 below. + 80 V1 _ 240V 60 Example: Use repeated voltage division to find the voltage V2 below. 100 120 + 40V 600 80 V2 _ Chapter 3 EGR 271 – Circuit Theory I 10 Example: Find the voltage V1 using repeated voltage division. 40 80 300 + 480V 240 600 900 V1 _ Chapter 3 EGR 271 – Circuit Theory I Current Division • Applies to parallel circuits only. • Current divides between parallel R’s with the smallest R getting the most current. R total • Show that: I i IS R i (general form) 11 Chapter 3 EGR 271 – Circuit Theory I 12 Example: Find the current I1 using current division. I1 20A 40 150 600 Chapter 3 EGR 271 – Circuit Theory I 13 Current Division - Special Case: Two Resistors Only Show that for two resistors only the general form of current division can be expressed as follows. R Ii IS total Ri R opposite I i IS R R i 2 (general form) (form for 2 resistors only) Example: Find the current I1 using current division (special form for two R’s only). I1 12mA 100 300 Chapter 3 EGR 271 – Circuit Theory I 14 Example: Find the current I1 using repeated current division. 40 200V 80 240 300 600 I1 900 Chapter 3 EGR 271 – Circuit Theory I 15 Example: Find I1, V2, I3, I4, V5, and V6 in the circuit shown below. I1 200 + V2 _ I4 160 300 800 260 + V6 _ 200V 500 I3 2000 + V5 _ 100 420 EGR 271 – Circuit Theory I Chapter 3 16 Bridge Circuits One type of resistive circuit that cannot be simplified through series and/or parallel combinations is the “bridge circuit.” A bridge circuit is shown below (drawn twice). Study the circuit to verify that there are no series resistors and no parallel resistors. R0 R0 R1 R2 R1 R2 R5 R5 Vs Vs R3 R4 R3 R4 Chapter 3 EGR 271 – Circuit Theory I 17 Delta-to-Wye (-Y) and Wye-to-Delta (Y-) Transformations Bridge circuits contain resistors that are connected in delta () and wye (Y) configurations. One way to analyze this circuit is to use a -Y or a Y- transformation. Y and connections of resistors are shown below: a b a b R R1 R 2 R R c R b 3 c c Wye Circuit Delta Circuit a Chapter 3 EGR 271 – Circuit Theory I 18 If the wye and delta circuits to be equivalent, then they should provide the same resistance between each pair of terminals (a-b, b-c, and c-a). Development: Determine the resistance seen at each set of terminals and equate them as follows: Ra-b (Delta) = Ra-b (Wye) Rb-c (Delta) = Rb-c (Wye) Rc-a (Delta) = Rc-a (Wye) a b a b R R1 R R R c 2 R b 3 c c Wye Circuit Delta Circuit a EGR 271 – Circuit Theory I Chapter 3 19 Solving the equations on the previous page yields the following relationships: Y- Conversion Equations: -Y Conversion Equations: R R R 2 R 3 R 3 R1 Ra 1 2 R1 R1 R R R 2 R 3 R 3 R1 Rb 1 2 R2 R2 R1 R 2 R 2 R 3 R 3 R1 R3 R3 Rc a Ra Rb Rc Rb Rc Ra Rc Ra Rb Rc Ra Ra Rb Rb Rc b a b R R1 R 2 R R c R b 3 c c Wye Circuit Delta Circuit a Note: The equations above must be used along with the circuit diagrams shown. The labeling of the resistors and nodes in the diagrams is critical. Chapter 3 EGR 271 – Circuit Theory I 20 Example: Determine I in the circuit shown below using: a) -Y conversion I 30 10 100 V 50 + _ 25 20 40 Chapter 3 EGR 271 – Circuit Theory I 21 Example: Determine I in the circuit shown below using: b) Y- conversion I 30 10 100 V 50 + _ 25 20 40 Chapter 3 EGR 271 – Circuit Theory I 22 The “balanced bridge” A bridge circuit is “balanced” when the following relationship exists: R1R4 = R2R3 When the bridge is balanced, it can be shown that I5 = 0 R0 R1 R2 R5 Vs I5 R3 R4 Chapter 3 EGR 271 – Circuit Theory I 23 Applications of bridge circuits An unknown resistance can be determined using a bridge circuit as follows: • Replace R4 with an unknown resistor • Place an ammeter in series with R5 • Adjust R1 until the bridge is balanced (i.e., I5 = 0) • Solve for R4 + _ R1 R 4 = R 2 R 3 R2 R1 or I5 R5 R3 Runknown R unknown R 2 R3 = R4 = R1 A bridge circuit with resistive values, sometimes called a Wheatstone bridge, can be used to determine unknown resistor values. Other bridge circuits (such as the Scherring bridge) can be used to determine unknown values of capacitors and inductors in a similar manner. Chapter 3 EGR 271 – Circuit Theory I 24 Impedance Bridge A bridge circuits used to find unknown values of resistance, inductance, or capacitance are sometimes called an impedance bridge. An impedance bridge is a common piece of test equipment and will be used in lab classes such as EGR 262 and EGR 270. A newer microprocessor-controlled impedance bridge. An older style impedance bridge that involved adjusting knobs until the needle indicated that the bridge was balanced. Ref: http://www.testequipmentconnection.com/products/908 http://www.testequipmentconnection.com/products/914 Chapter 3 EGR 271 – Circuit Theory I Strain gauges Strain gauges are often attached to surfaces to measure forces as the surface moves (such as in the deflection of an airplane wing or a beam). The force can be determined from the amount of stretch in the wire by measuring the resistance of the wire with a bridge circuit. As a wire stretches note that resistance increases since R l A Reference: http://www.allaboutcircuits.com/vol_1/chpt_9/7.html 25 Chapter 3 EGR 271 – Circuit Theory I Strain gauge mounted on a vehicle suspension system component for fatigue and durability testing Ref: http://blog.prosig.com/2006/05/17/fatigue-durability-testing/ 26 Chapter 3 EGR 271 – Circuit Theory I Strain gauge mounted on a human bone to study walking and running impact Biomechanics lab at Iowa State University Ref: http://www.kin.hs.iastate.edu/research/labs/biomechanics_lab.htm 27