E E 2320 Circuit Analysis. Calculating Resistance

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1 E E 30 Circuit Analysis Lecture 03 Simple esistive Circuits it and Applications Calculating esistance l A cm cu 6 al.7010 Area, A When conductor has uniform crosssection cm l 1

2 Temperature Coefficient of esistance Metallic conductors have a linear increase of resistance with increased temperature. ( T) o 1 T T T o is the reference temperature (usually 0 o C) and o is the resistance at the reference temperature. is the temperature coefficient of resistance for the material. At 0 o C, some values for are: Material 0 o C Aluminum Copper o esistors in Series I 1 1 I I s V 1 I V s V V s I s V s eq By KCL: I s = I 1 = I By Ohm s Law: V 1 = 1 I 1 and V = I Combine: V s = 1 I 1 I = ( 1 ) I s = eq I s In General: eq = 1 n

3 esistors in Parallel (1/) I I 1 I s V s V 1 1 V I s V s eq By KVL: V s = V 1 = V By KCL: I s = I 1 I By Ohm s Law: Combine: I s V 1 V I 1 and I 1 V1 V 1 1 Vs Vs 1 1 eq esistors in Parallel (/) 1 eq For two resistors: For many resistors: eq 1 In terms of conductance: G G G G eq 1 n n 3

4 Voltage Divider Circuit V 1 I 1 V s V Measure V V s V I s I 1 1 Vs V I V 1 1 s Loaded Voltage Divider 1 V s V o L eq L eq V o V s L eq V L o V s 1 L L 1 4

5 Unloaded: Loaded: Voltage Divider Equations If L >> : V V o V o s 1 V s 1 1 L Vo Vs 1 v i i Current Divider Circuit i 1 i I s v G 1 G I v o 1 1 s o i Is Is G1 G G1 G 1 1 G1 G i I s I s If there are only two paths: In general: G G i n n I s G 1 G G n 5

6 D Arsonval Meter Movement Permanent Magnet Frame Torque on rotor proportional to coil current estraint spring opposes electric torque Angular deflection of indicator proportional to rotor coil current S N D Arsonval Voltmeter Small voltage rating on movement (~50 mv) Small current rating on movement (~1 ma) Must use voltage dropping resistor, v v I d'a V x V v v V d'a 6

7 Example: 1 Volt F.S. Voltmeter ma 0.95 V 1.0 V 50 mv Note: d Arsonval movement has resistance of 50 Scale chosen for 1.0 volt full deflection. Example: 10V F.S. Voltmeter ma 9.95 V 10 V 50 mv Scale chosen for 10 volts full deflection. 7

8 D Arsonval Ammeter Small voltage rating on movement (~50 mv) Small current rating on movement (~1mA) Must use current bypass conductor, G a I Ga I x G a V d'a I d'a Example: 1 Amp F.S. Ammeter 999 ma 1 ma 1.0 A S 50 mv Note: d Arsonval movement has conductance of f00s 0.0 G a = S has ~ m resistance. Scale chosen for 1.0 amp full deflection. 8

9 Example: 10 Amp F.S. Ammeter 10 A S 9999A mA 50 mv G a = S has ~ m resistance. Scale chosen for 10 amp full deflection. Measurement Errors Inherent Instrument Error Poor Calibration i Improper Use of Instrument Application of Instrument Changes What was to be Measured Ideal Voltmeters have Infinite esistance Ideal Ammeters have Zero esistance 9

10 Example: Voltage Measurement V 100 V o 10 k voltmeter 100 True Voltage: Vo 45V 9V 500 (If voltmeter removed) Example: Voltage Measurement Measured Voltage: Vo V k 8.986V % Error 1 100% 0.794% 9.0V 10

11 Another Voltage Measurement (1/) 40 k 45 V 10 k V o 10 k voltmeter 10k True Voltage: Vo 45V 9V 50k (If voltmeter removed) Another Voltage Measurement (/) Measured Voltage: 10k Vo 45V 5.0V 10k 40k1 10k 10k 5.0V % Error 1 100% 44.44% 9.0V 11

12 Example: Current Measurement (1/) 100 5A 5 I o 50 m Ammeter 5 True Current: Io 5A 1.0A 15 (If ammeter replaced by short circuit) Example: Current Measurement (/) Measured Current: 5 Io 5A A A % Error 1 100%.04% 1.0 A 1

13 Another Current Measurement (1/) 100 m 5A 5 m I o 50 m Ammeter True Current: 5m Io 5A 1.0A 15m (If ammeter replaced by short circuit) Another Current Measurement (/) Measured Current: 5m Io 5A A 175m A % Error 1 100% 8.57% 1.0 A 13

14 Measuring esistance Indirect Measure Voltage across esistor Measure Current through esistor Calculate esistance (Inaccurate) d Arsonval Ohmmeter Very Simple Inaccurate Wheatstone Bridge (Most Accurate) D Arsonval Ohmmeter b V b x adj Need to adjust adj and zero setting each scale change. 14

15 Ohmmeter Example ma Full Scale (Outer Numbers) b adj d A =150 V b =1.5 V Inner (Nonlinear) Scale in Ohms Wheatstone Bridge c g 1 I 1 I Vab a b V ab = 0 and I ab = 0 V ad = V bd I = I x V g 3 I ab x 1 I 1 = I d I 3 I x 3 I 3 = x I x x

16 Example: Wheatstone Bridge c I a q b 1 kv d I = A 16

Chapter 26 Direct-Current and Circuits. - Resistors in Series and Parallel - Kirchhoff s Rules - Electric Measuring Instruments - R-C Circuits

Chapter 26 Direct-Current and Circuits - esistors in Series and Parallel - Kirchhoff s ules - Electric Measuring Instruments - -C Circuits . esistors in Series and Parallel esistors in Series: V ax I V