E E 2315 Lecture 03 - Simple Resistive Circuits and Applications

Calculating Resistance When conductor has uniform cross-section

Temperature Coefficient of Resistance Metallic conductors have a linear increase of resistance with increased temperature. T o is the reference temperature (usually 20 o C) and R o is the resistance at the reference temperature.  is the temperature coefficient of resistance for the material. At 20 o C, some values for  are: 20 o C Aluminum Copper

Resistors in Series By KCL: I s = I 1 = I 2 By Ohm’s Law: V 1 = R 1 ·I 1 and V 2 = R 2 ·I 2 Combine: V s = R 1 I 1 + R 2 I 2 = (R 1 + R 2 ) I s = R eq I s In General: R eq = R 1 + R 2 +···+ R n

Resistors in Parallel (1/2) By KVL: V s = V 1 = V 2 By KCL: I s = I 1 + I 2 By Ohm’s Law: and Combine:

Resistors in Parallel (2/2) For two resistors: For many resistors: In terms of conductance:

Voltage Divider Circuit

Loaded Voltage Divider

Voltage Divider Equations Unloaded: Loaded: If R L >> R 2 :

Current Divider Circuit (1/2)

Current Divider Circuit (2/2) In general: If there are only two paths:

D’Arsonval Meter Movement Permanent Magnet Frame Torque on rotor proportional to coil current Restraint spring opposes electric torque Angular deflection of indicator proportional to rotor coil current

D’Arsonval Voltmeter Small voltage rating on movement (~50 mV) Small current rating on movement (~1 mA) Must use voltage dropping resistor, R v

Example: 1 Volt F.S. Voltmeter Note: d’Arsonval movement has resistance of 50  Scale chosen for 1.0 volt full deflection.

Example: 10V F.S. Voltmeter Scale chosen for 10 volts full deflection.

D’Arsonval Ammeter Small voltage rating on movement (~50 mV) Small current rating on movement (~1 mA) Must use current bypass conductor, G a

Example: 1 Amp F.S. Ammeter Note: d’Arsonval movement has conductance of 0.02 S Scale chosen for 1.0 amp full deflection. G a = S has ~ m  resistance.

Example: 10 Amp F.S. Ammeter Scale chosen for 10 amp full deflection. G a = S has ~ m  resistance.

Measurement Errors Inherent Instrument Error Poor Calibration Improper Use of Instrument Application of Instrument Changes What was to be Measured –Ideal Voltmeters have Infinite Resistance –Ideal Ammeters have Zero Resistance

Example: Voltage Measurement True Voltage: (If voltmeter removed)

Example: Voltage Measurement Measured Voltage:

Another Voltage Measurement (1/2) True Voltage: (If voltmeter removed)

Another Voltage Measurement (2/2) Measured Voltage:

Example: Current Measurement (1/2) True Current: (If ammeter replaced by short circuit)

Example: Current Measurement (2/2) Measured Current:

Another Current Measurement (1/2) True Current: (If ammeter replaced by short circuit)

Another Current Measurement (2/2) Measured Current:

Measuring Resistance Indirect –Measure Voltage across Resistor –Measure Current through Resistor –Calculate Resistance (Inaccurate) d’Arsonval Ohmmeter –Very Simple –Inaccurate Wheatstone Bridge (Most Accurate)

D’Arsonval Ohmmeter Need to adjust R adj and zero setting each scale change.

Ohmmeter Example 10 mA Full Scale (Outer Numbers) R b +R adj +R d’A =150  V b =1.5 V Inner (Nonlinear) Scale in Ohms

Wheatstone Bridge V ab = 0 and I ab = 0 V ad = V bd R 1 I 1 =R 2 I 2 R 3 I 3 =R x I x

Example: Wheatstone Bridge