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ohm and power laws for Practical physics

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Methods (sketches of the circuits used, equipment, procedures,). Results (appropriately annotated and commented on, with appropriate uncertainties) Ohm's Law(s) - results from the investigation of the resistors, including the table(s), figure(s), and required calculation(s). Power Law - results from the investigation of the light bulb, including the table(s), figure(s), and the required calculation(s). Analysis & Discussion (Do the measurements confirm Ohm's law? Do the measurement agreed with the theoretical power law? What can you say about uncertainties in your values? What values did you manage to determine? Did you take enough data...) Indication of who did what for the lab/report. PYTHON CODE (uploaded separately)
Additional Instructions:
Model 130A/131 Publication Date: June 1984 Document Number: 130A-901-01C APPLICATIONS DEPT. mmm 28775 AURORA ROAD CLEVELAND, OHIO 44139 U.S.A. 1216) 248-0400 WARRANTY Keithley Instruments, Inc. warrants this product to be free from defects in material and workmanship for a period of two years from date of ship- ment. During the warranty period, we will, at our option, either repair or replace any product that proves to be defective. To exercise this warranty, write or call your local Keithley representative, or contact Keithley headquarters in Cleveland, Ohio. You will be given prompt assistance and return instructions. Send the instrument, transportation prepaid, to the indicated service facility. Repairs will be made and the instrument returned, transportation prepaid. Repaired products are warranted for the balance of the original warranty period, or at least 90 days. LIMITATION OF WARRANTY This warranty does not apply to defects resulting from unauthorized modification or misuse of any product or part. This warranty also does not apply to fuses, batteries, or damage from battery leakage. This warranty is in lieu of all other warranties, expressed or implied, in- cluding any implied warranty of merchantability or fitness for a particular use. Keithley Instruments, Inc. shall not be liable for any indirect, special or consequential damages. STATEMENT OF CALIBRATION This instrument has been inspected and tested in accordance with specifications published by Keithley Instruments, Inc. The accuracy and calibration of this instrument are traceable to the National Bureau of Standards through equipment which is calibrated at planned intervals by comparison to certified standards maintained in the Laboratories of Keithley Instruments, Inc. S P E C I F I CAT1 0 N S DC VOLTS ACCURACY (2 YEARS) +(%rdg + counts) RANGE RESOLUTION 1E0-2E0C 200mV 100 pv 2 v 1 mV 200 v 100mV 1000 v 1 v M A X I M U M ALLOWABLE INPUT: 1OOOV DC or peak AC non-switched, INPUT RESISTANCE: 10MR. NORMAL MODE REJECTION RATIO: Greater than 46dB at 50Hz, 60Hz. COMMON MODE REJECTION RATIO: Greater than lOOdB at DC, 50Hz AC VOLTS 20 v 10mV +(0.25% + 11 750V peak switched. and 6OHz (1kQ unbalance]. ACCURACY (2 YEARS)" f (Yordg + counts) FREQUENCY RANGE RESOLUTION 1Eo-28OC RANGE 200mV 100 pv 2 v 1 mV 200 v 100mV 750 V 1 v M A X I M U M ALLOWABLE INPUT: lOOOV peak non-switched, 750V peak switched: continuous except 200mV range: 15s max above 300V. INPUT IMPEDANCE: 10MQ shunted by less than 1OOpF. RESPONSE: Average responding, calibrated in rms of a sine wave. *Above 10 counts. OHMS 20 v 10mV + ( l % + 31 45Hz-500Hz ACCURACY (2 YEARS) f W r d g + counts) FULL SCALE RANGE RESOLUTION 1Eo-2E0C VOLTAGE 200 n lOOmn i (O .5% + 4) < 0.5V 2 kQ 1 n * (0.2% + 11 < 0.5V 20 k n 10 n f (0.2% + 1) > 0.7V 200 k f l 100 n f (0.2% + 1) > 0.7V 20MR 10 k f l + ( 2 % + 11 > 0.7V MAXIMUM OPEN CIRCUIT VOLTAGE: 1.5V. MAXIMUM ALLOWABLE INPUT: 300V DC or rrns. DC AMPS ACCURACY (2 YEARS) k(%rdg + counts) RANGE RESOLUTION 18"-28'C 2rnA 1 PA *(0.75% + 1) 20rnA 10 pA +(0.75% + 1) 200rnA 100pA :(0.75% + 1) 2000rnA 1 rnA *(2% + 1) 10 A 10rnA *(2% + 1) MAXIMUM FULL SCALE VOLTAGE BURDEN 0.25V 0.25V 0.25V 0.7 V 0.3 V OVERLOAD PROTECTION: rnA input: 2A fuse (250V), externally accessi- AC AMPS ble; 10A input: 20A for 15s unfused. ACCURACY (2 YEARS)' MAXIMUM *(%rdg + counts) FULL SCALE 18O-28'C VOLTAGE RANGE RESOLUTION (45HZ-500H~) BURDEN 2rnA 1 PA *(2% + 2) 0.25V 20mA 10 pA * ( 2 % + 2) 0.25V 200rnA 100pA f ( 2 % + 2) 0.25V 2000rnA 1 rnA +(3% + 5) 0.7 V 10 A 10rnA + (3% + 5) 0.3 V OVERLOAD PROTECTION: rnA input: 2A fuse (250V), externally accessi- *Above 10 counts. GENERAL DISPLAY: 3 % digit LCD, 0.6' height, with polarity and range indication. OVERRANGE INDICATION: 3 least significant digits blanked. MAXIMUM COMMON MODE VOLTAGE: 500V peak. OPERATING ENVIRONMENT: Oo to 5OOC; less than 80% relative humid- ble; 10A input 20A for 15s unfused. ity up to 35OC, linearly derate 3% RH/OC from 35OC to 5OOC. STORAGE ENVIRONMENT: -35OC to 6OOC. TEMPERATURE COEFFICIENT: (Oo t o 18% and 28O to 50°C): Less than POWER: 9V alkaline or carbon-zinc battery (NEDA 1604). BAlTERY LIFE: 100 hours typical with carbon-zinc cells, 200 hours with 0.1 x applicable accuracy specification per OC. alkaline cells. BAlTERY INDICATOR: Diplay indicates BAT when less than 10% of life DIMENSIONS, WEIGHT: 178mm long x 78mm wide x 42mm thick (7.0' ACCESSORIES SUPPLIED: Batten/, test leads and operating instructions. remains. x 3.1" x 1.6')). Net weigth 283gm (10 0 2 . ) . TABLE OF CONTENTS General Information ManualAddenda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Optional Accessories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preparation for Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Battery Installation/Replacement . . . . . . . . . . . . . . . Safety Symbols and Terms . . . . . . . . . . . . . . . . . . . . Safety Precautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Safety Precautions for High Energy Circuits . . . . . . . . . . . . . . . . . 6 Dissassembly Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Calibration Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 PartsList . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Operation Servicing Information GENERAL INFORMATION The Model 130A Digital Multimeter is supplied ready for use with a bat- tery. Descriptions of other available accessories, and other general in- formation concerning the instrument can be found below. MANUAL ADDENDA Information contained in this manual is believed to be accurate at the time of printing. Any improvements or changes to this manual will be documented in an addendum which will be included with the, instru- ment. OPTIONAL ACCESSORIES Model 1301 Temperature Probe is a rugged low cost temperature probe designed to allow precision temperature measurement from Model 1304 Soft Carrying Case and Stand Model 1306 Deluxe Case is a rugged DMM carrying case that is large enough to accomodate the Model 130A plus various other DMM articles such as a spare battery, test leads, etc. Model 1309 Spare Parts Kit is a collection of specially selected parts to maintain up to ten Model 130A DMMs for one year. Model 1600A High Voltage Probe extends the DMM to 40kV. It has a 1OOO:l division ratio, which means that 1V on the DMM corresponds to 1 kV. Model 1651 50-Ampere Shunt allows current measurements to be made up to 50-amperes. It is a 0.0010 1 % Cterminal shunt. When the DMM is set to the 2V range, a 50-ampere current will correspond to 50mV (.0500V). Model 1681 Clip-On Test Lead Set contains two leads, 1.2m(48 inch- es) long, terminated with banana plug and spring action clip-on probe. Model 1682A RF Probe allows voltage measurements from 100kHz to 250MHz. Model 1683 Universal Test Lead Kit consists of two test leads, 1.2177 (48 inches) long with 12 screw-in tips, two banana plugs, two spade lugs, two alligator clips with boots, two needle tips with chucks and four heavy-duty tip plugs. -55OC to 150OC. 1 Model 1685 Clamp-On AC Current Probe measures AC current by clamping onto a single conductor. Interruption of the current path is un- necessary. The Model 1685 detects current by sensing magnetic field produced by current. Model 1691 General Purpose Test Lead Set consists of two .91m (36 inches) test leads with probe tips terminated in banana plugs. PREPARATION FOR USE Carefully unpack the Model 130A from its shipping carton and inspect for any obvious signs of physical damage. Report any damage to the shipping agent at once. The following items are included with every Model 130A shipment. 1. Model 130A DMM 2. Model 130A Instruction Manual 3. 9V Battery NEDA 1604 4. Test Leads 5. Accessories as ordered. BATTERY INSTALLATION/REPLACEMENT The battery is accessible from the bottom of the instrument. Note the precautions on the case before installing or replacing the battery. WARNING Turn the Model 130A off and disconnect test leads before replacing the battery. Put the cover back into place on the compartment before resuming use of the instrument. A 9V battery is supplied with the instrument but not installed. To install or replace the battery, remove the cover from the battery compartment by sliding it off in the direction of the arrow located on the battery cover. The battery connector snaps on and off the terminal of the bat- tery. Improper installation of the battery will cause the connecting wires to be severed by excess strain. Proper installation requires that the bat- tery be positioned in such a manner (see drawing) that the leads pro- truding from the boot of the battery connector face toward the outside of the battery compartment. If the instrument is going to be stored for a long period of time or in a high temperature environment, remove the battery to prevent leakage damage. 2 R103 Figure 1. Battery Installation SAFETY SYMBOLS AND TERMS The symbol A on the instrument denotes that the user should refer to the operating instructions. The symbol r / on the instrument denotes that up to lOOOV may be present on the terminal(s). The WARNING used in this manual explains dangers that could result in personal injury or death. The CAUTION used in this manual explains hazards that could damage the instrument. 3 SAFETY PRECAUTIONS The following safety precautions should be observed before operating the Model 130A DMM. 1 . This instrument is intended for use by qualified personnel who recognize shock hazards and are familiar with the safety precautions required to avoid possible injury. Read over the manual carefully before operating this instrument. 2. Exercise extreme caution when a shock hazard is present at the in- strument's input. The American National Standards Institute (ANSI) states that a shock hazard exists when voltage levels greater than 30V rms or 42.4V peak are present. A good safety practice is to ex- pect that a hazardous voltage is present in any unknown circuit before measuring. 3. Inspect the test leads for possible wear, cracks or breaks before each use. If any defects are found, replace with test leads that have the same measure of safety as those supplied with the instrument. 4. For optimum safety do not touch the test leads or the instrument while power is applied to the circuit under test. Turn the power off and discharge all capacitors, before connecting or disconnecting the instrument. 5. Do not touch any objects which could provide a current path to the common side of the circuit under test or power line (earth) ground. Always make measurements with dry hands while standing on a dry, insulated surface, capable of withstanding the voltage being measured. 6. Exercise extreme safety when testing high energy power circuits (AC line of mains, etc.). Refer to the operation section. 7. Do not exceed the instrument's maximum allowable input as defined in the specifications and operation section. 4 OPERATION The following paragraphs contain information concerning basic opera- tion of the Model 130A. It is recommended that this information be reviewed before attempting to operate the Model 130A. Low Battery Indicator Minus sign indicates negative values. Plus sign implied. An over range condition is indicated by a “1“ followed by a blank display. ‘SET POWER ON (slide switch located on side of instrument). ELECT FUNCTION A N D fl RANGE DCV-20OmV, 2V, 20V, 200V or 1 ooov ACV-200mV, 2V, 20V, 200V, or 1 ooov NOTE: 750VAC is maximum allowable AC input. DCA-2mA, 20mA, 200mA, 2000mA or 10A ACA-2mA, 20mA, 200mA, 2000mA or 10A 0-2003, 2k0, 20k0, 200kR or 20M0 -INPUTS-Selects appropriate pair of input jacks. COM, V-0-for all voltage and resistance measurements. COM, mA-for current measure- ments up to 2000mA. 10A, COM, 10A, HI-for current measurements up to 10A. Figure 2. Front Panel Controls WARNING Do not apply more than 500V peak above earth ground to the COM input jack. 5 CAUTION Do not under any circumstance, use the 10A COM or 10A HI input jacks with the COM input jack for making measurements. This is a short and will damage the instrument. SAFETY PRECAUTIONS FOR HIGH ENERGY CIRCUITS To optimize safety when measuring voltage in high energy distribution circuits, read and use the directions in the following warning. WARNING Dangerous arcs of an explosive nature in a high energy circuit can cause severe personal injury or death. If the meter is connected to a high energy circuit, when set t o a current range, low resistance range or any other low impedance range, the circuit is virtually shorted. Dangerous arcing can result even when the meter is set to a voltage range if the minimum safety spacing is reduced. When making measurements in high energy circuits use test leads that meet the following requirements: 1. Test leads should be fully insulated. 2. Only use test leads than can be connected to the circuit (e.g. alligator or spade plugs) for a hands-off measurement. 3. Use test leads that do not reduce the arc protection by decreasing the voltage spacing. Use the following sequence when testing power circuits: 1. De-energize the circuit using the regular installed connect- disconnect device such as the circuit breaker, main switch, etc. 2. Attach the test leads to the circuit under test. Use appropriate safety rated leads for this application. 3. Set the DMM to thg proper function and range. 4. Energize the circuit using the installed connect-disconnect device and make measurements without disconnecting the DMM. 5. De-energize the circuit using the installed connect-disconnect device. 6. Disconnct the test leads from the circuit under test. 6 SERVICING INFORMATION This section contains servicing information for the Model 130A. WARNING All service information is intended for qualified electronic maintenance personnel only. FUSE CHECK With the instrument set to the 2kQ range, connect a jumper from the V-Q jack to the mA jack. The display should read approximately .loo. An overrange display would typically indicate a blown fuse. A display reading other than approximately .lo0 could indicate a defective current input circuit (see schematic). FUSE REPLACEMENT A 2-amp fuse protects the 2mA through 2000mA current ranges. To gain access to the fuse, remove the fuse Compartment cover in the same manner as removing the battery compartment cover. WARNING Turn off the Model 130A and disconnect the test leads before replacing the fuse. Reinstall fuse compartment cover before attempting to operate the instrument. Remove the fuse by pulling outward on the plastic tab that encircles the fuse body. Install the plastic tab on the new fuse and snap the fuse back into the fuse holder. Do not replace the fuse with a higher rated value or instrument damage that is not covered by the warranty may occur. DIODE TEST The 20kQ range can be used for testing of semiconductor junctions. A junction is probably good if the Model 130A indicates an overrange reading when the semiconductor is reversed biased, and an on-range reading, when the semiconductor is forward biased. (The V-fl jack is positive). 7 DISASSEMBLY INSTRUCTIONS Place the unit face down on a bench or other similar surface and remove the battery compartment cover. Disconnect and remove the battery. Remove the two #4-40 X 7/8 retaining screws. Grasp the bottom cover at the input jack end and with a lifting and for- ward pushing motion (see drawing), carefully remove the bottom cover. While removing the cover, feed the battery connector through the ac- cess hole in the bottom of the battery compartment. The component side of the PC board is now exposed and the battery can be reconnected for troubleshooting. To read the display, some light downward pressure at the top of the circuit board may be required in order to make contact through the elastomer contact strip between the circuit board and the LCD. The PC board and LCD assembly are not secured once the case retain- ing screws are removed. Be careful not to allow the PC board and LCD assembly to fall out or shift out of position during calibration. To remove the PC board from the top cover, grasp the function switch assembly and lift until the input jacks become disengaged from the cover. The PC board can now be removed using a slight clockwise mo- tion to free the two switch knobs from their normal positions in the case. The LCD assembly will remain in the top cover when the PC board is removed. Again, be careful not to allow the LCD assembly to fall out ac- cidentally. The two switch knobs and bushings can be removed from the PC board assembly by simply pulling them off the switch shafts. The LCD assembly, along with the zebra strip connector, lifts out of the case. To reassemble the Model 130A, remove the cover on the fuse cornpart- ment (to position fuse pulling tab) and reverse the above procedures. Be sure to replace the on-off switch cover. Common-mode voltage will be present on the switch, creating a possible hazard if the cover is not replaced. 8 1 t PUSH FORWARD ! LIFT c-- , I I I I I Figure 3. Rear Panel Removal TOP CASE Figure 4. Model 130A Exploded View 9 CALI BRATION Calibration should be performed every two years or whenever the in- strument is known to be out of specification. Calibration should be done at an ambient temperature of 23O f3OC ( 7 7 O f5OF), with a relative humidity of less than 80%. Equipment Needed: + 190mV DC voltage source with -05% accuracy or better. Model 130A Settings: DCV function, 200mV range, V-s2 and COM input jacks. CALIBRATION PROCEDURE Remove the battery cover to gain access to the calibration pot R103 (see Figure 1). Apply the + 190mV to the Model 130A input and adjust R103 for a display reading of 190.0 f 1 digit. This is the only adjustment need- ed to calibrate the instrument. Calibration should be performed every two years. 10 Model 130A Parts List Circuit Deslg. BT101 c101 c102 C103 C104 C105 C106 C107 C108 c109 c110 c111 c112 C113 CR101 CR102 CR103 CR104 CR105 CR106 CR107 CR108 CR109 DS101 F101 J1001 J 1002 J 1003 J1004 J1005 J1006 Q101 Q102 R101 R102 R103 R104 R105 R106 R107 Description Battery, 9V, NEDA 1604 Capacitor, 1 lOpF, 500V, Mica Capacitor, .047pF, lOOV, Polyester Capacitor, .047pF, lOOV, Polyester Capacitor, .IFF, lOOV, Polyester Capacitor, .lpF, 16OV, Polypropylene Capacitor, 1.5pF, 20V, Tantalum Capacitor, 4.7pF, 20V, Tantalum Capacitor, 1 .5pF, 20V, Tantalum Capacitor, .02pF, lOOOV, Ceramic Disc Capacitor, .IFF, 50V, Ceramic Capacitor, .1pF, 50V, Ceramic Capacitor, lpF, 20V, Tantalum Capacitor, 4.7pF, 20V, Tantalum Rectifier, lA, 800V Rectifier, 75mA, 75V Rectifier, 75mA, 75V Rectifier, 75mA, 75V Rectifier, 75mA, 75V Rectifier, 75mA, 75V Rectifier, 3A, 50V Rectifier, 3A, 50V Rectifier, 75mA, 75V Liquid Crystal Display Connector, Strip Fuse, 2A, 250V, 3AB, Ceramic Body Jack, Input Jack, Input Jack, Input Jack, Input Jack, Input Connector Battery Transistor, NPN, Switch, 2N3904 Transistor, NPN, Silicon, GES5818 Thick Film Resistor Network Thick Film Resistor Network Pot, 500Q Thick Film Resistor Network Resistor, lMQ, lo%, lW, Comp Resistor, .lo, .5%, lW, WW NOT USFD - Sch LOC c4 H3 F3 E4 G3 G3 04 D2 D4 c2 c2 D2 D2 F1 c4 C6 c2 c2 D2 D2 A4 84 83 G5 A4 A1 A4 A4 85 85 c4 E5 E3 SEV SEV G2 B3,B4 E4 84 - Keithley Pert No. BA-14 C-320-11OP C-308.047 C-305,047 C-305.1 C-306.1 C-3141.5 C-314-4.7 C-314-1.5 C-316.02 C-238.1 C-238.1 C-3251 C-3194. I R F-38 RF-28 RF-28 RF-28 RF-28 RF-28 RF-34 RF-34 RF-28 DD-27 CS-3762 FU-62 130A-303 130A-303 13OA-303 130A-303 130A-303 BH-29 TG-47 TG-138 TF-97 TF-119 RP-119-500 TF-94 R-2-1M R-279.1 11 Model 130A Parts List (Cont.) Circuit Desig. R108 R109 R110 R111 R112" R113 R114 R115 R116 R117 R118 R119 R120 R121 R122" R123" 4T101 s101 s102 S103 u101 u102 U103 VRlOl ~~ ~~ lescription aesistor, 1000, .1%, 1/1OW, MtF Resistor, 9000, . l%, l/ lOW, MtF Resistor, 9k0, , l%, 1/1OW, MtF Resistor, 90k0, . l%, l/ lOW, MtF Resistor, 900k0, . l%, 1/1OW, MtF Resistor, 100k0, lo%, lW, Comp NOT USED Resistor, 4.72k0, . l%, 1/1OW, MtF Resistor, 10k0, , l%, l/ lOW, MtF Thick Film Resistor Network Resistor, 0.010, 0.5%, lW, WW NOT USED NOT USED NOT USED Resistor, 9M0, .08%, 2W Thick Film Resistor Network Thermistor, 8mA, 500V, PTC Switch, SPDT, ON-OFF Switch, Rotary, Range Switch, Rotary, Function 3 1/2-Digit Single Chip A/D Converter CMOS Quad Exclusive OR Gate Low-Power JFet-Input OP-Amp Diode, Low Voltage Reference Supplied Test Lead Set Sch LOC 83 82 B2 82 82 c1 - D3 02 S EV B5 81 B1 E2 c4 S EV E2, F2 G4 F5, F6 c2 G1 ~ "R123 may be installed in your unit in place of R122 and R1 configuration). ALTERNATE CONFIGURATION Keithley Part No. R-283-100 R-283-900 R-283-9k R-283-90k R-283-900k R-2- 1 OOk R-283-4.71 k R-283-10k TF-96 R-280-.01 R-281-9M TF-104 RT-7 SW-417 SW-432 sw-433 32440 30847 IC-227 DZ-62 CA-8 (see alternate 12 COMPONENT LAYOUT 13 SCHEMATIC DIAGRAM INPUT $'" I I 15 SERVICE FORM Model No. Name Phone Company Address City State Zip Serial No. ~ P.O. No. date List all control settings and describe problem. (Attach additional sheets as necessary. 1 Show a block diagram of your measurement system including all instruments connected (whether power is turned on or notl. Also describe signal source. Where is the measurement being performed? (factory, controlled laboratory, out-of-doors, etc.1- What power line voltage is used? Variation? O F . Rel. Humidity? Other? Variation? Frequency? Ambient Temperature? OF. Any additional information (If special modifications have been made by the user, please describe below 1 ‘Be sure to Include your name and phone number an this sewice form Keithlsy Instruments. lnc./28775 Aurora RoadiCleveiand, Ohio 44139/U.S.A./(2161 2484400/Teiex: 985469 WEST GERMANY: Keithlsy Inatruments GmbH/Heiglhofstrasse 5/D-8000 Munchen 70/314 289/Telex: 1345000 GREAT BRITAIN: Keithlsy Instrurnenta. Ltd./l, Boulton RoadiReading, Berkshire RG2 ONL110734) 88 12 87/Telex: 847047 FRANCE: Keithlsy Instruments SARL/2 Bis, Rue LBon Blum/B.P. 60/91121 Palaiseau Cedexll61 0115 155/Telex: 600933 NETHERLANDS: Keithley lnatrumentr BV/Arkelsedijk 4/NL4206 AC Gorinchem/(01830l 25577/Telex: 24 684 SWITZERLAND: Keithlsy Inatruments SAIKriesbachstr. 4/CH-8660 DubendorfiOl 821 94 44/Telex: 57 536 AUSTRIA: Keithley Instruments Gss.m.b.H/Doblinger Hauptstr. 32/A-1190 Wien/0222 314 289/Telex: 134500
Find us at www.keysight.com Page 1 U1270 Series Handheld Digital Multimeters Be Ready for Harsh Environments and Sub-zero Temperatures The U1273AX, the latest addition to the U1270 Series is capable of operating down to –40 °C in temperature. Even in extremely cold conditions, the U1273AX handheld DMM delivers immediate and accurate results — no warm-up time required. All models are ergonomically built providing useful functions such as ZLOW, which eliminates stray voltages, and Smart Ohm that minimizes false readings from residual voltage induced by leakage current. All of this is designed into a case that fulfills the needs of today's industrial handheld users. Find us at www.keysight.com Page 2 Features – OLED display with 2000:1 contrast ratio and 160 degrees viewing angle3,4 – 30,000-count resolution – Measure up to 1000 V AC and DC – Measure up to 10 A (20 A for 30 s) – Resistance, diode test, temperature, capacitance – Low Impedance mode2,3,4 and Low Pass Filter – Peak detection of up to 250 µs – Continuity test with beeper and backlight1,2 – Seven readings/s measurement rate for voltage and current – Smooth function for accurately stable readings – Up to 10,000 points internal memory for data logging – Bluetooth wireless connectivity with optional U1177A Bluetooth – PC connectivity with optional U1173A IR-USB cable – IP 54 certified – water and dust resistant – CAT III 1000 V, CAT IV 600 V safety rating – Up to 3000m operating altitude – –40 to 55 °C operating temperature4 1. U1271A 2. U1272A 3. U1273A 4. U1273AX Operational down to –40 °C temperature The U1273AX OLED handheld digital multimeter, the latest addition to U1270 Series, is capable of operating in winter weather down to –40°C temperature. Even in frigid conditions, the U1273AX enables you to achieve immediate and accurate results without the need to warm up in advance. Increase productivity with Bluetooth® wireless connectivity For wireless connectivity to smartphones and tablets, the U1270 Series is compatible with the U1177A infrared-to-Bluetooth adapter for maximum efficiency and productivity in completing measurements tasks. Adding the optional U1177A to a U1270 Series you can easily perform remote monitoring and data logging via Android devices or Windows- based PC. OLED for more display clarity Designed with OLED display, you can experience crystal-clear measurement readings with its outstanding 2000:1 contrast ratio. The display also allows wider viewing angles up to 160 degrees ensuring you get the right readings at the first glance even in poorly lit environments. Find us at www.keysight.com Page 3 Features Water and dust resistance (IP54) The series’ tightly sealed design helps protect against water, dust and damage. Each handheld DMM is certified with IP 54 ratings so that you can carry out tests and measurements with confidence, even in harsh working conditions. Operational up to 3000 meters altitude For high altitude applications such as wind farm maintenance, you can measure with confidence using the U1270 Series, capable of measuring up to 3000 meters above sea level. High measurement rate at seven readings per second for Voltage and Current You can detect even the slightest change in your sensitive signals (Voltage and Current) with its high measurement rate capability. By clicking the resettable smooth function button, you may customize the readings’ sensitivity suitable for various tests. Visual alert for continuity test (for U1271A and U1272A only) Continuity detection in noisy and dark environments is made easy with U1270 Series’ loud beeper and flashing backlight that indicates continuity and thus improves safety. Up to 10,000 recording points for manual, auto and event logging Record measurements on-the-go and transfer data to PC conveniently with the huge internal memory of up to 10,000 recording points. The GUI Data Logging software and optional U1173A IR-USB cable are required to transfer data or perform real time data logging on a PC. Built-in Low Pass Filter The U1270 Series offers a 1 kHz LPF or Low Pass Filter to provide accurate Variable Frequency Drive (VFD) output measurements. This function eliminates high frequency noise and harmonics, ensuring motor filter efficiency. Figure 1. Comparison of voltage output from industrial motor VFD without and with Low Pass Filter functionality. Key Functions Key Functions Find us at www.keysight.com Page 4 Low impedance mode Stray voltages are usually found in non-energized electrical wiring adjacent to powered wires due to capacitive or inductive coupling between these wires. The low impedance mode serves to eliminate false readings by dissipating these stray voltages thus improves safety and measurement efficiency during voltage measurement. Peak detect at 250 µs The peak detect function allows you to capture the engine or motor startup transient as fast as 250 µs. Figure 2. U1272A helps you identify the presence of stray voltage on a disconnected wire running parallel with the wire powering up the VFD to an industrial motor. The image on the right shows the U1272A in low impedance mode. Find us at www.keysight.com Page 5 Front and Back Panel Description 1. U1272A, U1273A and U1273AX only Front Panel Figure 3. Once connected to any HH DMM via Bluetooth adapter you are able to log and view measurements graphically from smart phones and tablets. Peak detect records transients as fast as 250 μs Measure up to 1000V AC and DC Low Pass Filter (LPF) removes unwanted high frequency signals ZLOW1provides both high and low impedance modes to eliminate stray voltages Smart Ω 1 removes residual voltage of up to 1000 mV 30,000 counts resolution OLED display with 2000:1 contrast ratio and 160 degree viewing angle Auto diode 1 automatically determines diode polarity Temperature measurement: J type thermocouple –210 to 1200 °C 1, K type thermocouple –200 to 1732 °C Measure up to 10 A (20 A for 30 s) Front and Back Panel Description Find us at www.keysight.com Page 6 Back Panel Hang hole Probe holder/storage IR-USB connectivity with optional IR-USB cable Slim, ergonomic design for better grip Easy fuse and battery access from battery cover beneath the stand Choose Among These Four Models Find us at www.keysight.com Page 7 Basic Features U1271A U1272A U1273A U1273AX Display resolution 30,000 counts 30,000 counts 30,000 counts 30,000 counts Display LCD LCD OLED OLED Backlight Yes Yes N/A N/A True RMS AC AC + DC AC + DC AC + DC Measurements Voltage Up to 1000 V AC, DC Up to 1000 V AC, DC Up to 1000 V AC, DC Up to 1000 V AC, DC Basic dcV accuracy 0.05% + 2 counts 0.05% + 2 counts 0.05% + 2 counts 0.05% + 2 counts Current Up to 10 A Up to 10 A Up to 10 A Up to 10 A (20 A for 30 s) (20 A for 30 s) (20 A for 30 s) (20 A for 30 s) Resistance Up to 100 MΩ Up to 300 MΩ Up to 300 MΩ Up to 300 MΩ Other measurements Frequency, capacitance, temperature, continuity, diode test Frequency, capacitance, temperature, continuity, diode test Frequency, capacitance, temperature, continuity, diode test Frequency, capacitance, temperature, continuity, diode test AC bandwidth 20 kHz 100 kHz 100 kHz 100 kHz Low pass filter Yes Yes Yes Yes Low impedance mode — Yes Yes Yes Smart Ohm — Yes Yes Yes Safety and Regulatory Over-voltage safety protection CAT III 1000 V, CAT IV 600 V CAT III 1000 V, CAT IV 600 V CAT III 1000 V, CAT IV 600 V CAT III 1000 V, CAT IV 600 V General Logging memory 200 points 10,000 points 10,000 points 10,000 points Connectivity Optional IR-USB and Bluetooth Optional IR-USB and Bluetooth Optional IR-USB and Bluetooth Optional IR-USB and Bluetooth Operating temperature –20 to 55 °C –20 to 55 °C –20 to 55 °C –40 to 55 °C Altitude 3000 meters 3000 meters 3000 meters 3000 meters Water and dust ingress protection IP 54 IP 54 IP 54 IP 54 Battery life Up to 300 hours 4X AAA Alkaline Up to 300 hours 4X AAA Alkaline Up to 60 hours 4X AAA Alkaline Up to 100 hours 4X AAA Lithium General Specifications (continued) Find us at www.keysight.com Page 8 Display U1271A and U1272A: Liquid crystal display (LCD) (with maximum reading of 33,000 counts) U1273A/U1273AX: Organic LED (OLED) display (with maximum reading of 33,000 counts) (Note: OLED is made of organic materials and it has its lifespan.) Power consumption U1271A/U1272A: 460 mVA maximum (with backlight enabled) U1273A/U1273AX: 180 mVA maximum (with maximum brightness) Battery type 4 × 1.5 V Alkaline battery (ANSI/NEDA 24A or IEC LR03), or 4 × 1.5 V Zinc Chloride battery (ANSI/NEDA 24D or IEC R03) 4 × 1.5 V Lithium battery (ANSI/NEDA 24LF or IEC FR03) Battery life U1271A and U1272A: 300 hours typical (based on new Alkaline batteries for DC voltage measurement) U1273A/U1273AX: Based on new Alkaline batteries for DC voltage measurement: 30/45/60 hours typical at High/Medium/Low brightness, respectively Based on new Lithium batteries for DC voltage measurement: 50/100 hours typical at High/Low brightness, respectively Low battery indicator will flash when the battery voltage drops: For non-rechargeable batteries: 4.4 V (approximately) For rechargeable batteries: 4.5 V (approximately) Fuse 10 × 35 mm 440 mA/1000 V 30 kA fast-acting fuse 10 × 38 mm 11 A/1000 V 30 kA fast-acting fuse Input impedance at off mode 1.67 kΩ (protected by positive temperature coefficient resistor) (U1272A, U1273A and U1273AX only) Operating environment Operating temperature: U1271A/ U1272A/U1273A: –20 to 55 °C, 0% to 80% RH U1273AX: –40 to 55 °C, 0% to 80% RH (using Lithium batteries) Full accuracy up to 80% RH for temperatures up to 30 °C, decreasing linearly to 50% RH at 55 °C Altitude up to 3000 meters Pollution degree II Safety compliance –40 to 70 °C, 0 to 80% RH Storage compliance IEC 61010-1:2010 / EN 61010-1:2010, IEC 61010-2-033:2012 / EN 61010-2- 033:2012 Canada: CAN/CSA-C22.2 No.61010-1-12, CAN/CSA-C22.2 No. 61010-2- 033-12 USA: ANSI/UL Std. No. 61010-1 (3rd Edition), ANSI/UL Std. No. 61010-2-033 (1st Edition) Measurement category CAT III 1000 V/CAT IV 600 V General Specifications (continued) Find us at www.keysight.com Page 9 Electromagnetic compatibility (EMC) Commercial limits compliance with EN61326-1 Influence of radiated immunity; in RF electromagnetic fields of 3 V/m DC voltage measurement typical accuracy All ranges; ± 0.03% of range DC current measurement typical accuracy 300 uA, 3000 uA, 30 mA, 300 mA & 3 A range; ± 0.22% of range 10 A range; ± 0.66% of range Note: – The measurement accuracy is applied only when DC Low Pass Filter (LPF) is ON (factory default). – The use of LPF is recommended to improve the accuracy of measurements in the presence of RF fields.  If used in close proximity to an RF transmitter or when subjected to continuously present electromagnetic phenomena, some recoverable degradation of performance may occur. Ingress protection rating IP-54 Temperature coefficient U1271A/U1272A/U1273A: 0.05 × (specified accuracy)/°C (from –20 to 18 °C, or 28 to 55 °C) U1273AX: 0.05 x (specified accuracy/ °C (from –40 to 18 °C, or 28 to 55 °C) Common Mode Rejection Ratio (CMRR) > 120 dB at DC, 50/60 Hz ± 0.1% (1 kΩ unbalanced) Normal Mode Rejection Ration (NMRR) > 60 dB at 50/60 Hz ± 0.1% Dimensions (W x H x D) 92 × 207 × 59 mm Weight U1271A: 518 grams (with batteries) U1272A: 520 grams (with batteries) U1273A: 500 grams (with batteries) U1273AX: 500 grams (with batteries) Calibration cycle One year Specification Assumptions • Accuracy is given as ± (% of reading + counts of least significant digit) at 23 °C ± 5 °C, with relative humidity less than 80% RH. • AC V and AC μA/mA/A specifications are AC coupled, true RMS and are valid from 5% of range to 100% of range. • Crest factor ≤ 3 at full-scale and decrease reciprocally for overange as 3 x Full Scale / Input; except for the 1000 V range, where this range has a crest factor ≤ 1.5 at full scale and decrease reciprocally for overange as 1.5 x Full Scale / Input. • For non-sinusoidal waveforms, add (2% of reading + 2% of full scale) typical. • After ZLOW voltage measurements, wait at least 20 minutes for thermal impact to cool before proceeding with any other measurement. Electrical Specifications Find us at www.keysight.com Page 10 DC specifications for U1271A, U1272A, U1273A and U1273AX Function Range Resolution Accuracy ± (% of reading + counts of least significant digit) Test current / Burden voltage U1271A U1272A U1273A / U1273AX Voltage1 30 mV 0.001 mV — 0.05 + 20 0.05 + 20 — 300 mV 0.01 mV 0.05 + 5 0.05 + 5 0.05 + 5 — 3 V 0.0001 V 0.05 + 5 0.05 + 5 0.05 + 5 — 30 V 0.001 V 0.05 + 2 0.05 + 2 0.05 + 2 — 300 V 0.01 V 0.05 + 2 0.05 + 2 0.05 + 2 — 1000 V 0.1 V 0.05 + 2 0.05 + 2 0.05 + 2 — ZLOW (low impedance) enabled, applicable for 1000 V range and resolution only 0.1 V — 1 + 20 1 + 20 — Resistance2 30 Ω 0.001 Ω — 0.2 + 10 0.2 + 10 0.65 mA 300 Ω 0.01 Ω 0.2 + 5 0.2 + 5 0.2 + 5 0.65 mA 3 kΩ 0.0001 kΩ 0.2 + 5 0.2 + 5 0.2 + 5 65 µA 30 kΩ 0.001 kΩ 0.2 + 5 0.2 + 5 0.2 + 5 6.5 µA 300 kΩ 0.01 kΩ 0.2 + 5 0.2 + 5 0.2 + 5 0.65 µA 3 MΩ 0.0001 MΩ 0.6 + 5 0.6 + 5 0.6 + 5 93 nA/10 MΩ 30 MΩ 0.001 MΩ 1.2 + 5 1.2 + 5 1.2 + 5 93 nA/10 MΩ 100 MΩ 0.01 MΩ 2.0 +10 — — 93 nA/10 MΩ 300 MΩ 0.01 MΩ — 2.0 + 10 @ < 100 MΩ 8.0 + 10 @ > 100 MΩ 2.0 + 10 @ < 100 MΩ 8.0 + 10 @ > 100 MΩ 93 nA/10 MΩ 300 nS 0.01 nS 1 + 10 1 + 10 1 + 10 93 nA/10 MΩ Current3 300 µA 0.01 µA 0.2 + 5 0.2 + 5 0.2 + 5 < 0.04 V/100Ω 3000 µA 0.1 µA 0.2 + 5 0.2 + 5 0.2 + 5 < 0.4 V/100 Ω 30 mA 0.001 mA 0.2 + 5 0.2 + 5 0.2 + 5 < 0.08 V/1 Ω 300 mA 0.01 mA 0.2 + 5 0.2 + 5 0.2 + 5 < 1.00 V/1 Ω 3 A 0.0001 A 0.3 + 10 0.3 + 10 0.3 + 10 < 0.1 V/0.01 Ω 10 A 0.001 A 0.3 + 10 0.3 + 10 0.3 + 10 < 0.3 V/0.01 Ω Diode Test4 3 V 0.0001 V 0.5 + 5 0.5 + 5 0.5 + 5 Approximately 1 to 2 mA Auto 0.0001 V — 0.5 + 5 0.5 + 5 Approximately 1 to 2 mA Electrical Specifications (continued) Find us at www.keysight.com Page 11 Notes for DC specifications (previous page) 1. Notes for voltage specifications:  The accuracy of the 30 to 300 mV range is specified after the Null function is used to subtract the thermal effect (by shorting the test leads).  ZLow impedance: 2 kOhm (nominal). For ZLow measurements, autoranging is disabled and the multimeter's range is set to 1000 volts in the manual ranging mode. 2. Notes for resistance specifications:  Overload protection: 1000 Vrms for short circuits with < 0.3 A current.  Maximum open voltage is < +3.3 V.  Built-in buzzer beeps when the resistance measured is less than 25 Ω ± 10 Ω. The multimeter can capture intermittent measurements longer than 1 ms.  U1272A/73A/73AX only: The accuracy of the 30 Ω to 3 kΩ range is specified after the Null function is used to subtract the test lead resistance and thermal effect (by shorting the test leads).  U1271A only: The accuracy of the 300 Ω to 3 kΩ range is specified after the Null function is used to subtract the test lead resistance and thermal effect (by shorting the test leads).  U1273AX only: The accuracy for all resistance ranges is specified after the Null function is used when measuring at temperatures below –20 °C. The Null function is used to subtract the test lead resistance and thermal effect (by shorting the test leads).  For the ranges of 30 MΩ and 100 MΩ, the RH is specified for < 60%.  The accuracy for ranges < 50 nS is specified after the Null function is used on an open test lead.  The temperature coefficient of the 100 MΩ and 300 MΩ range is 0.1 × (specified accuracy)/°C (from –40 to 18 °C or 28 to 55 °C). 3. Notes for current specifications:  Overload protection for 300 μA to 300 mA range: 0.44 A/1000 V; 10 × 35 mm 30 kA fast-acting fuse.  Overload protection for 3 A to 10 A range: 11 A/1000 V; 10 × 38 mm 30 kA fast-acting fuse.  Specification for 300 mA range: 440 mA continuous.  Specification for 10 A range: 10 A continuous. Add 0.3% to the specified accuracy when measuring signals > 10 to 20 A for 30 seconds maximum. After measuring currents > 10 A, cool down the multimeter for twice the duration of the measured time before proceeding with low current measurements. 4. Notes for diode specifications:  Overload protection: 1000 Vrms for short circuits with < 0.3 A current.  Built-in buzzer beeps continuously when the voltage measured is less than 50 mV and beeps once for forward-biased diode or semiconductor junctions measured between 0.3 V and 0.8 V (0.3 V ≤ reading ≤ 0.8 V).  Open voltage for diode: < +3.3 V DC.  Open voltage for Auto diode: < +2.5 V DC and > –1.0 V DC. Electrical Specifications (continued) Find us at www.keysight.com Page 12 AC specifications for U1271A Accuracy ± (% of reading + counts of least significant digit) Function Range Resolution 45 Hz to 65 Hz 30 Hz to 1 kHz 1 kHz to 5 kHz 5 kHz to 20 kHz True RMS AC Voltage1 300 mV 0.01 mV 0.7 + 20 1.0 + 25 2.0 + 25 2.0 + 40 3 V 0.0001 V 0.7 + 20 1.0 + 25 2.0 + 25 2.0 + 40 30 V 0.001 V 0.7 + 20 1.0 + 25 2.0 + 25 2.0 + 40 300 V 0.01 V 0.7 + 20 1.0 + 25 2.0 + 25 — 1000 V 0.1 V 0.7 + 20 1.0 + 25 — — LPF (low pass filter) enabled, applicable for all voltage ranges and resolution0.1 V 0.7 + 20 1.0 + 25@ 10 A, cool down the multimeter for twice the duration of the measured time before proceeding with low current measurements. Electrical Specifications (continued) Find us at www.keysight.com Page 13 AC specifications for U1272A/U1273A and U1273AX Accuracy ± (% of reading + counts of least significant digit) Function Range Resolution 45 Hz to 65 Hz 20 Hz to 1 kHz 1 kHz to 5 kHz 5 kHz to 20 kHz 20 kHz to 100 kHz True RMS AC Voltage1 30 mV 0.001 mV 0.6 + 20 0.7 + 25 1.0 + 25 1.0 + 40 3.5 + 40 300 V 0.01 mV 0.6 + 20 0.7 + 25 1.0 + 25 1.0 + 40 3.5 + 40 3 V 0.0001 V 0.6 + 20 1.0 + 25 1.5 + 25 2.0 + 40 3.5 + 40 30 V 0.001 V 0.6 + 20 1.0 + 25 1.5 + 25 2.0 + 40 3.5 + 40 300 V 0.01 V 0.6 + 20 1.0 + 25 1.5 + 25 2.0 + 40 — 1000 V 0.1 V 0.6 + 20 1.0 + 25 1.5 + 25 — — LPF (low pass filter) enabled, applicable for all voltage ranges and resolution 0.6 + 20 1.0 + 25 @ < 200 Hz 5.0 + 25 @ < 440 Hz — — — ZLOW 1000 V 2.0 + 40 2 + 40 @ < 440 Hz — — — Accuracy ± (% of reading + counts of least significant digit) Function Range Resolution 45 Hz to 65 Hz 20 Hz to 2 kHz Burden voltage/Shunt True RMS AC Current2 300 µA 0.01 µA 0.6 + 25 0.9 + 25 < 0.04 V/100 Ω 3000 µA 0.1 µA 0.6 + 25 0.9 + 25 < 0.4 V/100 Ω 30 mA 0.001 mA 0.6 + 25 0.9 + 25 < 0.08 V/1 Ω 300 mA 0.01 mA 0.6 + 25 0.9 + 25 < 1.00 V/1 Ω 3 A 0.0001 A 0.8 + 25 1.0 + 25 < 0.1 V/0.01 Ω 10 A 0.001 A 0.8 + 25 1.0 + 25 < 0.3 V/0.01 Ω 1. Notes for voltage specifications:  Overload protection: 1000 Vrms. For millivolt measurements, 1000 Vrms for short circuits with < 0.3 A current.  Input impedance: 10 MΩ (nominal) in parallel with < 100 pF.  ZLOW impedance: 2 kΩ (nominal).  The input signal is lower than the product of 20,000,000 V×Hz.  For 20 to 100 kHz accuracy: Three counts of the LSD per kHz of additional error is to be added for frequencies > 20 kHz and signal inputs < 10% of range.  U1273AX only: For all AC voltage ranges, the accuracy is specified at 2.5% + 25 counts when measuring below –20 °C for 20 to 45 Hz AC signals. 2. Notes for current specifications:  Overload protection for 300 μA to 300 mA range: 0.44 A/1000 V; 10 × 35 mm 30 kA fast-acting fuse.  Overload protection for 3 A to 10 A range: 11 A/1000 V; 10 × 38 mm 30 kA fast-acting fuse.  Specification for 300 mA range: 440 mA continuous. Electrical Specifications (continued) Find us at www.keysight.com Page 14  Specification for 10 A range: 10 A continuous. Add 0.3% to the specified accuracy when measuring signals > 10 to 20 A for 30 seconds maximum. After measuring currents > 10 A, cool down the multimeter for twice the duration of the measured time before proceeding with low current measurements.  U1273AX only: The accuracy for the 300 μA range, 3000 μA range, and 30 mA is specified after the Null function is used when measuring at temperatures below –20 °C. The Null function is used to subtract the test lead resistance and thermal effect (by shorting the test leads).  U1273AX only: For all AC current ranges, the accuracy is specified at 2.5% + 25 counts when measuring below –20 °C for 20 to 45 Hz AC signals. Electrical Specifications (continued) Find us at www.keysight.com Page 15 AC + DC specifications for U1272A/U1273A and U1273AX Accuracy ± (% of reading + counts of least significant digit) Function Range Resolution 45 Hz to 65 Hz 20 Hz to 1 kHz 1 kHz to 5 kHz 5 kHz to 20 kHz 20 kHz to 100 kHz True RMS AC + DC Voltage1 30 mV 0.001 mV 0.7 + 40 0.8 + 45 1.1 + 45 1.1 + 60 3.6 + 60 300 mV 0.01 mV 0.7 + 25 0.8 + 30 1.1 + 30 1.1 + 45 3.6 + 45 3 V 0.0001 V 0.7 + 25 1.1 + 30 1.6 + 30 2.1 + 45 3.6 + 45 30 V 0.001 V 0.7 + 25 1.1 + 30 1.6 + 30 2.1 + 45 3.6 + 45 300 V 0.01 V 0.7 + 25 1.1 + 30 1.6 + 30 2.1 + 45 — 1000 V 0.1 V 0.7 + 25 1.1 + 30 1.6 + 30 — — Accuracy ± (% of reading + counts of least significant digit) Function Range Resolution 45 Hz to 65 Hz 20 Hz to 2 kHz Burden voltage/Shunt True RMS AC + DC Current2 300 µA 0.01 µA 0.8 + 30 1.1 + 30 < 0.04 V/100 Ω 3000 µA 0.1 µA 0.8 + 30 1.1 + 30 < 0.4 V/100 Ω 30 mA 0.001 mA 0.8 + 30 1.1 + 30 < 0.08 V/1 Ω 300 mA 0.01 mA 0.8 + 30 1.1 + 30 < 1.00 V/1 Ω 3 A 0.0001 A 0.9 + 35 1.3 + 35 < 0.1 V/0.01 Ω 10 A 0.001 A 0.9 + 35 1.3 + 35 < 0.3 V/0.01 Ω 1. Notes for voltage specifications: - Overload protection: 1000 Vrms. For millivolt measurements, 1000 Vrms for short circuits with < 0.3 A current. - Input impedance: 10 MΩ (nominal) in parallel with < 100 pF. - The input signal is lower than the product of 20,000,000 V×Hz. - For 20 to 100 kHz accuracy: Three counts of the LSD per kHz of additional error is to be added for frequencies > 20 kHz and signal inputs < 10% of range. - U1273AX only: For all AC voltage ranges, the accuracy is specified at 2.5% + 25 counts when measuring below –20 °C for 20 to 45 Hz AC signals. 2. Notes for current specifications: - Overload protection for 300 μA to 300 mA range: 0.44 A/1000 V; 10 × 35 mm 30 kA fast-acting fuse. - Overload protection for 3 A to 10 A range: 11 A/1000 V; 10 × 38 mm 30 kA fast-acting fuse. - Specification for 300 mA range: 440 mA continuous. - Specification for 10 A range: 10 A continuous. Add 0.3% to the specified accuracy when measuring signals > 10 to 20 A for 30 seconds maximum. After measuring currents > 10 A, cool down the multimeter for twice the duration of the measured time before proceeding with low current measurements. - U1273AX only: The accuracy for the 300 μA range, 3000 μA range, and 30 mA is specified after the Null function is used when measuring at temperatures below –20 °C. The Null function is used to subtract the test lead resistance and thermal effect (by shorting the test leads). - U1273AX only: For all AC current ranges, the accuracy is specified at 2.5% + 25 counts when measuring below –20 °C for 20 to 45 Hz AC signals. Electrical Specifications (continued) Find us at www.keysight.com Page 16 Temperature specifications1 - 4 Thermocouple type Range Resolution Accuracy ± (% of reading + as specified below) U1271A U1272A U1273A/U1273AX K –200 to 1372 °C 0.1 °C 1% reading + 1 °C 1% reading + 1 °C 1% reading + 1 °C –328 to 2502 °F 0.1 °F 1% reading + 1.8 °F 1% reading + 1.8 °F 1% reading + 1.8 °F I –210 to 1200 °C 0.1 °C — 1% reading + 1 °C 1% reading + 1 °C –346 to 2192 °F 0.1 °F — 1% reading + 1.8 °F 1% reading + 1.8 °F 1. The specifications above is specified after 60 minutes of warm-up time. 2. The accuracy does not include the tolerance of the thermocouple probe. 3. Do not allow the temperature sensor to contact a surface that is energized above 30 Vrms or 60 V DC. Such voltages pose a shock hazard. 4. The temperature calculation is specified according to the safety standards of EN/IEC-60548-1 and NIST175. Capacitance specifications5, 6 Range Resolution Accuracy ± (% of reading + counts of least significant digit) U1271A U1272A U1273A/U1273AX 10 nF 0.001 nF 1 + 5 1 + 5 1 + 5 100 nF 0.01 nF 1 + 2 1 + 2 1 + 2 1000 nF 0.1 nF 1 + 2 1 + 2 1 + 2 10 μF 0.001 μF 1 + 2 1 + 2 1 + 2 100 μF 0.01 μF 1 + 2 1 + 2 1 + 2 1000 μF 0.1 μF 1 + 2 1 + 2 1 + 2 10 mF 0.001 mF 1 + 2 1 + 2 1 + 2 5. Overload protection: 1000 Vrms for short circuits with < 0.3 A current. 6. The accuracy for all ranges is specified based on a film capacitor or better, and after the Null function is used to subtract the test lead resistance and thermal effect (by opening the test leads). Electrical Specifications (continued) Find us at www.keysight.com Page 17 Frequency specifications1, 2 Range Resolution Accuracy ± (% of reading + counts of least significant digit) Maximum input frequency 99.999 Hz 0.001 Hz 0.02 + 5 0.5 Hz 999.99 Hz 0.01 Hz 0.005 + 5 9.9999 Hz 0.1 Hz 0.005 + 5 99.999 kHz 1 Hz 0.005 + 5 999.99 kHz 0.01 Hz 0.005 + 5 > 1 MHz 0.1 Hz 0.005 + 5 @< 1 MHz 1. Overload protection: 1000 V; input signal is < 20,000,000 V × Hz (product of voltage and frequency). 2. The frequency measurement is susceptible to error when measuring low-voltage, low-frequency signals. Shielding inputs from external noise pickup is critical for minimizing measurement errors. Turning on the low pass filter may help you to filter out the noise and achieve a stable reading. Duty Cycle3 Mode Range Accuracy at full scale DC coupling 99.99% 0.3 % per kHz + 0.3 % AC coupling 99.99% 0.3 % per kHz + 0.3 % 3. Notes for duty cycle specifications: - The accuracy for duty cycle and pulse width measurements is based on a 3 V square wave input to the DC 3 V range. For AC couplings, the duty cycle range can be measured within the range of 10% to 90% for signal frequencies > 20 Hz. - The range of the duty cycle is determined by the frequency of the signal: {10 μs × frequency × 100%} to {[1 – (10 μs × frequency)] × 100%}. - The pulse width (positive or negative) must be > 10 μs. The range of the pulse width is determined by the frequency of the signal. Pulse Width4 Range Resolution Accuracy at full scale 999.99 ms 0.01 ms (duty cycle accuracy/frequency) + 0.01 ms 2000.0 ms 0.1 ms (duty cycle accuracy/frequency) + 0.1 ms 4. Notes for pulse width specifications: - The accuracy for duty cycle and pulse width measurements is based on a 3 V square wave input to the DC 3 V range. - The pulse width (positive or negative) must be > 10 μs. The range of the pulse width is determined by the frequency of the signal. Electrical Specifications (continued) Find us at www.keysight.com Page 18 U1271A and U1272A frequency sensitivity for voltage measurements 1, 2, 3 Minimum sensitivity (RMS sine wave) Trigger level for DC coupling Input range 15 Hz to 0.5 Hz to 200 kHz Up to 1 MHz 0.5 Hz to 200 kHz 100 kHz U1271A U1272A 30 mV 3 mV 3 mV — — 5 mV 300 mV 6 mV 8 mV 40 mV 10 mV 15 mV 3 V 0.12 V 0.2 V 0.4 V 0.15 V 0.15 V 30 V 0.6 V 0.8 V 2.6 V 1.5 V 1.5 V 300 V 6 V 8 V @ < 100 kHz — 9 V @ < 100 kHz 9 V @ < 100 kHz 1000 V 50 V 50V @ < 100 kHz — 90 V @ < 100 kHz 90 V @ < 100 kHz 1. Maximum input for specified accuracy, refer to “AC specifications” on page 12. 2. 30 mV range applicable for U1272A only. 3. 200 kHz to 1 MHz range applicable for U1272A only. U1273A/U1273AX sensitivity for voltage measurements4 Input range Frequency sensitivity and trigger level Minimum sensitivity (RMS sine wave) Trigger level for DC coupling Maximum input for specified accuracy, refer to AC voltage 15 Hz to 100 kHz 0.5 Hz to 200 kHz Up to 1 MHz 0.5 Hz to 200 kHz 30 mV 3 mV 3 mV — 5 mV 300 mV 7 mV 8 mV 38 mV 15 mV 3 V 0.12 V 0.2 V 0.48 V 0.15 V 30 V 0.8 V 0.8 V 3.5 V 1.5 V 300 V 6.7 V 8 V < 100 kHz — 11 V < 100 kHz 1000 V 67 V 67 V < 100 kHz — 110 V < 100 kHz 4. Maximum input for specified accuracy, refer to “AC specifications” on page 13. Frequency sensitivity for current measurements5 Minimum sensitivity (RMS sine wave) 2 Hz to 30 kHz Input range U1271A/U1272A U1273A/U1273AX 300 µA 100 µA 70 μA 3000 µA 70 µA 120 μA 30 mA 1.2 mA 1.2 mA 300 mA 12 mA 12 mA 3 A 0.12 A 0.12 A 10 A 1.2 A 1.2 A 5. Maximum input for specified accuracy, refer to “AC specifications” on page 12 and 13. Electrical Specifications (continued) Find us at www.keysight.com Page 19 Peak hold Signal width Accuracy for DC Voltage and Current Single event >1 ms Specified accuracy + 400 Repetitive >250 µs Specified accuracy + 1000 Decibel (dB) for U1272A and U1273A1, 2, 3 dB Reference Default reference 1 mW (dBm) 1 to 9999 Ω 50 Ω 1 V (dBV) 1 V 1 V 1. The reading of dBm is indicated in decibels of power above or below 1 mW, or decibels of voltage above or below 1 V. The formula is calculated according to the voltage measurement and specified reference impedance. Its accuracy is depended on the accuracy of the voltage measurement. See Decibel (dBV) accuracy table below. 2. Auto-ranging mode is used. 3. The bandwidth is according to voltage measurement. Decibel (dBV) accuracy dBV range Accuracy Range Minimum Maximum 45 Hz to 65 Hz 20 Hz to 1 kHz 1 Hz to 5 kHz 5 kHz 20 kHz 20 kHz to 100 kHz 30 mV –56.48 –30.46 0.06 0.07 0.09 0.1 0.32 300 mV –36.48 –10.46 0.06 0.07 0.09 0.1 0.32 3 V –16.48 +9.54 0.06 0.09 0.14 0.19 0.32 30 V +3.52 +29.54 0.06 0.09 0.14 0.19 0.32 300 V +23.52 +49.54 0.06 0.09 0.14 0.19 — 1000 V +33.98 +60 0.06 0.09 0.14 — — Measurement rate (approximate) Function Times / second U1271A U1272A/U1273A/U1273AX ACV 7 7 DCV 7 7 Ω 14 14 Ω with offset compensation — 3 Diode 14 14 Auto diode — 3 Capacitance 4 (< 100 µF) 4 (< 100 µF) DCA 7 7 ACA 7 7 Temperature 7 7 Frequency 2 (> 10 Hz) 2 (> 10 Hz) Duty cycle 1 (> 10 Hz) 1 (> 10 Hz) Pulse width 1 (> 10 Hz) 1 (> 10 Hz) Find us at www.keysight.com Page 20 Ordering Information Standard shipped accessories Standard test leads, test probes with 4-mm tips, K-type thermocouple and adapter, 4x AAA alkaline batteries (4x AAA lithium batteries for U1273AX only), Certificate of Calibration, UK 6 (test report), Quick Start Guide Optional accessories U1271A U1272A U1273A U1273AX Measuring accessories (non-temperature) U1161A Extended test lead kit Includes two test leads (red and black), two test probes, medium- sized alligator clips and 4-mm banana plugs. – Test leads: CAT III 1000 V, CAT IV 600 V, 15 A – Test probes (4-mm tips): CAT III 1000 V, CAT IV 600 V, 15 A – Medium-sized alligator clips: CAT III 1000 V/CAT IV 600 V, 15 A – 4-mm banana plugs: CAT II 600 V, 10 A U1162A Alligator clips – One pair of insulated alligator clips (red and black). Recommended for use with Keysight standard test leads. – CAT III 1000 V, CAT IV 600 V, 15 A U1163A SMT grabbers – One pair of SMT grabbers (red and black). Recommended for use with Keysight standard test leads. – Rated CAT II 300 V, 3 A U1164A Fine-tip test probes – One pair of fine-tip test probes (red and black). Recommended for use with Keysight standard test leads. – Rated CAT II 300 V, 3 A U1168A Standard test lead kit Includes two test leads (red and black), 4-mm test probes, alligator clips, fine-tip test probes, SMT grabbers and mini grabber (black). – Test leads: CAT III 1000 V, CAT IV 600 V, 15 A – Test probe (19-mm tips): CAT II 1000 V, 15 A – Test probe (4-mm tips): CAT III 1000 V, CAT IV 600 V, 15 A (highly recommended for CAT IV environment) – Alligator clips: CAT III 1000 V, CAT IV 600 V, 15 A – Fine-tip test probes: CAT II 300 V, 3 A – SMT grabber: CAT II 300 V, 3 A – Mini grabber: CAT II 300 V, 3 A U1583B AC current clamp – Dual range: 40 A and 400 A – Rated CAT III 600 V – BNC-to-banana-plug adapter provided for use with DMMs – -40 to 55 °C operating temperature Find us at www.keysight.com Page 21 Ordering Information (continued) Optional accessories (continued) Measuring accessories (non-temperature) U1180A Thermocouple adapter + lead kit, J and K types Includes thermocouple adapter, thermocouple bead J-type and thermocouple bead K-type. – T/C adapter J/K-type – T/C bead J-type: –20 to 200 °C – T/C bead K-type: –20 to 200 °C U1181A Immersion temperature probe – Type-K T/C for use in oil and other liquids – Measurement range: –50 to 700 ºC – Includes adapter U1184A for connection to DMM U1182A Industrial surface temperature probe – Type-K T/C for use on still surfaces – Measurement range: –50 to 400 ºC – Includes adapter U1184A for connection to DMM U1183A Air temperature probe – Type-K T/C for use in air and non-caustic gas – Measurement range: –50 to 800 ºC – Includes adapter U1184A for connection to DMM U1184A Temperature probe adapter – Mini-connector-to-banana-plug adapter for use with DMM U1185A J-type thermocouple and adapter – T/C adapter J/K-type – T/C bead J-type: –20 to 200 °C U1186A K-type thermocouple and adapter – T/C adapter J/K-type – T/C bead J-type: –20 to 200 °C Find us at www.keysight.com This information is subject to change without notice. © Keysight Technologies, 2017, 2019, 2020, Published in USA, August 22, 2020, 5990-6425EN Page 22 Learn more at: www.keysight.com For more information on Keysight Technologies’ products, applications or services, please contact your local Keysight office. The complete list is available at: www.keysight.com/find/contactus Ordering Information (continued) Optional accessories (continued) U1171A Magnetic hanging kit For fastening the DMM to a steel surface so both hands are free U1173A IR-to-USB cable – For remote control and data logging to PC – Maximum baud rate: 19,200 bits per second U1174A Soft carrying case The convenient way to carry your DMM and essential accessories – Dimension: 9 inches (H) x 5 inches (W) x 3 inches (D) U1177A Bluetooth Adapter – Enables Bluetooth connection to Keysight handheld digital multimeters – Support the U1230, U1240, U1250 and U1270 Series handheld multimeters – Remote monitoring and data logging capabilities via Android devices or Windows-based PC – Recommended to use Lithium battery in lower than -20°C in temperature for long hours of operation
Ohm and Power laws Resistance Box Wires Power Source Voltmeter and Ammeter Revisions 2022 T. Vahabi, E. Horsley, A. Harlick 2017 C. Lee 2016 J. Ladan and R. M. Serbanescu © 2016-2022 University of Toronto This work is licensed under the Creative Commons Attribution-NonComercial-ShareAlike 4.0 Unported License (http://creativecommons.org/licenses/oby-nc-sa/4.0/) 1 http://creativecommons.org/licenses/oby-nc-sa/4.0/ Introduction to fitting methods Background knowledge for Part I • Python: arrays, numpy, pyplot, scipy, curve fit() • Error Analysis Chi squared (�2), goodness of fit • Physics: Ohm’s law. Review appropriate sections of your textbook. Introduction The function curve fit() is introduced for fitting linear data. You will perform voltage and current measurements on a resistor to find the resistance via Ohm’s law. You are expected to maintain a notebook (physical or electronic) where you record your measurements, create plots, and write Python code appropriate for the lab. This lab contains explicit questions for you to answer labelled with “Q”. They are intended as prompts for analysis and discussion. The answers to the questions should be organically interwoven into your report. In this exercise, we will introduce how to fit experimental data to a certain function, and get quantitative observations from the experiment. The package “Notes on Error Analysis” can be found at https://www.physics. utoronto.ca/~phy224_324/web-pages/Notes_Error.pdf. It introduces linear and non- linear regression methods, aimed at finding the values of parameters which give the closest agreement between the data and the theoretical model. You should read the section on least-squares fitting. Suppose our data consisted of N measurements (yi, xi) with i = 1, 2, ..., N . A first step is to plot the data with error bars representing the reading error (or the standard deviation �i) on each point. This gives a visual check for what relation our data has. Next, assume that we want to make a function f that will predict a value ȳi if we call the function with the measurement xi and some set of parameters p0, p1. . . that represents the experiment (e.g. resistance, gravity, spring constant could all be parameters). Our goal is to find the value of the parameters that will allow the function f to make the best predictions given the available data. We could use this function f to make predictions for each measurement pair (xi, yi), and make guesses about the parameters until we find ‘good’ agreement, but a more robust method is to use the data fitting capabilities of Python to find the parameter values for us, using a form of linear regression. 2 https://www.physics.utoronto.ca/~phy224_324/web-pages/Notes_Error.pdf https://www.physics.utoronto.ca/~phy224_324/web-pages/Notes_Error.pdf Curve fitting To make use of Python in our regression, we need to write our mathematical relationship between x and y as a Python function. In this experiment we’ll be looking at an experiment that we think is modeled by a linear equation y = ax+ b, where a and b are the parameters we are trying to determine. In Python code this becomes Note that the independent variable x must be the first argument. In order for your model function to work it must be able to take a numpy array and return an array. This function only makes a prediction of the measurement of y given x. The job of finding the parameters is done by the scipy package in Python, using the curve fit() function from the scipy.optimize module. Every time you call curve fit() you will give it 3 arguments, and usually more. The first argument is your model function that curve fit() will use to make predictions, followed by the x data, and the y data. curve fit() can also take three additional arguments — you can provide an initial guess for your parameters using the p0 keyword argument, the uncertainties in the dependent data in the sigma keyword argument, and you should (almost always) and force curve fit() to consider uncertainties as absolute values (instead of relative values) using the absolute sigma=True keyword. Your code will then look like this: curve fit() returns two values. The first value p opt contains the estimates of the parameters you used in the model (in this case p opt has two elements). The second value p cov contains the covariance matrix giving the uncertainties in estimates of the parameters you used in the model (in this case p cov is a 2 ⇥ 2 matrix). In most experiments the important values in p cov will be the diagonal elements, which represent the variance (the uncertainty–squared) of the parameters. An easy way to extract these values and convert them to uncertainties is : where sqrt and diag are functions from numpy that calculate the square root, and extract the diagonal elements, respectively. p std will now contain your parameter uncertainties. 3 Curve fitting internals Internally, curve fit() uses an algorithm to find the best parameters. It does this by defining a metric called �2 that it tries to minimize. Starting with your initial guess for the parameters, curve fit() will calculate the value of �2 based on the experimental data and your model, and adjust the parameters systematically until it finds the minimum value of �2 within a (very small) uncertainty. For simple equations, like the linear model above, curve fit() will take approximately seven tries to find the best set of parameters. With more complex non–linear functions it can take about 50 tries if you make a good initial guess, and more than 1, 000 tries if you make a bad initial guess. In the default setup curve fit() is configured to stop trying after 800 tries, where it will report that it failed and stop the program. It is, however, possible for curve fit() to stop ‘successfully’ and have an ill fitting model if your model function is inappropriate, your data is inconsistent, or your initial parameter guesses were bad. In this case you might see that the parameters haven’t been changed from your initial guess, or the variances are infinite (inf) or “not a number”(NaN). We will use the metric �2 later to measure how good the model function fits the data, but you don’t need to define it for curve fit() to work. In words, �2 can be written as �2 = X measurements ✓ dependent data value - predicted value dependent data measurement error ◆2 (1) For a ‘good’ fit we want our model prediction to be about as close to each measurement (the numerator) as the uncertainty in each measurement (the denominator). We square the function because a model that is too high is just as bad as a model that it too low. Mathematically, �2 is written as �2 = NX i=1 ✓ yi � y(xi) u(xi) ◆2 (2) where xi and yi are the data from the ith measurement, f(xi) is the prediction from the model of the value yi, and u(xi) is the uncertainty of the ith measurement. There are many keyword arguments to curve fit(), which can be found in the documen- tation. Commonly, you may use maxfev to control the maximum number of function calls. The default value is 800. 4 https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html The experiment We will be testing Ohm’s Law – the linear relation between voltage and current in elec- trical circuits. The voltage is our independent variable, current is the dependent variable, and resistance is the parameter we would like to measure. There are two subparts to the experiment: testing a known resistor and testing a potentiometer. Uncertainty from multimeters Our later analysis requires an idea of the uncertainty in the measurements. Digital multi- meters have two types of uncertainty: • Error of accuracy which depends on function (DCV, ACV, DCA, ACA and Ohm (⌦)). This error is provided by the instrument manufacturer. This error is expressed as a percentage of the reading. Please check the link below for complete instrument specifications. The reading can be done on di↵erent scales of the same function, which changes the number of digits on display and also the error of accuracy. • Error of precision which is due to fluctuations at the level of the last digit. These errors are both present at the same time and vary depending on your reading, the range of values it falls int, and the type of multimeter used. Use the instructions provided in the multimeter manual to determine the values of uncertainties for your voltage, current, and resistance (where appropriate). Be sure to track the uncertainty in measurements from the multimeter. Also, keep in mind that there are other sources of uncertainty that are harder to measure, like the quality of electrical connections, change in the temperature of the resistor, etc. Experimental Procedure You will be provided with two multimeters, a board with electrical components (similar to that shown in Figure 1, and enough wires to connect everything. Choose one resistor, and one potentiometer for the experiment. 5 Figure 1: The board of electrical elements used in this experiment. The set of resistors is on the bottom-right, the potentiometers are on bottom-left, lightbulb (needed for Part II of this experiment) is in the top-right of the board. For the known resistor, perform the following. 1. Connect the ammeter, voltmeter, and power supply to the chosen resistor as shown schematically in Figure 2). Ask the TA if you’re in doubt. Note that Figures 3a and 3b show the visual representation for the set up and the closeup of the power supply indicating appropriate terminals for the connection. Figure 2: Circuit diagram of the experimental setup. 6 Ammeter Voltmeter Resistance Box Power Supply (a) Power Supply Wire Setup (b) Figure 3: Visual representation of the connections for step 1. Figure (a) shows the set up for the resistor, the voltmeter and the ammeter. Figure (b) is a close up of the power supply wire set up 2. Vary the voltage on the power supply. 3. Record the voltage and current from the multimeters, along with uncertainty. Make sure to choose the most appropriate range on the multimeters that allows you to measure your values with the highest precision. 4. Change the voltage, and record the new values – repeat su�cient number of times to have enough data to confidently fit. 5. Save the data (on a memory stick or in a file you store online) as a text file (ex- tension.txt) with the data in two columns: then first column being the independent variable (voltage). 6. After performing all the above measurements, disconnect the power, and switch the voltmeter to become a resistance meter. This will give you a reference value to compare against. 7. Repeat these measurements for a potentiometer, making sure not to move the dial be- tween measurements. If your resistor board has no potentiometer, use another resistor. Resistors are marked with coloured bars to indicate their nominal resistance through a code (there are many calculators for this online). As part of this exercise, you may also compare your results against the nominal resistance ± the tolerance. 7 iiiiii o 060 0.340 0.047 0.080 0 U4 0.055 0.100 0.645 01064 0.120 0ㄒ99 0.072 01 01A 㐅 140kfno 160 1.104 0叫 80 ⼈255 a 98 0.200 1.408 0.107 Analyzing the data We will analyze the dependency of the current on the voltage using a linear fitting program. Ohm’s law for resistors sates I = V R . (3) Thus, the resistance of a linear component can be determined by plotting I vs. V , and measuring the slope. There are other electrical components that have nonlinear voltage-current relations, like diodes, transistors, and amplifiers. Using a linear model for these would result in a bad fit. In Part II of this exercise, we will perform a nonlinear regression with a lightbulb. Build a linear fitting program Linear fitting is done via the Levenberg-Marquedt algorithm (an advanced least-squares method), as implemented in the scipy.optimize module. Today, you will use the curve fit() function (i.e. textttscipy.optimize.curve fit). Calculation of the statistics from the output of textttcurve fit() will be done in the next exercise. Here is the outline of the Python program for Part I: • import the necessary modules and functions (e.g. numpy and curve fit()) • Read the data file into an array with the loadtxt() function. • Extract the variables from the array columns. • Define the model function f(x) = ax+ b. • Call curve fit() with the function, data, and initial guess for the parameters. • Output the calculated parameters. • Create relevant plots (this includes plotting the residuals). Write a program to perform a linear fit for the data you collected in this experiment. It should create a plot of current vs. voltage with error bars and the line of best fit, and output the calculated value of resistance (the slope of the line of best fit). Run this program using the data for the resistor, and again using the data for the potentiometer. Q1 In Ohm’s law, V = IR, the line should pass through I = V = 0. Did your linear fit pass through zero as well? If it did not, why do you think that happened? Q2 What result do you find for resistance if you force the line to pass through zero? (i.e. try the model function f(x, a) = ax) Q3 How does your resistance from using curve fit() compare to the value measured with the multimeter? 8 Goodness of fit - reduced chi-squared Recall that the �2 distribution gives a measure of how unlikely a measurement is. The more a model deviates from the measurements, the higher the value of �2. But if �2 is too small, it is also an indication of a problem: usually that there were not enough samples. This best (most likely) value of �2 depends on the number of degrees of freedom of the data, v = N�n, where N is the number of observations and n is the number of parameters. This dependence is removed with the reduced chi-squared distribution, �2red = 1 v NX i=1 ✓ yi � y(xi) u(yi) ◆2 (4) where yi is the dependent data you measured, xi is the independent data, u(yi) is the measurement error in the dependent data. For �2red, the best value is 1: �2red � 1 indicates that the model is a poor fit; �2red > 1 indicates an incomplete fit or underestimated error variance; �2red < 1 indicates an over-fit model (not enough data, or the model somehow is fitting the noise). Q4 Add a function to your program to calculate �2red. What values were computed? How would you interpret your results? Resistors for electrical circuits are designed to be very linear and reliable, so the three measurements for resistance will likely be very close. The lines may even be indistinguishable on the plot. 9 Nonlinear fitting methods Nonlinear circuits We continue with the curve fitting from the previous exercises, but extend it to power law models. A lightbulb demonstrates various power laws; resistance and radioactive power are both dependent on temperature for blackbodies. In this lab, we will examine these relations with the voltage-current curve for a lightbulb. Background knowledge for Part II • Python: lists, arrays, numpy, scipy, pyplot, curve fit() • Error Analysis Chi squared (�2), goodness of fit • Physics: Power Law. You may need to review appropriate section of your textbooks to analyze and interpret your data. Introduction An incandescent lightbult works by heating a tungsten filament until it glows. All matter emits electromagnetic radiation, called thermal radiation, when it has a temperature above absolute zero. The simple theoretical material, a blackbody, is one that absorbs all radiation and follows a strict power law: P = A�T 4, (5) where � is the Stefan-Boltzmann constant (5.670373 ⇥10�8 W m�2K�4). Real materials are not ideal blackbodies. The relation for these “grey bodies” adds an emissivity, ✏ < 1, P = A�✏T 4, (6) The emissivity itself is typically not constant, and depends on temperature through a power law. For tungsten this relation is ✏(T ) = 1.731⇥ 10�3T 0.663 (7) Including the emissivity leads to a power law between power and temperature. There is also a power law for the resistance of the bulb. In the simplest case, it is treated as linear (R / T ), but for tungsten it is more accurately, R(t) / T 1.209 (8) Combining the dependencies on temperature with P = V I and V = IR, a power law can be found between current and voltage. For an ideal black body with a linear relation between resistance and temperature, 10 I / V 35 (9) For most of this experiment, we are not concerned with the constant of proportionality. However, when comparing the theoretical curve to your data you might need to define a suitable value. Analysis of power laws with logarithms The tool for analyzing power laws is, again, logarithms. Start with the general power law, y = axb, (10) and take the logarithm of both sides, log(y) = blog(x) + log(a). (11) Thus, log(y) depends linearly on log(x), with slope b. We can view this linear relation on a log-log plot (logarithmic scales for both x and y). Linear regression works again, with the same linear model function, except using log(xi) and log(yi) as the input data. Note: The same downfalls of using transformations apply. Again, using curve fit(), we can use a nonlinear function directly as our model function. The experiment We will observe the power law for blackbody radiation through a lightbulb’s voltage-current graph. Set up the Ohm’s Law experiment as you did in Part I, but with a lightbulb instead of a resistor, as shown in Figure 4. Ammeter Voltmeter Power Supply Lightbulb Figure 4: Visual representation of the experimental set up used in this part of the experiment. After setting up the apparatus do the following: 11 power Xiii's 11 1. Adjust the power supply voltage to its lowest value. 2. Wait for the voltage and current to stabilize (it should not take long). 3. Record the voltage and current values. 4. Record the uncertainties in your measurements. 5. Repeat this process for at least 15 di↵erent values of voltage. Pay attention to the frequency with which you collect the data - if the dependent values change quickly, decrease the “step”. 6. Save the values in a text file (.txt) and save it on an external memory device or online. The Python program You will use your programs to compare the transformation method and the non-linear least- squares method. The best parameters will be found with both methods, and used to plot best fit curves over the data. The program should be organized as follows: • Import the required functions and modules (numpy, scipy.optimize.curve fit(), matplotlib.pyplot). • Define the model functions(linearized: f(x, a, b) = ax+ b,non–linear: g(x, a, b) = axb) • Load the data and uncertainty measurements using loadtxt(). • Perform the linear regression on (log(xi),log(yi)) using f(x, a, b) as the model func- tion. • Perform the nonlinear regression on (xi,yi) using g(x, a, b) as the model function. • Output both of the power law relations you calculated. • Plot the error bars, both curves of best fit, and the theoretical curve. Include a legend for the di↵erent curves. • Plot all the same things on a log–log plot. One way of doing this is with the pylab.loglog() function, another way is to call pylab.yscale(‘log’) and pylab.xscale(‘log’) after you make the plot. The a and b parameters in the two model functions represent di↵erent parameters in the original, un-transformed, model function. Make sure you can convert between the functions correctly to compare the parameters accurately. Write the program, and run it using the data gathered in the experiment. Save all plots and the parameters you determined. Q5 Which regression method gave an exponent closer to the expected value? Can you see the di↵erence on the plots? 12 Analyzing the quality of the fit As covered in previous exercises, there are two ways that we can assess the quality of the fit of our model: variance of the calculated parameters, and the reduced chi-squared statistic. The variance of the parameters is returned by curve fit() as the diagonal entries in the covariance matrix, p cov. Recall that the uncertainty in measurements is understood as the standard deviation, a = ā± �a, (12) where �a = p V ar(a) In the python program, the variance of the first parameter in your model function is given in curve fit()p cov[0, 0], the variance of the second parameter in curve fit()p cov[1, 1], and so on. Modify your program from the Python program section to calculate the stan- dard deviation of the parameters. Q6 What values did you find? Does the value of your fitted exponent fall within the range of the blackbody values, 35 , with your calculated standard deviation? What about in comparison to the expected value for tungsten? Reduced chi-squared Recall that the �2 distribution gives a measure of how unlikely a measurement is. The more a model deviates from the measurements, the higher the value of �2. But if �2 is too small, it’s also an indication of a problem: usually that there were not enough samples. Add a function to your program to calculate �2red. Q7 What values were computed? How would you interpret your results? 13 Your submission for this exercise should include: • All sets of data, clearly labeled, presented in an appropriate manner (tables/figures). • All uncertainties, uncertainty budgets, sample uncertainty calculations • Important information about the set up (models of the multimeters, band colours of the resistor used, any other details that might be relevant (variations, disturbances, changes made (e.g. equipment swaps, heating noticed, lightbulb stops working, etc.) • Answers to all the questions, weaved throughout the report. • All relevant current vs voltage plots with su�cient captions • All obtained values of the resistance with uncertainties • All obtained values for power fit parameters with uncertainties • Plots of residuals • The final version of your Python code (included separately as a .py file) • Analysis and Discussion of your results. 14
Ohmdata V(V) I(A) Uncertainty (V)Uncertainty (I) 0.000 0.000 0.001 0.001 0.099 0.010 0.001 0.001 0.200 0.020 0.002 0.001 0.299 0.030 0.002 0.001 0.400 0.039 0.002 0.001 0.499 0.049 0.002 0.001 0.600 0.059 0.003 0.001 0.699 0.069 0.003 0.002 0.800 0.079 0.003 0.002 0.899 0.089 0.003 0.002 1.000 0.099 0.004 0.002 1
power_law_data 2.35.03 PM Power voltage(V) Voltage(V) Current(A) uncertainty (V)uncertainty (I) 0.050 0.044 0.63 0.001 0.02 0.100 0.087 1.23 0.001 0.03 0.150 0.131 1.82 0.001 0.05 0.200 0.175 2.38 0.001 0.06 0.250 0.220 2.89 0.002 0.07 0.300 0.265 3.36 0.002 0.08 0.350 0.311 3.78 0.002 0.09 0.400 0.357 4.16 0.002 0.09 0.450 0.403 4.49 0.002 0.10 0.500 0.450 4.79 0.002 0.11 0.550 0.498 5.06 0.002 0.11 0.600 0.544 5.30 0.002 0.12 0.650 0.592 5.51 0.002 0.12 0.700 0.640 5.71 0.003 0.12 0.750 0.689 5.89 0.003 0.13 1
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