LM158,LM258,LM2904,LM358 LM158/LM258/LM358/LM2904 Low Power Dual Operational Amplifiers Literature Number: SNOSBT3G LM158/LM258/LM358/LM2904 Low Power Dual Operational Amplifiers General Description Advantages The LM158 series consists of two independent, high gain, internally frequency compensated operational amplifiers which were designed specifically to operate from a single power supply over a wide range of voltages. Operation from split power supplies is also possible and the low power supply current drain is independent of the magnitude of the power supply voltage. n Two internally compensated op amps n Eliminates need for dual supplies n Allows direct sensing near GND and VOUT also goes to GND n Compatible with all forms of logic n Power drain suitable for battery operation Application areas include transducer amplifiers, dc gain blocks and all the conventional op amp circuits which now can be more easily implemented in single power supply systems. For example, the LM158 series can be directly operated off of the standard +5V power supply voltage which is used in digital systems and will easily provide the required interface electronics without requiring the additional 15V power supplies. Features The LM358 and LM2904 are available in a chip sized package (8-Bump micro SMD) using National's micro SMD package technology. Unique Characteristics n In the linear mode the input common-mode voltage range includes ground and the output voltage can also swing to ground, even though operated from only a single power supply voltage. n The unity gain cross frequency is temperature compensated. n The input bias current is also temperature compensated. n Available in 8-Bump micro SMD chip sized package, (See AN-1112) n Internally frequency compensated for unity gain n Large dc voltage gain: 100 dB n Wide bandwidth (unity gain): 1 MHz (temperature compensated) n Wide power supply range: -- Single supply: 3V to 32V -- or dual supplies: 1.5V to 16V n Very low supply current drain (500 A) -- essentially independent of supply voltage n Low input offset voltage: 2 mV n Input common-mode voltage range includes ground n Differential input voltage range equal to the power supply voltage n Large output voltage swing Voltage Controlled Oscillator (VCO) 00778723 (c) 2005 National Semiconductor Corporation DS007787 www.national.com LM158/LM258/LM358/LM2904 Low Power Dual Operational Amplifiers October 2005 LM158/LM258/LM358/LM2904 Distributors for availability and specifications. Absolute Maximum Ratings (Note 9) If Military/Aerospace specified devices are required, please contact the National Semiconductor Sales Office/ LM158/LM258/LM358 LM2904 LM158A/LM258A/LM358A Supply Voltage, V+ 32V 26V Differential Input Voltage 32V 26V -0.3V to +32V -0.3V to +26V Molded DIP 830 mW 830 mW Metal Can 550 mW Small Outline Package (M) 530 mW micro SMD 435mW Input Voltage Power Dissipation (Note 1) 530 mW Output Short-Circuit to GND (One Amplifier) (Note 2) V+ 15V and TA = 25C Continuous Continuous 50 mA 50 mA 0C to +70C -40C to +85C Input Current (VIN < -0.3V) (Note 3) Operating Temperature Range LM358 LM258 -25C to +85C LM158 -55C to +125C Storage Temperature Range -65C to +150C -65C to +150C 260C 260C 300C 300C 260C 260C Vapor Phase (60 seconds) 215C 215C Infrared (15 seconds) 220C 220C Lead Temperature, DIP (Soldering, 10 seconds) Lead Temperature, Metal Can (Soldering, 10 seconds) Soldering Information Dual-In-Line Package Soldering (10 seconds) Small Outline Package See AN-450 "Surface Mounting Methods and Their Effect on Product Reliability" for other methods of soldering surface mount devices. ESD Tolerance (Note 10) 250V 250V Electrical Characteristics V+ = +5.0V, unless otherwise stated Parameter Conditions LM158A Min Typ LM358A Max Min Typ LM158/LM258 Max Min Typ Units Max Input Offset Voltage (Note 5), TA = 25C 1 2 2 3 2 5 mV Input Bias Current IIN(+) or IIN(-), TA = 25C, 20 50 45 100 45 150 nA 30 nA V+-1.5 V VCM = 0V, (Note 6) Input Offset Current IIN(+) - IIN(-), VCM = 0V, TA = 25C Input Common-Mode V+ = 30V, (Note 7) Voltage Range (LM2904, V+ = 26V), TA = 25C Supply Current Over Full Temperature Range 2 10 V+-1.5 0 5 30 V+-1.5 0 3 0 RL = on All Op Amps V+ = 30V (LM2904 V+ = 26V) V+ = 5V www.national.com 2 1 2 1 2 1 2 mA 0.5 1.2 0.5 1.2 0.5 1.2 mA V+ = +5.0V, unless otherwise stated Parameter Conditions LM358 Min Typ LM2904 Max Min Units Typ Max Input Offset Voltage (Note 5) , TA = 25C 2 7 2 7 mV Input Bias Current IIN(+) or IIN(-), TA = 25C, 45 250 45 250 nA 50 5 50 nA VCM = 0V, (Note 6) Input Offset Current IIN(+) - IIN(-), VCM = 0V, TA = 25C 5 + Input Common-Mode V = 30V, (Note 7) Voltage Range (LM2904, V+ = 26V), TA = 25C Supply Current Over Full Temperature Range + 0 V -1.5 + 0 V -1.5 V RL = on All Op Amps V+ = 30V (LM2904 V+ = 26V) V+ = 5V 1 2 1 2 mA 0.5 1.2 0.5 1.2 mA Electrical Characteristics V+ = +5.0V, (Note 4), unless otherwise stated Parameter Conditions Large Signal Voltage V+ = 15V, TA = 25C, Gain RL 2 k, (For VO = 1V LM158A Min Typ LM358A Max Min Typ LM158/LM258 Max Min Typ Units Max 50 100 25 100 50 100 V/mV 70 85 65 85 70 85 dB 65 100 65 100 65 100 dB -120 dB to 11V) Common-Mode TA = 25C, Rejection Ratio VCM = 0V to V+-1.5V Power Supply V+ = 5V to 30V Rejection Ratio (LM2904, V+ = 5V to 26V), TA = 25C Amplifier-to-Amplifier f = 1 kHz to 20 kHz, TA = 25C Coupling (Input Referred), (Note 8) Output Current -120 -120 Source VIN+ = 1V, VIN- = 0V, V+ = 15V, 20 40 20 40 20 40 mA 10 20 10 20 10 20 mA 12 50 12 50 12 50 A VO = 2V, TA = 25C Sink VIN- = 1V, VIN+ = 0V V+ = 15V, TA = 25C, VO = 2V VIN- = 1V, VIN+ = 0V TA = 25C, VO = 200 mV, V+ = 15V Short Circuit to Ground TA = 25C, (Note 2), 40 V+ = 15V Input Offset Voltage (Note 5) Input Offset Voltage RS = 0 IIN(+) - IIN(-) Input Offset Current RS = 0 7 IIN(+) or IIN(-) 60 40 5 15 7 30 Drift Input Bias Current 40 4 Drift Input Offset Current 60 20 60 mA 7 mV 7 75 V/C 100 10 200 10 300 10 40 100 40 200 40 nA pA/C 300 nA V+-2 V + Input Common-Mode V = 30 V, (Note 7) Voltage Range (LM2904, V+ = 26V) 0 3 V+-2 0 V+-2 0 www.national.com LM158/LM258/LM358/LM2904 Electrical Characteristics LM158/LM258/LM358/LM2904 Electrical Characteristics (Continued) V+ = +5.0V, (Note 4), unless otherwise stated Parameter Conditions Large Signal Voltage V+ = +15V Gain (VO = 1V to 11V) LM158A Min Typ LM358A Max Min Typ 25 LM158/LM258 Max 15 Min Typ Units Max 25 V/mV RL 2 k VOH V+ = +30V Output (LM2904, V+ = 26V) Voltage RL = 2 k 26 RL = 10 k 27 VOL V+ = 5V, RL = 10 k Swing Output Current Source VIN+ = +1V, VIN- = 0V, V+ = 15V, VO = 2V Sink VIN- = +1V, VIN+ = 0V, V+ = 15V, VO = 2V 26 28 5 27 26 28 20 5 V 27 28 20 5 V 20 mV 10 20 10 20 10 20 mA 10 15 5 8 5 8 mA Electrical Characteristics V+ = +5.0V, (Note 4), unless otherwise stated Parameter LM358 Conditions Large Signal Voltage V+ = 15V, TA = 25C, Gain RL 2 k, (For VO = 1V Min Typ 25 LM2904 Max Units Min Typ Max 100 25 100 V/mV 65 85 50 70 dB 65 100 50 100 dB -120 dB to 11V) Common-Mode TA = 25C, Rejection Ratio VCM = 0V to V+-1.5V Power Supply V+ = 5V to 30V Rejection Ratio (LM2904, V+ = 5V to 26V), TA = 25C Amplifier-to-Amplifier f = 1 kHz to 20 kHz, TA = 25C Coupling (Input Referred), (Note 8) Output Current -120 Source VIN+ = 1V, VIN- = 0V, V+ = 15V, 20 40 20 40 mA 10 20 10 20 mA 12 50 12 50 A VO = 2V, TA = 25C Sink VIN- = 1V, VIN+ = 0V V+ = 15V, TA = 25C, VO = 2V VIN- = 1V, VIN+ = 0V TA = 25C, VO = 200 mV, V+ = 15V Short Circuit to Ground TA = 25C, (Note 2), 40 V+ = 15V Input Offset Voltage (Note 5) Input Offset Voltage RS = 0 Input Offset Current IIN(+) - IIN(-) RS = 0 7 IIN(+) or IIN(-) Input Common-Mode V+ = 30 V, (Note 7) Voltage Range (LM2904, V+ = 26V) www.national.com 45 10 40 0 4 60 mA 10 mV 7 150 Drift Input Bias Current 40 9 Drift Input Offset Current 60 V/C 200 10 500 V+-2 40 0 nA pA/C 500 nA V+ -2 V (Continued) V+ = +5.0V, (Note 4), unless otherwise stated Parameter LM358 Conditions Large Signal Voltage V+ = +15V Gain (VO = 1V to 11V) Min Typ LM2904 Max 15 Min Typ Units Max 15 V/mV RL 2 k Output Voltage Swing Output Current VOH V+ = +30V (LM2904, V+ = 26V) RL = 2 k 26 RL = 10 k 27 VOL V+ = 5V, RL = 10 k 22 28 5 Source VIN+ = +1V, VIN- = 0V, V+ = 15V, VO = 2V Sink VIN- = +1V, VIN+ = 0V, V+ = 15V, VO = 2V 23 20 V 24 5 V 100 mV 10 20 10 20 mA 5 8 5 8 mA Note 1: For operating at high temperatures, the LM358/LM358A, LM2904 must be derated based on a +125C maximum junction temperature and a thermal resistance of 120C/W for MDIP, 182C/W for Metal Can, 189C/W for Small Outline package, and 230C/W for micro SMD, which applies for the device soldered in a printed circuit board, operating in a still air ambient. The LM258/LM258A and LM158/LM158A can be derated based on a +150C maximum junction temperature. The dissipation is the total of both amplifiers -- use external resistors, where possible, to allow the amplifier to saturate or to reduce the power which is dissipated in the integrated circuit. Note 2: Short circuits from the output to V+ can cause excessive heating and eventual destruction. When considering short cirucits to ground, the maximum output current is approximately 40 mA independent of the magnitude of V+. At values of supply voltage in excess of +15V, continuous short-circuits can exceed the power dissipation ratings and cause eventual destruction. Destructive dissipation can result from simultaneous shorts on all amplifiers. Note 3: This input current will only exist when the voltage at any of the input leads is driven negative. It is due to the collector-base junction of the input PNP transistors becoming forward biased and thereby acting as input diode clamps. In addition to this diode action, there is also lateral NPN parasitic transistor action on the IC chip. This transistor action can cause the output voltages of the op amps to go to the V+voltage level (or to ground for a large overdrive) for the time duration that an input is driven negative. This is not destructive and normal output states will re-establish when the input voltage, which was negative, again returns to a value greater than -0.3V (at 25C). Note 4: These specifications are limited to -55C TA +125C for the LM158/LM158A. With the LM258/LM258A, all temperature specifications are limited to -25C TA +85C, the LM358/LM358A temperature specifications are limited to 0C TA +70C, and the LM2904 specifications are limited to -40C TA +85C. Note 5: VO . 1.4V, RS = 0 with V+ from 5V to 30V; and over the full input common-mode range (0V to V+ -1.5V) at 25C. For LM2904, V+ from 5V to 26V. Note 6: The direction of the input current is out of the IC due to the PNP input stage. This current is essentially constant, independent of the state of the output so no loading change exists on the input lines. Note 7: The input common-mode voltage of either input signal voltage should not be allowed to go negative by more than 0.3V (at 25C). The upper end of the common-mode voltage range is V+ -1.5V (at 25C), but either or both inputs can go to +32V without damage (+26V for LM2904), independent of the magnitude of V+. Note 8: Due to proximity of external components, insure that coupling is not originating via stray capacitance between these external parts. This typically can be detected as this type of capacitance increases at higher frequencies. Note 9: Refer to RETS158AX for LM158A military specifications and to RETS158X for LM158 military specifications. Note 10: Human body model, 1.5 k in series with 100 pF. 5 www.national.com LM158/LM258/LM358/LM2904 Electrical Characteristics LM158/LM258/LM358/LM2904 Typical Performance Characteristics Input Voltage Range Input Current 00778734 00778735 Supply Current Voltage Gain 00778736 00778737 Open Loop Frequency Response Common-Mode Rejection Ratio 00778738 00778739 www.national.com 6 (Continued) Voltage Follower Pulse Response Voltage Follower Pulse Response (Small Signal) 00778741 00778740 Large Signal Frequency Response Output Characteristics Current Sourcing 00778742 00778743 Output Characteristics Current Sinking Current Limiting 00778744 00778745 7 www.national.com LM158/LM258/LM358/LM2904 Typical Performance Characteristics LM158/LM258/LM358/LM2904 Typical Performance Characteristics (Continued) Input Current (LM2902 only) Voltage Gain (LM2902 only) 00778746 00778747 Application Hints The LM158 series are op amps which operate with only a single power supply voltage, have true-differential inputs, and remain in the linear mode with an input common-mode voltage of 0 VDC. These amplifiers operate over a wide range of power supply voltage with little change in performance characteristics. At 25C amplifier operation is possible down to a minimum supply voltage of 2.3 VDC. Precautions should be taken to insure that the power supply for the integrated circuit never becomes reversed in polarity or that the unit is not inadvertently installed backwards in a test socket as an unlimited current surge through the resulting forward diode within the IC could cause fusing of the internal conductors and result in a destroyed unit. Large differential input voltages can be easily accomodated and, as input differential voltage protection diodes are not needed, no large input currents result from large differential input voltages. The differential input voltage may be larger than V+ without damaging the device. Protection should be provided to prevent the input voltages from going negative more than -0.3 VDC (at 25C). An input clamp diode with a resistor to the IC input terminal can be used. To reduce the power supply current drain, the amplifiers have a class A output stage for small signal levels which converts to class B in a large signal mode. This allows the amplifiers to both source and sink large output currents. Therefore both NPN and PNP external current boost transistors can be used to extend the power capability of the basic amplifiers. The output voltage needs to raise approximately 1 diode drop above ground to bias the on-chip vertical PNP transistor for output current sinking applications. For ac applications, where the load is capacitively coupled to the output of the amplifier, a resistor should be used, from the output of the amplifier to ground to increase the class A bias current and prevent crossover distortion. Where the load is directly coupled, as in dc applications, there is no crossover distortion. www.national.com Capacitive loads which are applied directly to the output of the amplifier reduce the loop stability margin. Values of 50 pF can be accomodated using the worst-case non-inverting unity gain connection. Large closed loop gains or resistive isolation should be used if larger load capacitance must be driven by the amplifier. The bias network of the LM158 establishes a drain current which is independent of the magnitude of the power supply voltage over the range of 3 VDC to 30 VDC. Output short circuits either to ground or to the positive power supply should be of short time duration. Units can be destroyed, not as a result of the short circuit current causing metal fusing, but rather due to the large increase in IC chip dissipation which will cause eventual failure due to excessive function temperatures. Putting direct short-circuits on more than one amplifier at a time will increase the total IC power dissipation to destructive levels, if not properly protected with external dissipation limiting resistors in series with the output leads of the amplifiers. The larger value of output source current which is available at 25C provides a larger output current capability at elevated temperatures (see typical performance characteristics) than a standard IC op amp. The circuits presented in the section on typical applications emphasize operation on only a single power supply voltage. If complementary power supplies are available, all of the standard op amp circuits can be used. In general, introducing a pseudo-ground (a bias voltage reference of V+/2) will allow operation above and below this value in single power supply systems. Many application circuits are shown which take advantage of the wide input common-mode voltage range which includes ground. In most cases, input biasing is not required and input voltages which range to ground can easily be accommodated. 8 LM158/LM258/LM358/LM2904 Connection Diagrams DIP/SO Package Metal Can Package 00778701 00778702 Top View Top View 8-Bump micro SMD 00778755 Top View (Bump Side Down) LM358BP micro SMD Marking Orientation LM2904IBP micro SMD Marking Orientation 00778757 00778756 Top View Top View LM358TP micro SMD Marking Orientation LM2904ITP micro SMD Marking Orientation 00778758 00778759 Top View Top View 9 www.national.com LM158/LM258/LM358/LM2904 Ordering Information Package Temperature Range -55C to 125C -25C to 85C SO-8 8-Pin Molded DIP 8-Pin Ceramic DIP TO-5, 8-Pin Metal Can 0C to 70C -40C to 85C LM358AM LM358AMX LM358M LM358MX LM2904M LM2904MX LM358AN LM358N LM2904N LM158AJ/883(Note 11) LM158J/883(Note 11) LM158J LM158AJLQML(Note 12) LM158AJQMLV(Note 12) LM158AH/883(Note 11) LM158H/883(Note 11) LM158AH LM158H LM158AHLQML(Note 12) LM158AHLQMLV(Note 12) LM258H N08E LM358H H08C LM358BP LM358BPX LM2904IBP LM2904IBPX 8-Bump micro SMD Lead Free LM358TP LM358TPX LM2904ITP LM2904ITPX LM158AWG/883 BPA08AAB 0.85 mm Thick TPA08AAA 0.50 mm Thick WG10A Note 11: LM158 is available per SMD #5962-8771001 LM158A is available per SMD #5962-8771002 Note 12: See STD Mil DWG 5962L87710 for Radiation Tolerant Devices www.national.com M08A J08A 8-Bump micro SMD 14-Pin Ceramic SOIC NSC Drawing 10 LM158/LM258/LM358/LM2904 Typical Single-Supply Applications (V+ = 5.0 VDC) Non-Inverting DC Gain (0V Output) 00778706 00778707 *R not needed due to temperature independent IIN DC Summing Amplifier (VIN'S 0 VDC and VO 0 VDC) Power Amplifier 00778709 VO = 0 VDC for VIN = 0 VDC 00778708 Where: VO = V1 + V2 - V3 - V4 AV = 10 (V1 + V2) (V3 + V4) to keep VO > 0 VDC 11 www.national.com LM158/LM258/LM358/LM2904 Typical Single-Supply Applications (V+ = 5.0 VDC) (Continued) "BI-QUAD" RC Active Bandpass Filter 00778710 fo = 1 kHz Q = 50 Av = 100 (40 dB) Fixed Current Sources Lamp Driver 00778712 00778711 www.national.com 12 LM158/LM258/LM358/LM2904 Typical Single-Supply Applications (V+ = 5.0 VDC) (Continued) LED Driver Current Monitor 00778713 00778714 *(Increase R1 for IL small) VL V+ -2V Driving TTL Voltage Follower 00778715 00778717 VO = VIN Pulse Generator 00778716 13 www.national.com LM158/LM258/LM358/LM2904 Typical Single-Supply Applications (V+ = 5.0 VDC) (Continued) Squarewave Oscillator Pulse Generator 00778718 00778719 Low Drift Peak Detector 00778720 HIGH ZIN LOW ZOUT www.national.com 14 LM158/LM258/LM358/LM2904 Typical Single-Supply Applications (V+ = 5.0 VDC) (Continued) High Compliance Current Sink Comparator with Hysteresis 00778722 00778721 IO = 1 amp/volt VIN (Increase RE for IO small) Voltage Controlled Oscillator (VCO) 00778723 *WIDE CONTROL VOLTAGE RANGE: 0 VDC VC 2 (V+ -1.5V DC) 15 www.national.com LM158/LM258/LM358/LM2904 Typical Single-Supply Applications (V+ = 5.0 VDC) (Continued) AC Coupled Inverting Amplifier 00778724 Ground Referencing a Differential Input Signal 00778725 www.national.com 16 LM158/LM258/LM358/LM2904 Typical Single-Supply Applications (V+ = 5.0 VDC) (Continued) AC Coupled Non-Inverting Amplifier 00778726 Av = 11 (As Shown) DC Coupled Low-Pass RC Active Filter 00778727 fo = 1 kHz Q=1 AV = 2 17 www.national.com LM158/LM258/LM358/LM2904 Typical Single-Supply Applications (V+ = 5.0 VDC) (Continued) Bandpass Active Filter 00778728 fo = 1 kHz Q = 25 High Input Z, DC Differential Amplifier 00778729 www.national.com 18 LM158/LM258/LM358/LM2904 Typical Single-Supply Applications (V+ = 5.0 VDC) (Continued) Photo Voltaic-Cell Amplifier Bridge Current Amplifier 00778730 00778733 High Input Z Adjustable-Gain DC Instrumentation Amplifier 00778731 19 www.national.com LM158/LM258/LM358/LM2904 Typical Single-Supply Applications (V+ = 5.0 VDC) (Continued) Using Symmetrical Amplifiers to Reduce Input Current (General Concept) 00778732 Schematic Diagram (Each Amplifier) 00778703 www.national.com 20 LM158/LM258/LM358/LM2904 Physical Dimensions inches (millimeters) unless otherwise noted Metal Can Package (H) NS Package Number H08C Cerdip Package (J) NS Package Number J08A 21 www.national.com LM158/LM258/LM358/LM2904 Physical Dimensions inches (millimeters) unless otherwise noted (Continued) SOIC Package (M) NS Package Number M08A Molded Dip Package (N) NS Package Number N08E www.national.com 22 LM158/LM258/LM358/LM2904 Physical Dimensions inches (millimeters) unless otherwise noted (Continued) Order Number LM158AWG/883 NS Package Number WG10A 23 www.national.com LM158/LM258/LM358/LM2904 Physical Dimensions inches (millimeters) unless otherwise noted (Continued) NOTES: UNLESS OTHERWISE SPECIFIED 1. EPOXY COATING 2. 63Sn/37Pb EUTECTIC BUMP 3. RECOMMEND NON-SOLDER MASK DEFINED LANDING PAD. 4. PIN A1 IS ESTABLISHED BY LOWER LEFT CORNER WITH RESPECT TO TEXT ORIENTATION REMAINING PINS ARE NUMBERED COUNTERCLOCKWISE. 5. XXX IN DRAWING NUMBER REPRESENTS PACKAGE SIZE VARIATION WHERE X1 IS PACKAGE WIDTH, X2 IS PACKAGE LENGTH AND X3 IS PACKAGE HEIGHT. 6. REFERENCE JEDEC REGISTRATION MO-211, VARIATION BC. 8-Bump micro SMD NS Package Number BPA08AAB X1 = 1.285 X2 = 1.285 X3 = 0.850 www.national.com 24 inches (millimeters) unless otherwise noted (Continued) NOTES: UNLESS OTHERWISE SPECIFIED 1. EPOXY COATING 2. RECOMMEND NON-SOLDER MASK DEFINED LANDING PAD. 3. PIN A1 IS ESTABLISHED BY LOWER LEFT CORNER WITH RESPECT TO TEXT ORIENTATION REMAINING PINS ARE NUMBERED COUNTERCLOCKWISE. 4. XXX IN DRAWING NUMBER REPRESENTS PACKAGE SIZE VARIATION WHERE X1 IS PACKAGE WIDTH, X2 IS PACKAGE LENGTH AND X3 IS PACKAGE HEIGHT. 5. REFERENCE JEDEC REGISTRATION MO-211, VARIATION BC. 8-Bump micro SMD Lead Free NS Package Number TPA08AAA X1 = 1.285 X2 = 1.285 X3 = 0.500 National does not assume any responsibility for use of any circuitry described, no circuit patent licenses are implied and National reserves the right at any time without notice to change said circuitry and specifications. For the most current product information visit us at www.national.com. LIFE SUPPORT POLICY NATIONAL'S PRODUCTS ARE NOT AUTHORIZED FOR USE AS CRITICAL COMPONENTS IN LIFE SUPPORT DEVICES OR SYSTEMS WITHOUT THE EXPRESS WRITTEN APPROVAL OF THE PRESIDENT AND GENERAL COUNSEL OF NATIONAL SEMICONDUCTOR CORPORATION. As used herein: 1. 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