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How water heater thermostats works
Short answer/ Gas water heater thermostat:
Gas valve control has a thermostat located on the back of valve inside a copper tube that protrudes into the tank.
The thermostat is a thermistor that responds to heat by producing a small electric current. The current increases when water temperature increases, and drops when temperature drops.
Changes in the electric current cause gas control to turn ON and OFF.
When gas control turns ON, it releases gas to burner.
Some gas water heaters are integrated so a thermostat, typically used for electric heater, controls tank temperature.
 Immersion Thermostats are used on some gas, and some electric heaters, and are not coved in this webpage.
Resources:
Add another thermostat to gas or electric heater
Troubleshoot gas water heater thermostats
Basic parts and operation of gas water heater
How to replace thermostat
Gas control valve thermostatMore detail: Gas water heater thermostat is located inside copper tube
Inside the copper tube is a temperature sensor or thermistor that acts as a signal switch. When water temperature inside tank drops below set point, a signal, in the form of small electric current, is intrepreted by the gas control valve to release gas to the burner where it is ignited by the pilot light or hot surface igniter. When temperature of water reaches set point, typically 120-130F, the thermistor signals gas valve to stop sending gas to burner.
Read about basic gas water heater parts and operation

You can change temperature of thermostat using dial or buttons located on front of gas control, or on a LED control board. Typical range for residential gas thermostat goes from lowest or vacation at 50-57, with highest around 160 for residential, and 180F for commercial heaters. Typical bath or shower is 104-110 and rarely higher. How to adjust thermostat temperature

Temperatures above 125F can pose risk of scalding. Above 135F and the risk of scalding is dramatically increased. Thermostat setting above 125F should generally be avoided. Read about scalding.  
Adding a mixing valve can temper very hot water and increase overall capacity of hot water supply:
Read about mixing valve and ways to increase hot water and water heater recovery  and first hour delivery
A new residential gas water heater thermostat is set at 125F per testing requirement of 2015 energy standards. The older 2012 testing standard was 135F.
Read about
Energy Factor and cost to run gas water heater.
Troubleshoot gas water heater
Gas control valve thermostatMore detail: gas water heater
The temperature sensor (copper tube) contains two things: "the ECO (energy cut off), and a temperature probe thermistor.
Thermistors are thermally (heat) sensitive variable resistors. A change in water temperature changes the resistance (measured in ohms) of the thermistor. The gas control valve interprets change in resistance as a change in temperature and uses information to activate and
deactivate call for heat."

 ECO (energy cut off) will trip if water inside residential tank exceeds 180-188F high limit.
 ECO is typically single-use and gas control must be replaced. The new electronic gas control, such as Intellivent control valve used on some power vent models, and the Honeywell electronic gas control valves are resettable after overheat event. Models vary, and reset is not possible for every condition.

 Check tank temperature in two places: Open drain valve to see if lower part of tank has hot water. Open TP valve and see if upper part of tank has hot water. Then compare with water temperature at tap. If water heater contains hot water at top and bottom of tank, but hot water at tap not as hot or never receives hot, then check plumbing problem such as crossover. Read about crossover

Typical bath shower is 105-108F.
Adjust temperature at tank
Crossover: single handle faucet has bad cartridge and other causes
Stacking: Several short water draws cause very hot water
How to replace gas control
How to install gas water heater
Thermostat Calibration/ applies to gas and electric thermostats
Thermostat settings are not exact. Thermostats are calibrated within a range, and not intended to be precise like a scientific instrument.
People frequently complain that thermostat lets water get too cold before turning on, or gets hotter or colder that set point on dial.

Thermostats have 2 ranges for calibration:
First is the actual temperature of water vs the setting you choose. You can set thermostat to 120 and water temperature might be 115 or 125. However, a wide temperature differential between setting and actual water temperature can mean thermostat needs to be replaced.

Second is the differential of water temperature before water heater turns on. For example, thermostat is set to 120 and tank is fully heated. Hot water is drawn for bath. How LOW will the water temperature get before the water heater turns ON? Answer: calibration range for thermostat can be around 10-20%. On gas water heater, 'water temperature around the thermostat probe can drop 15-25F before main burner is activated.' Water at bottom of tank can be 95 and top of tank still be near 120 thermostat set point.
If differential is greater than 20%, the thermostat should be replaced.
Other things affect the actual water heater. If a water heater is short cycling, where it turns on-and-off too frequently, it can mean thermostat calibration is too narrow, however thermostat are not 'fast-acting' and problem can be caused by other issues, such as water leak or stacking,

Deadband/ applies to gas and electric thermostats
Deadband is a range of input that doesn’t result in any output. In this range, the thermostat fails to trigger any activity. Deadband sometimes causes thermostat to overshoot its setpoint. The effect can be occasional or sporadic and indicates thermostat failure and need to replace.

Upper and lower thermostats

Electric Buy residential upper and lower thermostats
Thermostats are interchangeable/ each works with your water heater

Buy 90-150 degree F for residential tank:

Water heater thermostats at Amazon
Camco thermostats at Amazon
More thermostats
89T short/ dual-element thermostat at Amazon
Example Camco dual-element thermostat at Amazon
Short answer/ Electric water heater:
Electric water heater thermostats operate with a mechanical bi-metal switch located inside the thermostat body. The thermostat is installed flat against tank wall, and covered with insulation. When water inside tank is heated, it heats the the steel tank which heats the bi-metal switch.
When heat drops below set point, the bi-metal switch closes (turns on) electric circuit going to element. When heat reaches set point, for example 120F, the switch opens and electricity to element is turned off. Thermostat temperature is set using a screwdriver.

The operation for typical residential non-simultaneous 2-element tanks is more complex since the heater has two thermostats, the upper thermostat and lower thermostat. The thermostats are wired so that only one element is ON at any one time.
The upper thermostat heats top 1/3 of tank to set point, and then switches electricity to lower thermostat that finishes heating whole tank.
There are many different wiring combinations for electric water heater thermostats.
How to wire thermostats
 
There are other types of thermostats for electric water heaters. For example, the Energy Smart series uses a thermistor that is integrated with electronic control board that switch water heater ON-OFF.
Read about Energy Smart
thermostat temperature rangeMore detail: Electric water heater
A bi-metal switch is two different metals sandwiched together. Metal expands when heated, except one metal expands differently than other metal, causing a deformation or bend. Inside electric water heater thermostat, the bending metal opens and closes a mechanical switch.

Typical household non-simultaneous water heater has 2 heating elements called upper and lower elements.
Elements are controlled by upper and lower thermostats. Each element is connected to a thermostat.
Thermostats read temperature through side of tank wall and turn elements ON and OFF.
Residential water heater thermostats can be manually set to temperatures between 90F to 150F. Commercial water heater thermostats  120-180F.
How to adjust thermostats
How to replace thermostat on electric water heater
Higher thermostat settings use more electricity.

Both elements on ordinary residential water heater are never ON at same time unless specific wiring is changed. How to wire thermostats
Upper thermostat is main controller.
Upper thermostat turns on upper element until top 2/3 of tank reaches temperature setting.
After top of tank is heated, upper thermostat turns-off upper element and sends power to lower thermostat which turns on lower element. Lower element runs until tank reaches temperature setting.
Lower element turns on-and-off during standby hours to keep tank temperature at thermostat set point.
Thermostat operation
See Larger

Some thermostats, the L2 and T1 screws are empty slot or rivet since wires do not connect to either L2 or T1.

Brands of thermostat vary slightly, screws are located differently, some numbers are slightly different, but basic concept is same from brand to brand. Water heater manuals available on-line. Label on side of tank shows watts and volts of appliance.

Each thermostat comes with instruction sheet.
When replacing thermostat, take digital photo of water heater and old thermostat to hardware store or plumbing store for quick identification.

Back of thermostat
Thermostat must sit flat against tank wall so temperature reading is accurate and water heater is protected from overheating
More detail: Electric water heater thermostat operation sequence
Residential water heaters can be wired 4 basic ways. Non-simultaneous found in most residential water heaters.
Plus simultaneous, redundant, and off-peak. The key is knowing how a thermostat works.
How to wire thermostats

Diagram on left illustrates basic non-simultaneous thermostat function. This thermostat is found on 240 Volt water heaters that have 2 elements. Check label on side of tank for voltage info. Figure volts, amps, watts

Power from breaker box connects to L1 and L3 screws > some thermostats, the screws are marked simply 1 and 3.

Top half of thermostat is the ECO or energy cut off > The ECO is heat sensitive, so if temperature reaches dangerously high 150-180F, then ECO trips and water heater no longer has power. This prevents water heater explosion.

Viewing the diagram, power from L1 and L3 pass through ECO red reset and go to L2 and L4 screws.

After power passes through ECO, it is available for heating water. To control heating, power arrives at the thermostat which begins at T1 screw.

Inside the thermostat a bi-metal switch. Two different metals are fused together. Each metal expands differently when heat is applied. Since the two metals are joined, the metal that expands most causes the bi-metal switch to bend. The bending action opens and closes the switch.

In the diagram, the bi-metal switch is connected to T1. When temperature inside tank is cool, the bi-metal switch rests against T2 so power travels from T1 to T2, and then on to the upper heating element. When temperature inside tank is hot, the bi-metal switch disengages from T2 and clicks over to T4. When this happens, T1 sends power to T4, so the upper heating element is turned off and power travels to lower element and thermostat through the T4 screw.

In a 240 Volt water heater, elements must receive power from 2 hot leads. The diagram shows the hot leads at L1 and L3 screws. In the description above, power from one hot lead was switched between T2 and T4. From there, power traveled to the elements but the circuit was not yet complete. To complete the 240 circuit, each element must connect back to the L4 screw.  See complete circuit diagram below.
 More detail Electric water heater thermostat specifications                                                Larger image
Upper thermostat is made of two parts >> the top part is the ECO, the bottom part is the thermostat.
Residential Therm-o-disc thermostats: Minimum temperature 90, Maximum 150. High Limit (ECO or reset button) trips at 170. All values +/- 5%
Aprom thermstats: 110F minimum, 160F maximum. High limit trips at 180F.
Camco commercial Therm-o-disc thermostats: Minimum temperature 120, Maximum 180. High Limit 200.
Rheem commercial Therm-o-disc thermostats: Minimum temperature 110, Maximum 170. High Limit 190.
Resources:
How to wire thermostats
Compare Rheem thermostats
Reset buttonECO (energy cut off) or Reset button on electric water heater is located on upper thermostat
Upper thermostat is made of two parts >> the top part is the ECO, the bottom part is the thermostat.
 Remove upper cover and insulation.
Firmly push in thermal reset button. If reset is not tripped out, then reset is not problem. If reset will not stay pushed in, then replace thermostat
Use nose and eyes to inspect for signs of high heat and burning. Replace burned thermostat or element.
Inspect for water seep, water leak, and signs of rusted parts. Tank may be leaking, or element is leaking. Leaking water heater
Must Re-install insulation and cover for thermostat to function correctly

Why does reset button trip? Bad themostat usually, replace thermostat, then if problem continues then test or replace both elements.
How-to replace thermostats
How-to test elements
Other causes: Stacking, missing insulation and cover over thermostat, loose wire on upper thermostat, burned thermostat.
Stacking
How to make lower thermostat using upper thermostat
If water heater has foam insulation, then replacing lower thermostat with ECO protected thermostat may not be easy.
Take apart upper thermostat and make new lower thermostat that will work for the specific two-water heater diagrams shown above.

When not to use:
Do not remove ECO protection on upper part of tank.
On simultaneous and single-tank off-peak water heaters, the ECO is require for lower thermostat operation.
ECO protects you in case of overheating and is required for safe operation of water heaters.
Typical residential non-simultaneous electric water heater
Water heater cycle
How to wire thermostats
Non-simultaneous off-peak electric water heater
Off peak water heater sequence
How to wire off-peak water heater
Very hot water poses risk of scalding.
Temperatures above 135 are extremely dangerous and will scald the skin in seconds.
Average bath-shower temperatures are 104-110 and rarely much higher.
120 F thermostat setting is considered best for optimal savings, 140F is for killing bacteria .

Hotter water heater temperatures increase pressures inside the tank, possibly shortening life-span of tank, causing rupture or drizzling TP valve.
Prevent rusted tank by replacing anode rod: 
Resource: How to replace anode rod

A tank that is connected to pipes in a closed system where expansion of water is confined by a check valve or pressure-reducing valve, can experience dangerous pressure increases when thermostat temperatures are set very high. 
Resource: Prevent damage from closed system
Additional links

Troubleshoot gas water heater
Numbering on upper thermostats varies slightly
How to replace thermostats
Upper thermostats
See larger

How to install gas water heater

How to install gas water heater

9 ways to save with water heater
How to test elements

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See basic 120V and 240V water heater circuits

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Stacking
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