RTD and Thermocouple Circuits

All answers based on ITS-90 thermocouple table values:

{\bullet} $T_1 = 589~^{\circ}F$ ; $T_2 = 63~^{\circ}F$ ; $T_3 = 70~^{\circ}F$ ; $V_{meter} = 0.463~mV$

{\bullet} $T_1 = 821~^{\circ}F$ ; $T_2 = 69~^{\circ}F$ ; $T_3 = 73~^{\circ}F$ ; $V_{meter} = 0.953~mV$

{\bullet} $T_1 = 1524~^{\circ}F$ ; $T_2 = 91~^{\circ}F$ ; $T_3 = 105~^{\circ}F$ ; $V_{meter} = 3.378~mV$

{\bullet} $T_1 = 1922~^{\circ}F$ ; $T_2 = 102~^{\circ}F$ ; $T_3 = 135~^{\circ}F$ ; $V_{meter} = 5.294~mV$

Note: all temperatures at $T_2$ are irrelevant because this is a junction between similar metals (type B wires connecting to corresponding type B wires). The reference (cold) junction is where the type B wires connect to copper, at temperature $T_3$.

The extension wire has been connected “backwards” to the thermocouple. Yellow should connect to yellow, and red to red!

Seeing how the four junctions’ voltages interact is best done by drawing the circuit with each junction explicitly labeled as to polarity. You will see that the process and reference (transmitter) junctions are aiding each other, and the two junctions formed at the head are opposing the first two (but aiding each other). The formula for calculating voltage at the transmitter screw terminals then becomes:

$$V_{terminals} = V_{592^oC} + V_{23^oC} - V_{31^oC} - V_{31^oC}$$

$$V_{terminals} = 24.57~mV + 0.92~mV - 1.24~mV - 1.24~mV$$

$$V_{terminals} = 23.01~mV$$

If the transmitter has CJC enabled, it will add 0.92 mV to compensate for the reference junction, making the interpreted (measurement junction) voltage equal to 23.93 mV. This equates to 577 °C, which is substantially cooler than the real process temperature.

These are type ‘T’ thermocouples, sending 1.174 mV to the transmitter’s input. Cold junction compensation (CJC) should not be enabled in the transmitter, because we want to measure the difference in temperature between the collector and the house, not compensate for one of those thermocouples’ voltages!

The diamond symbol is a logic device—most likely a PLC—used for on/off control of the fan.

Voltage at ADC: $35.948~mV$

According to an ITS-90 table for type N thermocouples, the measurement junction will generate 36.577 mV at 1847 °F, while the reference junction (at the NX-copper cable junction) will generate 0.629 mV at 75 °F. Since we know these two junctions’ voltages are opposed to each other, the voltage seen at the transmitter terminals will be 35.984 mV.

{\bullet} $T_{simulate} = 550~^{\circ}F$ ; $V_{input} = 14.516~mV$ ; $R_{pot} = 741.39~\Omega$

{\bullet} $T_{simulate} = 300~^{\circ}F$ ; $V_{input} = 6.815~mV$ ; $R_{pot} = 347.80~\Omega$

{\bullet} $T_{simulate} = 150~^{\circ}F$ ; $V_{input} = 2.278~mV$ ; $R_{pot} = 116.20~\Omega$

An equivalent circuit diagram shows the relationships between the thermocouple measurement junction (in the process), the two reference junctions formed where thermocouple wire meets copper wire, the digital multimeter, and the temperature transmitter (when the DMM is connected at the head):

$T_{process} = 388~^{\circ}F$ at 7.825 mV (measured at 92 °F head)

Another equivalent circuit diagram shows the relationships between the thermocouple measurement junction (in the process), the two reference junctions formed where thermocouple wire meets copper wire, the digital multimeter, and the temperature transmitter (when the DMM is connected at the transmitter):

$T_{process} = 385~^{\circ}F$ at 8.332 mV (measured at 66 °F transmitter terminals)

All answers based on ITS-90 thermocouple table values:

{\bullet} 5.05 mV ; $T = 112~^{\circ}C$

{\bullet} 17.82 mV ; $T = 346~^{\circ}C$

{\bullet} 31.44 mV ; $T = 586~^{\circ}C$

{\bullet} 40.29 mV ; $T = 73~^{\circ}2C$

All answers based on ITS-90 thermocouple table values:

{\bullet} 2.55 mV ; $T = 195~^{\circ}F$

{\bullet} 6.21 mV ; $T = 357~^{\circ}F$

{\bullet} 10.93 mV ; $T = 567~^{\circ}F$

{\bullet} 18.83 mV ; $T = 904~^{\circ}F$

Each voltmeter forms its own reference junction where the copper wires of each voltmeter connect to the chromel and alumel thermocouple wires. Each voltmeter’s reference junction produces its own voltage, opposed in polarity to the voltage of the measurement junction (at 1050 °F). This means we must perform a separate voltage calculation for each voltmeter based on each meter’s reference junction temperature. An equivalent schematic diagram shows how the four dissimilar metal junctions relate to one another, and to the three voltmeters:

The chromel and alumel wire types identify this as a type K thermocouple, and so we may use a type K thermocouple table to look up voltages for all four junction temperatures:

{\bullet} Type K at 1050 °F = 23.439 mV

{\bullet} Type K at 71 °F = 0.865 mV

{\bullet} Type K at 85 °F = 1.181 mV

{\bullet} Type K at 60 °F = 0.619 mV

Since we know the measurement and reference junctions stand opposed to one another in each voltmeter loop, we may calculate each voltmeter’s reading by subtracting the reference junction’s voltage from the measurement junction’s voltage for each voltmeter:

Voltage registered by the left-hand voltmeter (at 71 °F):

$$23.439~mV - 0.865~mV = 22.574~mV$$

Voltage registered by the center voltmeter (at 85 °F):

$$23.439~mV - 1.181~mV = 22.258~mV$$

Voltage registered by the right-hand voltmeter (at 60 °F):

$$23.439~mV - 0.619~mV = 22.820~mV$$

The voltage read by each voltmeter is a simple function of Kirchhoff’s Voltage Law, with the measurement junction and the voltmeter’s reference junction being the only voltage sources in the KVL loop. Therefore, connection of a fourth voltmeter will have no effect whatsoever on the other three voltmeters. Likewise, connection of a thermocouple transmitter to this same chromel/alumel wire pair will have no effect on the voltmeters’ readings, with or without reference junction compensation.

All answers based on ITS-90 thermocouple table values:

{\bullet} $T_{1} = 233~^{\circ}C$ ; $T_{2} = 31~^{\circ}C$ ; $T_3 = 25~^{\circ}C$ ; $V_{meter} = 14.395~mV$

{\bullet} $T_{1} = 348~^{\circ}C$ ; $T_{2} = 40~^{\circ}C$ ; $T_3 = 16~^{\circ}C$ ; $V_{meter} = 23.856~mV$

{\bullet} $T_{1} = -161~^{\circ}C$ ; $T_{2} =-4~^{\circ}C$ ; $T_3 = 23~^{\circ}C$ ; $V_{meter} = $-$9.039~mV$

{\bullet} $T_{1} = 836~^{\circ}C$ ; $T_{2} = 34~^{\circ}C$ ; $T_3 = 19~^{\circ}C$ ; $V_{meter} = 62.701~mV$

Note: all temperatures at $T_2$ are irrelevant because this is a junction between similar metals (type E wires connecting to corresponding type EX extension wires). The reference (cold) junction is where the type EX wires connect to copper, at temperature $T_3$.

All answers shown here based on tabulated values for the RTD’s resistance (rather than values calculated by formula):

{\bullet} $T = 0~^{\circ}C$ ; $V_{AB} = -0.6989~mV$

{\bullet} $T = 81~^{\circ}C$ ; $V_{AB} = 0.9568~mV$

{\bullet} $T = -45~^{\circ}C$ ; $V_{AB} = -1.6215~mV$

{\bullet} $T = 320~^{\circ}F$ ; $V_{AB} = 2.4831~mV$

The voltmeter should read $-3.908$ mV. Counting all the junction voltages (with polarities shown in reference to whether they match or oppose the meter’s test lead polarity):

{\bullet} 193 °F junction = $-3.789~mV$

{\bullet} 71 °F junction = $+0.857~mV$

{\bullet} 102 °F junction = $-1.565~mV$

{\bullet} 59 °F junction = $+0.589~mV$

{\bullet} Loop total voltage = $-3.908~mV$

Hint: any junction pushing conventional flow in a counter-clockwise direction is regarded here as a “positive” figure. Any junction pushing in a clockwise direction is regarded as a “negative” figure.

The last junction (at 59 °F) may be treated as a single copper-constantan junction because the metal type of the meter’s test leads acts as an intermediate metal. The fact that both the copper and constantan test lead junctions are at the same temperature allows us to disregard the test lead metal altogether and treat it as a single copper-constantan junction at 59 °F.

The Law of Intermediate Metals tells us that a series chain of dissimilar metal junctions at the same temperature are electrically equivalent to a single junction comprised of the outer two metals (ignoring all the intermediate metal types between) at that temperature. Thus, the iron-copper-constantan reference junction pair is electrically equivalent to a single iron-constantan reference junction, so long as both screw terminals are at the same temperature (isothermal). This principle renders the meter’s metal type irrelevant.

The voltmeter should read -0.643 mV, because the 105 °F junction has a potential of 1.635 mV, the 77 °F junction has a potential of 0.992 mV, and the two junctions’ voltages are opposing one another in polarity with the positive terminal of the voltmeter connected to the negative-most copper wire.

$V_{RTD} = 140.1~mV$

$R_{RTD} = 1167.5~\Omega$

$T = 43.5~^{\circ}C = 110.31~^{\circ}F$ (both values calculated from formula, not table)

$R_{RTD} = 1.617~k\Omega$

$T_{oven} = 160.4~^{\circ}C = 320.7~^{\circ}F$ (calculated from formula, not from a table)

The voltage measurements must be taken very soon after the switch is thrown to avoid measurement errors due to “self-heating” of the RTD.

$V_{RTD} = 88.2~mV$

$V_{wire} = 0.5~mV$ (each current-carrying conductor)

$R_{RTD} = 215.12~\Omega$

$T = 309~^{\circ}C$ (from table)

$T = 299.02~^{\circ}C$ (from formula)

$R_{RTD} = 107.256~\Omega$

$T = 65~^{\circ}F$ (from table)

$T = 65.32~^{\circ}F$ (from formula)

The $\alpha$ value is most likely 0.00385, resulting in a calculated temperature of 218.96 °C or 426.13 °F.