About Instrumentation

The principle of resistance temperature detectors (RTD) is based on the variation of electrical resistance of metals with temperature. For this purpose several metals are used, namely, platinum, copper, nickel. When temperature increases the resistance of these metals increases. Temperature function of resistance for metals in a narrow temperature interval can be expressed by a relationship:

(1)

where:

R_ϑ and R₀ - are the values of electrical resistance of a metal conductor at temperatures ϑ and 0,̊C, respectively, Ohm;

α - thermal coefficient of electrical resistance, 1/̊C .

For metals this coefficient is positive. Fig.1 shows relationship between resistance of platinum and copper RTD and temperature.

Platinum RTDs are used for temperature measurements from -220 to 850 ^oC (they are used as reference RTDs, as well), copper RTD - from -50 to 150 ^oC, and Nickel RTD - from -215 to 320 ^oC.

Figure 1: Resistance vs temperature for platinum and copper RTD.

Fig. 2 shows the assembly of RTDs. Sensitive elements of RTDs are made of a thin wire 1 with outside diameter equal to 0.025 mm (platinum RTD) and 0.1 mm (copper RTD) double wounded (non-inductive) on a micaceous or porcelain stem 2. For mechanical strength the sensitive element is placed in the ceramic insulator tube 3 filled by extremely fine granular powder; extension wires are placed in the ceramic insulator 4, and entire assembly is covered by a protective sheath of stainless steel 5. The space between the sheath and ceramic insulator is filled by ceramic packing powder 6. To avoid contact of sensitive element with environment, sensitive assembly is protected by high-temperature hermetic seal 7. The contact between the wire of the sensitive element and the ceramic encapsulation permits a rapid speed of response.

Figure 2: RTD assembly.

For measurements of resistance of RTDs several methods are used. Among them the most widely used is a method employing a Wheatstone bridge (Figure 3). In this case RTD is connected to the bridge by two connecting cables (conductors). The bridge is powered by direct current power supply in the points "a" and "b". RTD is immersed in the media, which temperature to be measured. When this bridge is in balance, then there is no voltage between points "c" and "d", and zero-indicator (ZI) shows no current. For this condition we can write the following equation:

(2)

(3)

where:

R₁ and R₂ - electrical resistance of two invariable resistors, usually, R₁=R₂, Ohm;

R_ϑ - electrical resistance of RTD, Ohm;

R_c - electrical resistance of a connecting cable, Ohm;

R_sr - electrical resistance of a slide (variable) resistor, Ohm.

Figure 3.: Two-conductor connecting. Figure 4: Three-conductor connecting.

As follows from these equations, the certain position of a slide of the variable resistor (ie, certain value of R_sr) corresponds to each value of a measuring temperature (ie, for each value of R_ϑ) at any balance condition of the bridge. Therefore, the scale of the bridge may be calibrated in degrees Celsius.

For a two-conductor connecting scheme (see Fig.3), variation of an ambient temperature will effect values of an electrical resistance of connecting cables R_cand, therefore, the results of measurements will be erroneous.

With all resistance temperature detectors a three-conductor connection scheme is recommended (see Fig. 4). One conductor is common to both sides of the bridge, while other two connect the RTD to each side of the bridge. Any change in the cable temperature (as the result of variations in ambient temperature) will be cancelled because the resistance of both sides of the bridge change by the same value (providing three connecting cables are at the same temperature).

A four-conductor connection scheme is used when very accurate measurements of temperature are required, up to ±0.01 °C of accuracy.

Derivation of a balanced condition for a Wheatstone Bridge

Below I give you the derivation of the balanced condition for a Wheatstone bridge. Figure 4a shows a circuit of five electrical resistors connected to form the Wheatstone bridge.

Figure 4a.: Wheatstone bridge.

Here we use two Kirchoff’s Laws. The first Kirchoff’s law says: the total current flowing into any junction is equal to the total current flowing out of this junction. The second Kirchoff’s law states: the total change in potential around any closed circuit loop is equal to zero.

For the second Kirchoff’s law we should be sure to include the sign of potential energy correctly: the potential decreases around a resistor in the direction of current flow and increases in the direction opposite to current flow; potential increases from negative to positive terminals of the battery.

Below I give you two different methods for the derivation of the balanced condition for a Wheatstone bridge.

Now, let’s consider several closed circuit loops and apply the above law to them.

a. First method

1. Loop e-a-b-c-f-e :

(4)

2. Loop a-b-g-d-a :

(5)

3. Loop b-c-d-g-b :

(6)

Here, U is a dc voltage, and U≠ 0.

So, we have three equations with 3 unknowns - I₁; I₃ and I₅.

(7)

(8)

(9)

Now we combine terms with I₁; I₃ and I₅.

(10)

(11)

(12)

Now we evaluate the determinant of the matrix developed using this set of equations:

(13)

In the case of the Wheatstone bridge the resistance R₅ is substituted by an ampmeter. Ampmeters have a negligibly small resistance, so we can write that R₅= 0. With this condition we can re-write (13) as follows:

(14)

Because all R_i > 0 we have that ∆ > 0 and ∆ ≠ 0.

For current I₅ we have:

(15)

Now we can evaluate the value of the current, which flows through an ammeter as follows:

(16)

We know that a balanced condition is when current I₅ = 0, and because ∆ > 0, and ∆ ≠ 0 and U ≠ 0, we can give an expression for the balanced condition for the Wheatstone bridge as follows: