Wheatstone bridge circuit

The Wheatstone bridge is used to measure the resistance of the sensor by means of the deflection method [101], i.e., by measuring the difference of the drop voltage between the two arms of the bridge. The bridge circuit is shown in Figure 2.6.

R₁ R₂

R_ref R(T) v (T)_o V_b

Figure 2.6: Wheatstone bridge circuit scheme. R(T ) is the sensor and Rref determines

the centre of the temperature scale in the absolute measurements. When differential measurements are needed Rref is substituted by another temperature sensor.

The resultant output voltage of the Wheatstone bridge is

vo(T ) = Vb  _R ref Rref+ R1 − R(T ) R(T ) + R2  (2.8) where

ˆ Vbis the voltage supply of the bridge, ˆ R1and R2are fixed resistors,

ˆ Rref is an array of resistors that allows to centre the temperature scales to some specific values,

5_{The potential excess noise in the Wheatstone bridge is included in x(t) since it dependents on the voltage (or}

ˆ R(T ) is the resistance of the temperature sensor.

The Wheatstone bridge circuit allows centring the zero output for different temperature scales by only changing the resistor Rref and, also, permits to perform differential measurements by simply substituting Rref by another temperature sensor. The value of R2 is calculated by substituting Eq. (2.6) into Eq. (2.8) and looking for the value of R2that maximises the sensitivity of the bridge. This leads to

R2(T ) = R(T ) β − 2T

β + 2T . (2.9)

Assuming a sensor of nominal resistance of 10 kΩ and β=3694 K, the value of R2 (≡ R1) is set to 10 kΩ in order to simplify the design6. The value of Rref depends on the temperature scale chosen and it can be selected by means of the multiplexers placed at the output of the bridge. Six scales have been defined with centre temperatures: 12, 15, 20, 22.5, 25 and 27.5‰ —see Table 2.3 and§2.3.6. This solution has been adopted due to the high sensitivity and resolution needed in a temperature span of 20‰ (from 10 ‰ to 30 ‰ —cf. §1.5.2) and to maintain the output of the bridge as close to zero as possible and, thus, minimise the effect of the gain errors and gain temperature coefficient along the measurement chain —see Figure 2.17 and appendix§A.

Another important parameter that must be considered is the dissipated power in the sensor, which is limited to a maximum of 10 µW not to disturb other nearby LTP subsystems such as the GRS and the OMS [83] (an estimation of the effect of the power dissipated in the thermistors in the temperature stability of the GRS is given in§6.2.2) and to minimise the errors due to the self-heating effect in the sensor [71] —see§2.3.2.3 and appendix §B. This limits the bridge voltage excitation, Vb, to

Vb= (4R2Pmax)1/2 Pmax =10 µW

→ Vb= 0.632 V. (2.10)

The sensitivity of the Wheatstone bridge is defined as

sb(T ) = dvo(T )

dT [V K

−1_{] . (2.11)}

By substituting Eqs. (2.3), (2.6) and (2.8) into Eq. (2.11) we obtain an expression for the sensitivity of the bridge for each of the sensor types, i.e.,

sb, NTC(T ) = Vb R2 [R2+ R(T )]2 R(T ) T2 β , (2.12a) sb, RTD(T ) = R2Ro R2+ R(T )  Pmax R(T ) 1/2 αPt. (2.12b)

These expressions are plotted in Figure 2.7 for Vb=0.632 V (P =10 µW). Figure 2.7 is useful to compare both sensors under the same conditions: with the same nominal resistance (Ro=10 kΩ) and the same dissipated power. The value of R2 (=10 kΩ) for the thermistor bridge has been chosen using Eq. (2.9) in order to have a maximum sensitivity around 20‰. For the case of the platinum RTD, the sensitivity of the bridge increases with the value of R2 to an asymptotic value for R2' 100Ro. However, such solution entails an increase of the common mode voltage [102] at the output of the bridge which translates into an error at the output of the amplification stage due to the finite common-mode rejection ratio (CMRR) and its temperature dependence. Anyway, we have used also R2=10 kΩ.

6_{Actually for each T there is an optimum value of R}

2, for instance, for T =283 K is 14 kΩ and for T =303 K is

12 6.2 6.4 6.6 6.8 s b,NTC (T) [mV K −1 ] 10 14 16 18 20 22 24 26 28 300.5 0.55 0.6 0.65 temperature [oC] sb,PRTD (T) [mV K −1 ] PRTD NTC

Figure 2.7: Wheatstone bridge sensitivity for NTC thermistors (solid trace) and for plat- inum RTD sensors (dashed trace). Dissipated power is 10 µW and both sensors have a nominal resistance of 10 kΩ. Note that the scale of the thermistor is one order of mag- nitude higher than the RTD one. The resistor R2 in the Wheatstone bridge is 10 kΩ

which implies a maximum sensitivity at '18‰ when using thermistors —see Eq. (2.9). In the case of the platinum sensors if R2 is set to 100 kΩ instead of 10 kΩ the sensitivity

is '1 mV K−1, still a factor of 6 lower than the thermistor one. The sensitivity shown is, though, with R2=10 kΩ [106].

The NTC thermistor bridge sensitivity is about one order of magnitude larger than the platinum RTD one, as expected since a thermistor has a typical sensitivity of ' −0.04 K−1while the platinum sensor one is 0.00385 K−1 —see §2.2.

The bridge circuit is the first element of the measuring chain. Therefore, the noise introduced by this stage contributes to the overall system performance. In order to estimate the levels of noise introduced by this stage we consider the Johnson noise model for the three resistors (R1, R2 and Rref) and the same for the sensors7. The expression of the noise power spectral density (in units of V2_Hz−1_{) introduced by the bridge is (assuming T = T}

NTC' TFEE8) SV, b(T, ω) = 4kBT  R1Rref R1+ Rref + R2R(T ) R2+ R(T )  (2.13)

where kB = 1.38 · 10−23J K−1 is Boltzmann’s constant. A more convenient way to deal with this expression for our purposes is to express it in terms of noise equivalent temperature (K2Hz−1). This is readily done by dividing Eq. (2.13) by the bridge sensitivity, i.e.,

ST , b(T, ω) =

SV, b(T, ω) [sb(T )]2

. (2.14)

Manipulating Eq. (2.14) we obtain a useful generic equation for the equivalent temperature noise bridge contribution to the system,

S_{T , b}1/2(T, ω) = S 1/2 V, b(T, ω) R2 R2+R(T )α[P R(T )] 1/2 (2.15)

where α (the relative sensitivity of the sensor) is 0.04 K−1 and 0.00385 K−1 for the NTC thermistor and the platinum sensor, respectively. Equation (2.15) leads to important conclusions:

7_{The hypothesis of modelling the sensors as pure Johnson noise sources even at frequencies around the milli-Hertz}

region has been confirmed by experimental results, i.e, no excess noise have been detected in the Wheatstone bridge components —see§5.

ˆ the greater the dissipated power in the sensor, P , the lower the noise, ˆ the greater the relative sensitivity, α, the lower the noise,

ˆ the value of the nominal sensor resistance, Ro, does not reduce or increase the noise,

ˆ in the case of the platinum sensors the increase of R2 translates into a lower noise, although the noise is not further reduced for a relationship ofR2_{/Ro> 100.}

Evaluation of Eq. (2.15) for both sensors is shown in Figure 2.8. The results confirm that the noise levels when using a thermistor are one order of magnitude lower than those of the platinum RTD. Furthermore, the noise levels of the latter are non-compliant with the requirement. The thermistor noise levels are five times lower than the requirement. Consequently, at this point, platinum RTDs were discarded as an option capable of achieving the requirements and the NTC thermistor was the option chosen for the LTP TMS although some concerns had first to be cleared —see§2.2.2. 10 12 14 16 18 20 22 24 26 28 30 1.94 1.96 1.98 2 2.02 2.04 2.06 2.08 temperature [oC] S T,NTC [ K Hz −1/2 ] 21 21.5 22 22.5 23 23.5 24 24.5 S T,PRTD [ K Hz −1/2 ] NTC PRTD µ µ

Figure 2.8: Wheatstone bridge noise equivalent temperature when using a thermistor and a platinum RTD. The latter is already above the requirement —see Eq. (2.1)— and, therefore, it is discarded. This noise figure is valid for all the frequency range since Johnson noise, i.e., white noise, is considered. The dissipated power in the sensor is 10 µW. The noise of the bridge when using a platinum sensor can be slightly reduced by using a higher value of R2. For R2=100 kΩ the noise goes down a factor of '2, still not meeting the

requirements. The noise of the bridge when using thermistors is five times lower than the requirement.

Another important issue to take into account with respect to the Wheatstone bridge is its temperature coefficient (TC), αb. Temperature fluctuations in the resistors forming the bridge show up as an error in the measurement. The coefficient for the whole bridge is (in V K−1) —see appendix§A.3, αb(T ) = VbR  _2R ref (Rref+ R)2 + RNTC(T ) (RNTC+ R)2  αR (2.16)

where R = R1= R2= 10 kΩ and αR is the temperature coefficient of the resistors. The maximum bridge resistors’ TC, αR, permitted in the MBW can be calculated by using the following expression

αb(T ) sb, NTC

S_{T , FEE}1/2 (ω) ≤ S_{T , req}1/2 (ω) (2.17)

where S_{T , FEE}1/2 stands for the temperature fluctuations in the electronics, i.e., in the Wheatstone bridge and S_{T , req}1/2 = 10−5K Hz−1/2. Thus, the maximum permitted resistors’ TC, αR, depends

on the temperature fluctuations of the electronics on board the satellite. In the design we have assumed to be less than 0.1 K Hz−1/2 which appears to be comfortably met in the satellite. This leads to a maximum value of αRof 2 ppm K−1. The resistors of the bridge (R1, R2 and Rref) used in the system have a lower temperature coefficient (Vishay Metal Foil resistors of 0.6 ppm K−1) in order to be on the safe side.

In summary, the Wheatstone bridge designed consists of two high stability resistors (R1 = R2=10 kΩ), six high-stability reference resistors (Rref0 to Rref5) to centre the output of the bridge at different temperatures and a NTC thermistor as the sensing element. Table 2.3 summarises the references used and the associated centre for each of the scales and their temperature span considering a gain of 200 in the amplification stage —see§2.3.2.4.

reference label resistance [kΩ] centre of scale [‰] Tmin—Tmax [‰]

0 17.5 12 8.28—15.87 1 15 15 11.79—19.29 2 12.5 20 15.99—23.41 3 11 22.65 18.97—26.54 4 10 25 21.21—28.89 5 9.1 27.5 23.47—31.29

Table 2.3: References used and their corresponding centres of scale in the Wheatstone bridge circuit. All the resistors have very low temperature coefficient (0.6 ppm K−1). The selection of the references is done by means of multiplexers —see Figure 2.10. Tmin and

Tmaxare calculated assuming the gain of the amplification stage is 200. The temperature

expected in the LCA of the LTP is from 10‰ to 30 ‰.

The bridge exhibits a sensitivity of '6.5 mV K−1 in the temperature range from 10‰ to 30 ‰ with P '10 µW. In terms of noise equivalent temperature the Wheatstone bridge contributes with white noise9of amplitude '2 µK Hz−1/2(20% of the requirement). Finally, we calculate the needed number of bits of the ADC by using the dynamical range (DR) definition [102], i.e.,

DR = 20 logtemperature span

resolution ' 6Nbit (2.18)

where the temperature span is Tmax− Tmin'7 K and the resolution is set to 10−6K. These numbers yield a needed number of bits, Nbit, of '22 —see§2.3.2.5.