# Design Issues for Low Power Integrated Thermal Flow Sensors with Ultra-Wide Dynamic Range and Low Insertion Loss

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Design Parameters and Optimization

#### 2.1. Description of the Sensor Structure and Main Performance Parameters

_{T}thermocouples with the hot and cold junctions placed on the cantilever tip and the substrate, respectively. A variable number of heaters are placed in the space between the thermopiles. The heaters are deposited over thermally insulating membranes (rectangular), anchored to the cavity edges through a series of arms. The particular shape of the suspending arms is dictated by the properties of the wet anisotropic etch used to open the cavity. The considerations made in this paper can be easily extended to different sensor layouts. The basic operating principle of the sensor is simple: as the flow progressively increases, heat transfer in the downwind direction increases as well, producing a temperature difference that is sensed by the thermopiles. The typical configuration includes a single heater whereas multi-heater structures can be used to perform particular functions [17,22,23,24] or simply extend the effective heater area with no adverse consequences on the etching time required to release the suspending membranes.

**Figure 1.**(

**a**) Schematic view of a thermal flow sensor, based on a calorimeter type configuration with the main dimensions indicated; (

**b**) Representation of the flow channel.

_{S}using the heating power P

_{H}(total power delivered to the heaters).

**Figure 2.**(

**a**) Signal flow path representation of the transduction mechanism, with the main non-idealities indicated; (

**b**) Effect of the offset and noise on the sensor detection limit. Q

_{meas}is the flow sensor reading under still fluid conditions.

_{tp}, convert the temperature difference into the output voltage. Note that ΔT

_{S}is considered free of random phenomena such as offset and noise, which are taken into account with the terms V

_{nS}and V

_{OS}(sensor noise and offset voltage), added to the output voltage. The sensor noise is mainly thermal noise from the thermopile resistance, while the sensor offset is dominated by geometrical asymmetries of the sensor structure, producing a temperature difference even at zero flow rate. Finally, a noise and offset contribution comes also from the amplifier, as represented in the figure.

- (a)Sensitivity, defined as S = dV
_{out}/dQ. The sensitivity depends generally on the input flow rate, although differential calorimeters exhibit a nearly linear behavior at small flow rates, where a constant sensitivity S(0) can be used. The sensitivity alone is not a real figure of merit, but it is a fundamental parameter for the design of the interface and, together with other parameters, contributes to the resolution and detection limit, defined below. - (b)Resolution, defined as the minimum variation of the input flow that can be detected. The resolution is expressed in terms of the equivalent noise flow rate, Q
_{np−p}= v_{np−p}/S, where v_{np−p}is the peak to peak amplitude of the total output noise voltage. - (c)Offset flow rate, defined as Q
_{O}= v_{O}/S(0), where v_{O}is the total offset voltage. The offset flow rate coincides with the reading of the flow sensor when the fluid velocity is zero. - (d)Detection limit Q
_{min}, defined as the minimum (unsigned) flow value that can be reliably detected by the flow sensor. This parameter is particularly important for leakage sensors, which should be able to determine if very small flow rates are present. Figure 2(b) shows a sketched time diagram of the flow sensor readings in conditions of zero flow. If calibration of the offset is not performed, Q_{0}is an unknown quantity and we only have statistical information about it, i.e., we can presume that it falls between –Q_{0max}and Q_{0max}, where Q_{0max}is the maximum offset for that kind of sensor. It is apparent that, in this case, if a reading is within –Q_{0max}− Q_{np−p}/2 and Q_{0max}+ Q_{np−p}/2, it is not possible to determine whether a flow rate is present or Q = 0 and the measurement is simply the result of a possible combination of offset and noise. Therefore, in the case of no offset calibration, the detection limit is Q_{min}= Q_{Omax}+ Q_{np−p}/2. Even if an offset calibration procedure is applied, a residual offset will still be present. The detection limit coincides with the theoretical limit Q_{np−p}/2 only if the residual offset is negligible with respect to noise. - (e)Bandwidth, B, defined as the frequency range of the flow velocity in which the sensor sensitivity remains over nearly 70% of the value for constant flow. The sensor response time is inversely proportional to the bandwidth. Note that the intrinsic bandwidth, due to the sensor thermal masses and resistances, can be further reduced in the interface to limit the noise and improve resolution.
- (f)Full scale flow rate, Q
_{max}. The response of thermal flow sensors tends to saturate at high flows, for the effect of several concomitant causes. As a result, the sensitivity progressively drops, degrading the resolution. The maximum value Q_{max}can be conventionally set at the point where the sensitivity drops below a given fraction of the initial S(0) value. Response saturation is mainly due to fluid-dynamics and forced convection mechanisms that can be studied only by means of simulations or experiments. In this work we will provide some indications based on tests performed on different sensor configurations. - (g)Dynamic range, DR=Q
_{max}/Q_{np−p}. The DR coincides with the number of distinct flow levels that can be distinguished by the flow sensor. This parameter is particularly important for flow sensors used to measure large flow rates with high resolution. High dynamic range flow sensors are required in flow control units designed to deliver precise fluid flows to reaction chambers (as in semiconductor processing equipments [9]) or ionic thrusters for fine handling of satellite attitude and orbital parameters [11]. The DR of traditional macroscopic flow sensors is typically lower than 10^{2}while it may exceed 10^{3}in MEMS flow sensors. - (h)Insertion loss, p
_{loss}, defined as the pressure drop across the sensor flow channel, measured at Q_{max}. - (i)Power consumption P
_{H}. This is an extremely important parameter for battery powered applications.

#### 2.2. Lumped Element Model of the Transduction Principle

_{A}(ambient temperature). The temperature difference of the sensor elements with respect to T

_{A}will be indicated as “overheating”. The model is based on the hypothesis that the relationship between overheating and the heater powers is linear (linearity hypothesis). This approximation derives from considering that: (i) heat transfer is dominated by conduction and forced convection; and that (ii) the fluid parameters are independent of temperature.

_{Hi}while all the other heaters are not powered. The overheating of the heater is given by:

_{Hi}, indicated with P

_{k,i}that reaches hot junctions of the k-th thermopile can be expressed as:

_{k,i}is a flow dependent coupling coefficient. With Equation (2) we have considered that the linear power density across the device width (W) is constant (2-dimensional approximation). Several geometrical parameters of the sensing structure affect coefficients g

_{i,k}(Q), including the depth of the insulating cavity underneath the sensor elements. The temperature overheating of the thermopile, ΔT

_{k,i}is given by:

_{c}is the cantilever thickness and k

_{eq}the effective conductivity that takes into account the cantilever and the thermopile material [20].

_{k,i}, defined as:

_{Hi}and voltage V

_{k,i}:

_{H}heaters. Therefore:

_{2}-V

_{1}, resulting in:

_{i}are defined as follows:

_{Hi}of the individual heaters to the total power P

_{H}, through a set of dimensionless parameters x

_{i}, indicated as “power distribution”:

#### 2.3. Sensor Resolution

_{np−p}. The peak-to-peak noise can be derived from the rms value, considering that, for Gaussian noise, it can be assumed that v

_{np−p}= 4v

_{nrms}. With this choice, the noise voltage exceeds the conventional peak-to-peak value only for 4.6% of the observation time. The sensor noise is essentially thermal noise from the thermopile resistance (R). This contribution is clearly independent of the amplifier noise. With these considerations:

_{nA}is the amplifier rms input noise voltage, k

_{B}the Boltzmann constant, T the absolute temperature, B the sensor bandwidth, considered coincident with the equivalent noise bandwidth. Note that the thermopile overheating is generally limited to a few Kelvin in this type of sensors, so that we can assume that T coincides with T

_{A}. In order to find the degrees of freedom on the sensor design that actually affect the resolution, it is necessary to obtain an expression for the thermopile resistance.

_{P}= 2(W

_{S}+ W

_{T}), where W

_{S}is the minimum spacing between two lines, while W

_{T}is the dimension of a single conductor forming the thermocouple. For the sake of simplicity, we have considered that the two materials A and B have identical sheet resistance R

_{S}, otherwise the mean sheet resistance can be used.

_{P}= W/N

_{T}. The line width W

_{T}is then given by:

_{S}is the thermopile layer sheet resistance, equal to the resistivity to thickness ratio. The total thermopile sensitivity (k

_{tp}) is simply given by N

_{T}·s

_{AB}, where s

_{AB}is the Seebeck coefficient of the A-B couple. The spacing W

_{S}should be kept at the minimum value allowed by the technology, W

_{SMIN}, in order to reduce the resistance, so that we assume W

_{S}= W

_{SMIN}. The W

_{SMIN}/W ratio can be expressed in term of the maximum number of thermocouples, N

_{TMAX}, given by:

_{TMIN}is the minimum width of conductors A and B allowed by the technology. Generally, in microelectronic processes, W

_{TMIN}≥ W

_{SMIN}, so that m = (1 + W

_{TMIN}/W

_{SMIN}) ≥ 2. Using Equation (16) and the above consideration, the following expression can be derived:

_{NT}= (1-N

_{T}/mN

_{TMAX})

^{−1}is a factor that monotonically increases with N

_{T}. Now, let us make a few considerations about the amplifier noise. The fact that the output signal of a flow sensor includes a DC component dictates the use of dynamic techniques, such as chopper stabilization or autozeroing to cancel the amplifier offset and offset drift [26]. These techniques also strongly reduce the low frequency (Flicker) components, so that the amplifier noise consists only of thermal noise. For a given amplifier topology it is possible to express the rms thermal noise as in [27]:

_{T}is the thermal voltage (nearly 26 mV at 300 K), I

_{A}the total current absorbed by the amplifier and NEF (noise efficiency factor) is a figure of merit associated to the amplifier architecture.

#### 2.4. Offset Flow Rate

_{OS}should be nominally zero. Unavoidable random asymmetries due to the fabrication process add errors to the a

_{k,i}(0) terms in Equation (21). Introducing the relative errors e

_{k,i}associated to the quantities a

_{k,i}, it is possible to write the offset in the case of uniform power distribution (x

_{i}= 1/N

_{H}) as:

_{k,i}(0) are larger. In turn, considering Equation (7), the larger the a

_{k,i}(0) values, the larger the voltages V

_{1}and V

_{2}, individually produced by the two thermopiles at zero flow (static thermopile voltages). On the other hand, the denominator includes the derivative of the a

_{i,k}parameters with respect to flow. In deriving Equation (7), one finds that the larger the a

_{i,k}derivatives, the larger V

_{1}and V

_{2}derivatives with respect to flow. Then, as a general rule, in order to achieve a small offset, the thermopile voltage should exhibit a small static value and a large derivative with respect to flow rate.

#### 2.5. Insertion Loss vs. Sensitivity

_{H}the hydraulic diameter, defined as:

_{CH}and H are the channel width and height, respectively, as shown in Figure 1(b). Combining Equations (24) and (25) we get:

_{CH}or H, or both) for a given flow rate leads to an increase in pressure drop. Note that reducing the cross section is the most commonly used method to increase the sensitivity of flow sensors. This occurs through the sum of the a

_{i,k}derivatives with respect to flow, included in the parameter h

_{eff}. In turn, these derivatives are proportional to the flow dependent components of g

_{i,k}terms, i.e., to the forced convection contribution to heat transfer from the heaters to the thermopiles. This contribution is proportional to the velocity gradient perpendicular to the sensor element surfaces [2]. Therefore:

_{i,k}component due to forced convection. Note that the rightmost proportionality relationship in Equation (27) is valid if the shape of the velocity distribution does not change with flow (e.g., remains parabolic for a laminar regime). Considering Equations (5), (10) and (12) we finally get:

#### 2.6. Design Considerations

_{i}have been measured on a prototype and that the following parameters are fixed: (i) bandwidth, dictated by the application; (ii) s

_{AB}, R

_{S}, t

_{C}, W

_{S}, W

_{TMIN}, deriving from the fabrication technology; (iii) N

_{H}, dictated by the chosen sensor architecture. As a design constraint we will consider a target resolution indicated with Q*. Note that, if we neglect the amplifier noise term in Equation (18), we get the intrinsic resolution limit of the sensor. At this point the designer needs to impose that:

- (1)The sensor width W should be set at the maximum allowed value, imposed by the fabrication process (etching times) and by the channel width W
_{CH}. Increasing W_{CH}to make room for a further W increase is ineffective, since h_{eff}proportionally decreases through Equation (28). - (2)The power P
_{H}should be set to the maximum value, which can be due either to a power consumption constraint or a reliability issue, the latter deriving from the maximum allowed heater temperature. - (3)The effect of the thermopile number is rather weak, since it acts only on f
_{NT}, which varies in a narrow interval. From Equation (17) the optimum situation seems to be represented by N_{T}= 1. Nevertheless, we will discover later that such a choice can adversely affect the amplifier design and should generally be avoided. - (4)If the above operations are not sufficient to obtain the target resolution, parameter h
_{eff}should be improved. The relevant equations are (5), (10) and (12). Improvement of the heater insulation ( ) is effective only if the power constraint does not derive from reliability issues, since, in this case, an increase of with the maximum allowed power applied would simply result in exceeding the maximum heater temperature. Improving the thermopile insulation by increasing L_{C}produces significant h_{eff}improvements when L_{C}< L_{S}, so that L_{T}is not proportionally affected. Furthermore, L_{C}is often limited by the allowed etching times and mechanical robustness. Thermopile insulation can also be improved by reducing the cantilever thickness t_{c}, as clearly shown by Equation (4). Using the post-processing approach, this parameter is determined by the dielectric thickness deriving from the original process. Thickness reduction can be achieved in the post-processing phase by selective etching steps. The minimum thickness value is fixed by structural issues and by the necessity to maintain a sufficient margin with respect to the thermopile conductors, in order to prevent them from being damaged by the etching process. The last parameter to be taken into account to improve the sensitivity is ∂g_{i,k}/∂Q. Reducing the channel height (H) is strongly effective as shown by Equation (28). However, the concomitant pressure drop increase, deriving from Equation (26), should be carefully taken into account. - (5)If the previous steps are successful, then it is necessary to design an amplifier with an input noise just low enough not to significantly change the achieved resolution. To obtain that, the following condition should be met:

_{A}V

_{DD}, where V

_{DD}is the amplifier supply voltage, adds to the heater power P

_{H}, increasing the overall power consumption of the system. As anticipated earlier, it is generally convenient to renounce to the marginal advantage of the choice N

_{T}= 1 and adopt an N

_{T}number as close as possible to the maximum value N

_{TMAX}. In this way the noise constraint on the amplifier is relaxed, with proportional advantages in terms of power consumption.

## 3. Experimental Results and Discussion

#### 3.1. Test Chip Architecture and Technology

_{out}= V

_{2}− V

_{1}) or the individual thermopile voltages V

_{1}and V

_{2}. The electronic interface includes also a driver for the sensor heaters. More details on the interface can be found in [21]. The sensing structures share a series of common parameters, shown in Table 1, regarding either the structure and channel dimensions. The differences between the devices are described in next sub-section.

**Figure 5.**(

**a**) Optical micrograph of the test chip, including the read-out electronics and the flow sensors before the post-processing; (

**b**) Magnification of the sensing structures after post-processing, showing the three different configurations, namely the single heater, the double heater and the triple heater.

**Table 1.**Parameter values common to all sensing structures. The Seebeck coefficient was measured with a test structure on the chip.

N_{T} | s_{AB} | L_{C} | L_{S} | L_{H} | L_{GH} | W | L | W_{CH} |
---|---|---|---|---|---|---|---|---|

10 | 315 µV/K | 35 µm | 90 µm | 46 µm | 71 µm | 121 µm | 2 mm | 0.5 mm |

_{T}= 10 n + poly/p + poly thermocouples with the hot contacts at the tip of a cantilever beam. Each thermopile has an electrical resistance of 50 kΩ. The thermal insulation of the heaters and the thermopiles from the substrate has been obtained applying a post-processing technique based on front-side bulk micromachining, as schematically shown in Figure 6.

**Figure 6.**Schematic view of the post-processing: (

**a**) passivation openings performed by the silicon foundry; (

**b**) dielectric openings by reactive ion etching (RIE); (

**c**) silicon anisotropic etching in TMAH solution.

#### 3.3. Measurements

_{1−3}is shown in Figure 8. Clearly, only the case where the central heater (H

_{2}) is activated produces an ideally symmetrical power distribution, resulting in a relatively low offset (30 µV), due only to random asymmetries, while in the other two cases a large systematic offset is present.

**Figure 8.**Response to nitrogen flow rate of a triple heater sensor with only one heater powered at a time. The powered heater is indicated in the plots.

**Figure 9.**(

**a**) Measured (symbols) and calculated (line) response of the triple heater sensor to a nitrogen flow with all the heaters powered with 1.25 mW; (

**b**) Overheating of the 2nd thermopile with respect to H

_{3}heater.

_{3}(downstream heater). The measurements have been performed with zero flow and 20 sccm and the results are shown in Figure 9(b). In both cases an excellent linearity can be observed.

_{np−p}= 1.6 µV) and the sensitivity at zero flow rate, S(0), also reported in Table 2. Numerical derivation of the sensor responses was used to calculate the sensitivity. The full scale flow rate, Q

_{max}, was set at the point the sensitivity dropped below 50% of S(0). The results, all referring to nitrogen flows, are summarized in Table 2, where the parameters that distinguish the various samples are also reported. Note that in the single heater samples (first three lines) metal interconnections (Aluminum) are used to access the heaters through the suspending arms (heater connection column). The other samples in the table use silicided polysilicon (Si-Poly), providing a low enough sheet resistance with much less thermal conductivity than Aluminum. As a result, the heater thermal resistance is nearly 50% higher for the samples using silicided interconnections, with a proportional benefit in terms of h

_{eff}, and, consequently, low power performances.

**Table 2.**Main parameters and performances of the sensors used in this work. The sensor resolution (Q

_{np−p}) has been estimated considering a 100 Hz bandwidth. All figures refer to nitrogen flows.

Sensor parameters | Sensor performance (P_{Hi} = P_{H}/N_{H} = 1.25 mW) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

N_{H} | Heater | H | L_{G} | S(0) | k_{tp}h_{eff} at Q = 0 | V_{K}(0) | Q_{np−p} | Q_{max} | DR | |

connection | mm | µm | µV/sccm | K/mW | µV/sccm/mW | mV | sccm | sccm | ||

1 | Metal | 0.5 | 40 | 87 | 30.8 | 69.6 | 3.28 | 1.8 × 10^{−2} | 30 | 1.7 × 10^{3} |

1 | Metal | 0.5 | 60 | 100 | 30.8 | 80 | 1.88 | 1.6 × 10^{−2} | 21.5 | 1.3 × 10^{3} |

1 | Metal | 0.5 | 80 | 96 | 30.8 | 76.8 | 1.14 | 1.7 × 10^{−2} | 20 | 1.2 × 10^{3} |

2 | Si-poly | 0.25 | 60 | 940 | 46.7 | 376 | 3.87 | 1.7 × 10^{−3} | 3.6 | 2.1 × 10^{3} |

2 | Si-poly | 0.5 | 60 | 313 | 46.7 | 125 | 3.61 | 5.1 × 10^{−3} | 12 | 2.3 × 10^{3} |

2 | Si-poly | 1 | 60 | 100 | 46.7 | 40 | 3.73 | 1.6 × 10^{−2} | 42 | 2.6 × 10^{3} |

3 | Si-poly | 0.5 | 40 | 278 | 46.7 | 74 | 4.9 | 5.6 × 10^{−3} | 20 | 3.5 × 10^{3} |

_{G}. The test chip includes single heater structures with L

_{G}= 40, 60 and 80 µm. The measured sensor responses are shown in Figure 10. Note that the curves are similar, with a zero flow sensitivity that exhibit variations less than 13%. This experiment suggests that variation of the thermopile-heater gap in this range does not constitute a viable method to improve sensitivity. This result is in contrast with [15,16], where the sensitivity was found to increase with larger thermopile-heater spacing. A possible reason is that in [15,16] all sensor elements were placed on a single membrane, so that the distance between the heater and the thermopile strongly affects the thermal insulation of the thermopile. On the other hand, L

_{G}strongly affects the static thermopile voltage V

_{K}(0) that in Table 2 indicates the average value of V

_{1}(0) and V

_{2}(0). This quantity is significantly higher for the smallest L

_{G}value, proving that the a

_{k,i}(0) parameters are larger. Considering that the sensitivity (and then h

_{eff}) does not change with L

_{G}, a proportionally higher sensor offset should be expected for the smallest L

_{G}value. A higher offset is actually visible in Figure 10 for L

_{G}= 40 µm.

**Figure 10.**Response to a nitrogen flow of the three single heater structures, differing for the dimension of the thermopile-heater gap L

_{G}, indicated in the figure together with the sensitivity at zero flow.

**Figure 11.**Responses of double heater structures to a nitrogen flow. The structures were packaged with different channel heights, H, indicated in the figure for each curve.

^{3}, which is an excellent result for this kind of sensors. Considering the single heater sensors, the effect of reducing the thermopile-heater gap (L

_{G}) produces an interesting DR improvement, due to higher full scale flow rate. On the contrary, the resolution increase obtained reducing the channel height is accompanied by a range (Q

_{max}) reduction, resulting in a slight deterioration of the DR. Finally, it can be observed that increasing the number of heaters improves the DR at the cost of increased power consumption.

**Figure 12.**(

**a**) Noise voltage spectral density (total sensor noise) referred to the amplifier input, measured on double heater samples with a total heater power (PH) of 2.5 mW, kept in still nitrogen. The heater power distribution on the two heaters was slightly unbalanced to compensate for the sensor offset. The amplifier background noise and ideal thermopile thermal noise are also indicated; (

**b**) Time diagram of the total voltage noise, referred to the amplifier input.

## 4. Conclusions

_{i,k}(Q) that, with their derivatives, determine the sensor sensitivity. Indications about the sensitivity dependence on parameters that have a direct effect on the g

_{ik}factors, such as the flow channel height and heater thermopile gap, are extrapolated by the experiments performed on the test chip.

## Acknowledgments

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**MDPI and ACS Style**

Bruschi, P.; Piotto, M.
Design Issues for Low Power Integrated Thermal Flow Sensors with Ultra-Wide Dynamic Range and Low Insertion Loss. *Micromachines* **2012**, *3*, 295-314.
https://doi.org/10.3390/mi3020295

**AMA Style**

Bruschi P, Piotto M.
Design Issues for Low Power Integrated Thermal Flow Sensors with Ultra-Wide Dynamic Range and Low Insertion Loss. *Micromachines*. 2012; 3(2):295-314.
https://doi.org/10.3390/mi3020295

**Chicago/Turabian Style**

Bruschi, Paolo, and Massimo Piotto.
2012. "Design Issues for Low Power Integrated Thermal Flow Sensors with Ultra-Wide Dynamic Range and Low Insertion Loss" *Micromachines* 3, no. 2: 295-314.
https://doi.org/10.3390/mi3020295