The general structure of the flow sensors analyzed in this work is shown in Figure 1(a). Two temperature sensors (thermopiles) are placed on cantilevers suspended over a cavity etched into the substrate in order to provide thermal insulation. The thermopiles are composed of N_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.

Possible alternatives to the structure shown in Figure 1(a) are often used. A simple modification is the replacement of the thermopiles with resistive temperature detectors (RTD) [14,17,24], with the drawback of an additional offset term derived from resistor mismatch. Another variant that adds robustness to the device, but requires front-to-back alignment equipments and significantly longer etching times, is the placement of all sensor elements on a single closed membrane [25]. The devices are placed into channels where they are directly exposed to the flow. The shape of the flow channels is shown in Figure 1(b) with the most relevant dimensions indicated.

Several phenomena, regarding the fluid-dynamic, thermal and electrical domains, are involved in the operating principle of a flow sensor. In order to simplify the analysis, it is useful to introduce a block diagram such as Figure 2(a), representing the successive conversions of the original quantity (flow rate) resulting in the output voltage. The amplifier (A), used to read the output voltage, is also included, since it contributes to the overall device accuracy. Fluid-dynamic and thermal mechanisms are confined to block “h”, which converts the volumetric flow rate Q into a temperature difference ΔT_S using the heating power P_H (total power delivered to the heaters).

The thermopiles, represented by the multiplication of the constant k_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.

Several parameters are used to specify the performance of a flow sensor. These parameters are strictly correlated and, generally, it is meaningless to boost a single characteristic (e.g., high sensitivity) without clearly indicating at which cost it has been achieved. Some of the most important performance parameters are listed below.

Analytical solution of the fluid-dynamic and heat transfer equations involved in the operation of thermal flow sensors is feasible only for very simplified cases, that are poorly related to the complex three dimensional structures used in MEMS sensors. In this sub-section, a lumped element model for the whole transduction process will be introduced in order to provide some design indications. The fluid flow and the substrate will be assumed to be at the same temperature of the surrounding environment, indicated with T_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.

Figure 3 represents the interaction of the i-th heater with the k-th thermopile. The heater receives the power P_Hi while all the other heaters are not powered. The overheating of the heater is given by:

   (1)

where is the total thermal resistance between the heater and the environment. The fraction of the power P_Hi, indicated with P_k,i that reaches hot junctions of the k-th thermopile can be expressed as:

   (2)

where g_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:

   (3)

where is the thermal resistance between the thermopile hot junctions and the environment. For the structure of Figure 1(a) the thermal resistance is dominated by conduction through solid elements (cantilever), and is then given by:

   (4)

where t_c is the cantilever thickness and k_eq the effective conductivity that takes into account the cantilever and the thermopile material [20].

Introducing parameters a_k,i, defined as:

   (5)

It is possible to write the transduction law between power P_Hi and voltage V_k,i:

   (6)

Using the linearity hypothesis, we can consider that the total thermopile voltage is the superposition of the effects of the N_H heaters. Therefore:

   (7)

The output signal is the difference V₂-V₁, resulting in:

   (8)

The sensor sensitivity can be calculated from Equation (8) and written in the following compact form:

   (9)

where the coefficients h_i are defined as follows:

   (10)

It is convenient to relate the powers P_Hi of the individual heaters to the total power P_H, through a set of dimensionless parameters x_i, indicated as “power distribution”:

   (11)

Thus, Equation (9) becomes:

   (12)

In order to calculate the equivalent noise flow it is necessary to estimate the total noise voltage v_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:

   (13)

where v_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.

Figure 4 shows a single thermocouple, formed by conductors A and B. The pitch, by which the thermocouple are lined up over the cantilever, is indicated with W_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.

Considering that the thermopile spans the whole cantilever width, and neglecting the margin at the cantilever borders we get: W_P = W/N_T. The line width W_T is then given by:

   (14)

The total sensor electrical resistance, R, is the sum of the resistances of the two thermopiles. With the above simplifications:

   (15)

where R_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:

   (16)

where W_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:

   (17)

Substitution of Equation (17) in Equation (13) yields:

   (18)

where f_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]:

   (19)

where U_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.

Considering the sensor and amplifier contribution to the total offset voltage it is possible to write:

   (20)

The second term can be made very small using the mentioned dynamic approaches to cancel the amplifier offset. Such techniques are ineffective with respect to the sensor offset, which constitutes the main contribution. The sensor offset can be calculated using Equation (8) in condition of zero flow

   (21)

In practice, the majority of flow sensors exhibit a thermopile and heater arrangement that is symmetric with respect to a plane perpendicular to the flow direction. Therefore, if the heaters are driven with identical power, the v_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:

   (22)

From the sensitivity expression Equation (12), finally we get:

   (23)

where the amplifier offset has been neglected. Since the entity of relative errors is a technological parameter, the numerator is larger in sensors where the static coupling coefficients a_k,i(0) are larger. In turn, considering Equation (7), the larger the a_k,i(0) values, the larger the voltages V₁ and V₂, 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 V₁ and V₂ 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.

The pressure drop across the flow channel sketched in Figure 1(b) can be approximated by the following expression [28]:

   (24)

where a laminar flow regime have been assumed, A is the channel cross sectional area, µ the gas dynamic viscosity and D_H the hydraulic diameter, defined as:

   (25)

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

   (26)

Equation (26) states that reducing the channel cross section (either reducing W_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:

   (27)

where y is a coordinate referred to an axis parallel to the channel dimension H (see Figure 1(b)) and is the g_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:

   (28)

Equation (28) shows that reducing the channel cross-section, and in particular the channel height, is a very effective way to improve the sensor sensitivity but at the cost of improved pressure drop, as pointed out by Equation (26).

The expressions found so far do not allow ab-initio prediction of the sensor performances but should be complemented with experimental data measured on test structures. It is beyond the aim of this work to examine all possible combinations of design constraints, so that this section is limited to a simple example where we will suppose that parameters h_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:

   (29)

The following design indications are straightforward:

   (30)

Equation (19) states that, once an optimum amplifier topology has been chosen, the amplifier noise can be reduced only by increasing the absorbed current and, as a consequence, the amplifier power consumption. The latter, given by I_AV_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.

In order to find additional information about the design of thermal flow sensors, measurements have been performed on sample structures included in a test chip. An optical micrograph of the latter is shown in Figure 5(a), while an enlargement of three different sensor structures is shown in Figure 5(b). The chip includes also a low noise chopper amplifier, properly designed to introduce negligible noise contributions with respect to the thermopile thermal noise. The amplifier can read the differential voltage (V_out = V₂ − V₁) or the individual thermopile voltages V₁ and V₂. 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.

The test chip has been designed with the BCD6s (Bipolar, CMOS, DMOS) process of STMicroelectronics. The heaters are polysilicon resistors placed over suspended dielectric membranes while the thermopiles consist of N_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.

In order to access the bulk silicon in selected areas of the chip, the dielectric layers have been selectively removed from the front-side. In particular, the passivation layer and a part of the inter-metal dielectric layers have been removed directly by the silicon foundry at the end of the chip fabrication (Figure 6(a)). This was obtained exploiting the same etching step used by the foundry to open the passivation layer over the pads [29]. Then, during the post-processing phase, residual dielectric layers have been completely removed by means of a photolithographic step followed by silicon dioxide reactive ion etching (RIE) (Figure 6(b)). A silicon anisotropic etch was then applied through the openings, using an aqueous solution of TMAH with silicic acid and ammonium persulfate [29] (Figure 6(c)). An optical micrograph of the sensing structures after the silicon removal is shown in Figure 5(b). The depth of the cavity was 80 µm for all devices used in this work.

After post-processing, the chips were glued to ceramic DIP28 cases by means of epoxy resin and wedge bonding was used to connect selected chip pads to the case pins. A recently proposed packaging technique [30] has been used to connect the integrated flow sensor to the gas lines. This technique is based on a purposely built poly-methyl-methacrylate (PMMA) conveyor which is aligned to the chip by means of a guide and is capable to convey multiple flows to different areas of the chip [31]. Trenches with different cross-section dimensions have been milled on the flat face of the conveyor by means of a precision computer controlled milling machine (VHF CAM 100) and cyanocrylate glue has been used to fix the conveyor to the chip surface, as schematically shown in Figure 7(a). In Figure 7(b) a photograph of a device after packaging is shown.

The first set of measurements has been performed on the triple-heater sensor and was devoted to check the validity of the linearity hypothesis. The responses to nitrogen flow have been measured delivering a constant power (1.25 mW) to one heater at a time. The channel height (H) is 0.5 mm. The other dimensions are specified in table 1. The response of the three heaters H₁₋₃ is shown in Figure 8. Clearly, only the case where the central heater (H₂) 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 9(a), shows the response to nitrogen flows when all the heaters are powered with 1.25 mW. The solid symbols represent the result of an actual measurement, while the line has been obtained by superposition of the three responses shown in Figure 8. A good agreement, confirming the validity of the linearity hypothesis, is visible. To test also the supposed linearity of the sensor element overheating with respect to the heater powers, we have measured the output voltage of the 2nd thermopile (downstream thermopile) as a function of the power delivered to the H₃ (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.

The second series of experiments, which involved samples with one, two and three heaters was devoted to test the effects of parameters change on the sensor resolution and dynamic range. The resolution is calculated considering the thermal noise of the two thermopiles over a bandwidth of 100 Hz (v_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.

The experiments performed on single heater structures have been devoted to investigate the role of the heater-thermopile gap, L_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 V₁(0) and V₂(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.

The last series of experiments was devoted to investigate the effect of channel height, H. The measurements, shown in Figure 11, were performed on double-heater structures, with both the heaters driven with 1.25 mW. These curves clearly show the sensitivity increase that can be obtained by reducing the channel height, as predicted by Equation (28). The sensitivity increase was obtained at the cost of increased pressure drop, which passed from 0.1 kPa at 100 sccm for the 0.5 mm channel to 0.4 kPa for the 0.25 mm channel, at the same conditions. The actual sensitivity gain obtained by halving the channel height is nearly 3, while a factor of 4 was expected from Equation (28). A not fully developed flow regime in the relatively short channel can be a reasonable explanation.

As far as the dynamic range is concerned, all tested sensors feature DR value greater than 10³, 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.

All considerations made in this work on the sensor resolution and DR are based on the hypothesis that the only significant contribution to the overall noise is thermal noise of the thermopile resistance. This hypothesis is confirmed by recording the sensor output signal, properly amplified by the on-chip amplifier (gain = 230). Data digitalization has been performed using a 16 bit acquisition system (Pico Technology Ltd, mod. ADC216) combined with a program for spectral analysis. Figure 12(a) shows the total noise voltage spectral density of the sensor and the amplifier background noise. The latter was measured by substituting the sensor with a short circuit.

The sensor is a double heater version, with 0.5 mm channel height, maintained in still nitrogen (zero flow). Both quantities are referred to the amplifier input, as in the scheme of Figure 2(a). A fifth order Butterworth low pass filter is inserted at the amplifier output port, to limit the signal bandwidth and eliminate aliasing phenomena. The actual frequency response of the filter has been experimentally determined and the result has been used to correct the measured spectra shown in Figure 12(a), which can be considered free of filtering effects in the whole displayed frequency interval. DC coupling is used in all measurements; spectra are averaged over 25 independent captures. The ideal thermal noise density (40 nV/sqrt(Hz) of the total thermopile resistance (R = 100 kΩ) is indicated by the dashed line. The results shown in Figure 12(a) prove that the approximation of considering only the thermal noise contribution is appropriate, at least for the sensors used in this work. This was confirmed also by measurement of the rms noise voltage, estimated by data recorded over intervals of 12.6 s, with a 100 Hz signal bandwidth. Figure 12(b) shows a time diagram of the signal measured from a double heater sample kept in still nitrogen. The original sensor offset (around 100 µV) was compensated by balancing the heater power as described in [21]; a residual offset of nearly −1.5 µV can be observed. The estimated rms noise voltage (i.e., the signal standard deviation) was 0.41 µV, in good agreement with the theoretical 0.4 µV value calculated by considering only thermal noise.