Louge, et al (10) offered a detailed summary of guarded capacitance instrumentation. Their article included governing equations of the electric field and its analogy with conduction heat transfer, operations of the processing electronics, algorithm optimization for the rapid evaluation of magnitude and phase of the signal, derivation of the real and imaginary parts of the dielectric constant from these quantities, and calibrations yielding instantaneous bulk density and solvent content of semi-conductive powders from the complex dielectric constant.

In this section, we illustrate principles that are essential to justify placement of conductive surfaces in the ST. We then derive estimates of its capacitance and measurement volume from an exact calculation of its electric field in a homogeneous medium of indefinite extent. Lastly, we consider the case where solvent content is inhomogeneous, so the complex dielectric constant varies along powder depth.

In the Smart Tray, the metal base features one or several flat capacitance probes, each consisting of the three essential conductive surfaces of this technology, namely the sensor, the guard, and the grounded target, which are insulated electrically from one another using thin dielectric borders made of pharmaceutical-grade material. An advantage of such permanent assembly is that it can be cleaned and uphold Good Manufacturing Practices (GMP). In the pharmaceutical environment, the three conductors can be made of stainless steel. An alternative is to bond a thin printed circuit board (PCB) featuring sensor and guard to the base of an existing metal tray, suitably separated from the powder using a laminated pharmaceutical-grade dielectric sheet that is equally straightforward to clean. We implemented both methods in a level sensor designed for GMP continuous manufacturing (10).

Figure 1 shows the setup of a proof-of-concept. Here, for simplicity, the base of the ST was conveniently made of a printed circuit board (Fig. 1A). In this configuration, we deployed two independent probes to investigate the possibility of horizontal powder cake non-uniformity. A production design can add probes to regions that may be prone to variations in solvent content.

As Louge, et al (12, 13) discussed, the processing electronics is driven by an oscillator (a.k.a. the clock) that provides a sinusoidal voltage of a few kHz to a control circuit feeding an alternating current (AC) of constant amplitude \(\tilde\) to the sensor. Then, a negative-feedback operational amplifier (op amp) of high input impedance keeps the guard voltage amplitude rigorously identical to the sensor’s \(\tilde_s\) without drawing an appreciable current (Fig. 2). In this way, the complex guard voltage remains proportional to the impedance Z between sensor and target. For example, if the dielectric material has no imaginary part, the guard amplitude is inversely proportional to the recorded capacitance.

Because this technique requires highly stable controls circuits generating sensor currents on the order of nanoAmps, it poses no electrical hasard, but its electronics remains the province of a few commercial companies that specialize in the ultra-accurate determination of variable gaps separated by air (14). Our contribution has been to fix the probe geometry and monitor dielectric properties of the medium it encompasses. As we showed for applications as diverse as snow packs (15), desert dunes (16, 17), and pharmaceutical powders (10), the guard voltage amplitude and its phase with respect to the clock provide enough information to calculate the real and imaginary parts of the dielectric constant of the effective medium within the measurement volume.

What distinguishes our guarded technique from other capacitance methods (19,20,21,22,23,24,25,26) is the judicious placement of active conductors directing the measurement volume where quantitative bulk density and solvent content are needed. Then, unlike conventional unguarded bridge circuits, the coaxial cable joining our probes to its processing electronics has no discernible effect on the signal. This is achieved by surrounding the sensor wire by an outer sheath driven at the guard voltage over the entire cable length. As a result, because the sheath acts as a Faraday cage, this perfect screening only exposes the sensor to the grounded target at the probe. In this design, the only limitation on cable length arises from the capacitance it adds between sensor and guard circuits, which lies in parallel to the input impedance of the voltage-follower op amp keeping sensor and guard voltages strictly equal. Specifically, for the commercial Capacitec processing electronics that we used, cable impedance must be kept \(\lesssim 500~}\) to avoid drawing appreciable current from the sensor circuit at the clock frequency \(f = 15.5762~}\). Equivalently, a typically coaxial cable with \(85~/\text }\) specific capacitance should have a length \(\lesssim 6~}\) for optimum measurement accuracy. If required by regulations or GMP, future designs can incorporate the coaxial cable, processing electronics and power supply within the tray.

Standard Capacitec electronics can be adjusted to detect capacitances as low as \(7~}\) and as high as \(3~}\). Because any capacitance in air (or vacuum) is the product of a length scale \(\ell \) and the dielectric permittivity of free space \(\epsilon _0 \simeq 8.854~/\text }\),

$$\begin C_0 = \epsilon _0 \times \ell \, , \end$$

(1)

such precision allows the design of instruments of millimetric to decimetric size (\(0.8~}\) \(\lesssim \ell \lesssim \) \(310~}\)), which may be embedded through solid walls of any typical production vessel without affecting process operations (10).

A powder homogeneously laced with solvent has a dielectric permittivity \(\epsilon _e = \epsilon _0 K_e\), where

$$\begin K_e = K_e^\prime -\imath K_e^ \end$$

(2)

is the complex dielectric constant and \(\imath ^2 = -1\). Dry powders typically possess only a real part \(K_e^ >1\), while their imaginary part \(K_e^ \simeq 0\). In contrast, solvent-laced powders possess a significant \(K_e^ > 0\) that is related to their intrinsic conductivity. Both \(K_e^\prime \) and \(K_e^\) rise with bulk density and solvent content (10).

If the powder has homogeneous solid volume fraction but inhomogeneous solvent content, the ST sheds an electric field between sensor and target that is affected by spatial variations of \(K_e\). The ST then produces the impedance

$$\begin Z = [2 \pi f C_0 (\bar_e^+ \imath \bar_e^\prime )]^ \end$$

(3)

between sensor and ground, where \(C_0\) is capacitance of the ST exposed to air alone. In this expression, the overbar denotes dielectric properties as they appear to the processing electronics. For the ST, \(\bar_e\) is the result of the electric field propagating through layers of the effective medium consisting of the powder and its solvent. We will discuss this further in Section 3.

In similar fashion than an amplitude modulation (AM) radio, the electronics ‘rectifies’ the guard to produce a voltage V (Fig. 2). Because guard and sensor voltages are proportional to Z, this output is

$$\begin V=g_a\, |Z|= g_a/[2\pi f C_0 \sqrt_e^ + \bar_e^ } ] \, , \end$$

(4)

where \(g_a\) is a ‘gain’ that sets the intensity of the current fed to the sensor. Then, the modulus \(|\bar_e | = \sqrt_e^ + \bar_e^ }\) of the apparent dielectric constant is obtained by forming the ratio of the rectified voltage \(V_0 = g_a/(2\pi f C_0)\) when the probe is exposed to air (\(K_e^\prime =1\), \(K_e^ = 0\)) and its counterpart V in the powder’s presence,

$$\begin |\bar_e| = V_0/V \, . \end$$

(5)

In principle, Eq. (4) implies that neither \(g_a\), nor f, nor \(C_0\) affect \(V_0/V\). However, to achieve stability of the processing electronics and best signal-to-noise ratio, the gain is set as high as possible to produce the largest unclipped sinusoidal guard voltage. In practice, this maximum is set by the DC power supply, typically operating at \(\pm 15~}\). Meanwhile, the phase between guard and oscillator voltages yields the apparent ‘loss tangent’

$$\begin \tan \bar = \bar_e^ / \bar_e^\prime \, , \end$$

(6)

thereby providing enough information to find \(\bar_e^\prime \) and \(\bar_e^\) separately using an algorithm available in reference (10). As we showed for a pharmaceutical powder, or for sand holding trace amounts of water (17), a calibration of loss tangent provides solvent content. The further use of a homogenization model (27) then yields the powder’s bulk density.

In this technology, uncontrollable parasite capacitances created by nearby objects are suppressed by judiciously placing guarded conductors around sensors arrayed on the probe surface in contact with the powder. For the non-invasive ST, another design requirement is to locate the measurement volume near the base, where the powder is last to dry. As Fig. 1 illustrates, this is achieved by flattening the electric field emanating from the sensor using two grounded target surfaces.

(A) Predicted electric field in the (x, y) plane of ST of Fig. 1 for a homogeneous powder of \(10~}\) depth and \(K_e = 2\) illustrating notation found in the text, including radial distances \(r_i\) and polar angles \(\theta _i\) from point M to the four singularities of guard-ground interfaces at \(x=-a\), \(g_1\), \(g_2\) and \(+a\). In this axisymmetric design, we choose \(g_2=-g_1=g\). The right sensor marked by a black rectangle lies in the range \(s_1< x < s_2\) wherefrom electric field lines define the measurement volume. Equipotential lines form its orthogonal network. They refract at the air-powder free surface. (B) Apparent dielectric constant \(\bar_e\) relative to \(K_e\) vs powder depth h relative to \(h_}\). From top to bottom, lines of Eq. (11) and symbols are \(K_e = 3\), 2, 1.5 and 1.25. (C) \((\text - \overline})\) vs \(\text \), where \(\text \) is the actual relative humidity at equilibrium with a homogeneous MCC of dielectric properties in Eq. (12) and \(\overline}\) is the apparent value returned by the ST. From top to bottom, relative depths \(h/h_} = 0.32\), 0.48, 0.64, 0.88, 1.04, and 1.20. Symbols are model predictions, and lines are obtained by inverting Eq. (13)

To analyze this arrangement and optimize its performance, Appendix A calculates the electric field above a ST of indefinite extent exposed to a homogeneous dielectric using complex variables in a plane perpendicular to the long axis of the sensor. In air, the resulting characteristic length is

$$\begin \ell = \frac \ln \left[ \left( \frac \right) \left( \frac \right) \left( \frac \right) \left( \frac \right) \right] \, , \end$$

(7)

where \(w_s\) is the transverse length of the sensor; a is the guard half-width; \(g_1\) and \(g_2\) are, respectively, x-coordinates of singularities at the left and right edges between inner ground and guard; and \(s_1\) and \(s_2\) are their counterparts between sensor and guard (Fig. 3A). The (x, y, z) coordinate system is sketched in Fig. 1B. Meanwhile, the apex of the outermost field line emerging from the sensor at \(x=s_2\) serves as a scale for the outer reach \(h_}\) of the measurement volume satisfying

$$\begin \nabla F \cdot \hat} = \frac (x=s_2,y=h_}) = 0 \, , \end$$

(8)

which we solve numerically once probe dimensions are specified.

To optimize the design while reducing the number of independent parameters, we impose an axisymmetric inner ground, \(g_2 = - g_1 \equiv g\). Next, we make sure that the outermost sensor-guard edge at \(x=s_2\) be \(< s_} = \sqrt\) where field lines never return to the base (Eq. 43), thus making \(s_2 = f_s \sqrt\) with factor of safety \(f_s = 0.95\). We also impose a minimum relative gap \(\gamma = 0.14 \times a\) between the right-most edge of the inner ground at \(x=g\) and the left-most edge of the sensor, such that \(s_1 = g + \gamma \). Lastly, we select the ratio \(g/a \simeq 0.275\) that maximize \(\ell /w_s\) in Eq. (7), achieving \(\ell \simeq 0.179 \, w_s\).

In this optimum configuration, a and \(w_s\) are the only remaining independent length scales. Because a cancels out from Eq. (7), all x-coordinates are relative to a, while \(w_s\) alone controls the magnitude of \(\ell \) and therefore \(C_0\). Lastly, we fix a such that the electric field line emanating from the outermost x-coordinate of the sensor reaches a distance on the order of the minimum powder depth that we expect to process. With \(f_s = 0.95\) and \(g/a \simeq 0.275\), the solution of Eq. (8) is \(h_}/a \simeq 0.346\). In the proof-of-concept design that we tested, \(a \simeq 36~}\), \(g \simeq 9.9~}\), \(s_1 \simeq 15.0~}\), \(s_2 \simeq 17.9~}\), and \(h_}\simeq 12.5~}\). With \(w_s \simeq 32~}\), Eq. (7) yields the nominal capacitance in air \(C_0 \simeq 51~}\). Depending on user’s requirements, other dimensions may be selected, for example to bring \(y=h_}\) closer to the base.