Introduction

This report outlines the research completed to develop a dynamic thrust measuring device to analyze mono-propellant engines in the 1 and 20 newton range. The development has been separated into stages with the first stage focused on calibration of a thrust stand at the desired force range. Ideally, the calibration should produce a non-contact force that is uniform with displacement and capable of 1kHz. Calibration systems capable of producing the previously mentioned conditions include but are not limited to electrostatic, electromagnetic, Helmoltz coils, and Maxwell coils. Specifically, this work has analyzed electrostatic and electromagnetic calibration systems.

Progress

Calibration Systems

Electrostatic

Electrostatic calibration is useful for calibrating thrusts stands because of the known relationship between voltage and force. Seldon and Ketsdever produced the Electrostatic Comb (ESC) calibration system which provides a consistent force with small displacements Selden. The force range of the ESCs is between 35nN and 1μN providing calibration forces adequate for a NanoNewton thrust stand. The force range can be scaled as shown by Allan Yan who desired a calibration force between 0.01 and 1mNs. This increase in magnitude was achieved by increasing the width of the combs on the ESC. The scaled ESC is known as the Electrostatic Fin (ESF) and is shown in Figure 1 Yan.

Figure 1: Set of Seldon et al.'s ESCs (left) and Yan's ESF (right) Yan.

Theory: Electrostatic Combs (ESC)

The actuation of ESCs can be modeled by separating the force into local and global forces Johnson. The local force is an attractive force while the combs are engaged and is mathematically modeled using the following equation where V is voltage, ε is the permittivity of vacuum, and c, d, and g are geometric constants defined in Figure 1.

F_x^{Local} \approx \frac{VQ}{2}

Q \approx 2 \epsilon_0 \frac{V}{\pi} (\ln([(c/g+1)^2-1](1+2g/c)^{1+c/g})+\pi d/g)

The global force repels the combs while they are engaged and can be modeled using:

F_x^{Global} = -4(c+g) \frac{\epsilon_0 V^2}{4\pi x_0}

To find the total force acting on the ESCs, the local and global forces must be summed. For ESCs it is assumed that c=d=g which simplifies the total force to the following:

F_x \approx 2\epsilon_0V^2\left[1.0245 - \frac{g}{\pi x_0}\right]

An important note is that this equation only models the ESCs while they are engaged.

Theory: Electrostatic Fins (ESF)

The attractive force of ESFs is calculated using a similar equation as ESCs. Instead of assuming c=d=g, just c=g was assumed and d was left as a variable. With this new assumption, the local and global forces sum to provide the total force as shown:

F_{x}\approx 2 \epsilon_0 V^2 \left[0.5245+\frac{d}{2g}-\frac{g}{\pi x_0}\right]

This equation shows that increasing the width of the fins, d, or decreasing the gap, g, will provide more force than the ESCs. The geometry of the ESCs and ESFs is shown in Figure \ref{Geometry} which defines c, d, x₀ and g.

Figure 2: (a) Front and (b) tob views of a pair of fins cut along symmetry Yan.

ESFs can provide more force than ESCs as shown by Pancotti2012. In theory, any magnitude of force can be achieved with ESFs by manipulating the voltage, width, gap, and number of fins.

Design of Electrostatic Fins (ESF)

Theoretically, forces of 1 and 20 newtons are possible with electrostatic fins by manipulating the geometry or applied voltage. In this research, ESFs were designed to achieve a 1 newton force while neglecting breakdown voltage. Figure \ref{Design} demonstrates the relationship of force as a function of voltage using Equation \ref{eq:ESF} for the designed ESF on the right. From the fins shown in Figure \ref{Geometry}, Yan's ESFs were modified by increasing the number of fins, reducing the gap distance, and increasing the width.

Figure 3: Force obtained from input voltage (left) and the geometry of the ESFs (right).

With electrostatic calibration, one limiting factor is the maximum voltage that can be applied before an arc jumps the gap, $2g$. From Paschen's Law, the breakdown voltage can be found as a function of pressure and distance ~\cite[p. 546]{Lieberman2005}. The thrust analysis of the mono-propellant engines will be performed in a vacuum chamber with pressures that will oscillate between 3 and 27 Torr. In Figure \ref{fig:Paschen} a range of $pd$ values is shaded to demonstrate the expected conditions in the vacuum chamber. The minimum $pd$ was set to 3 Torr$*$0.05~cm because the minimum expected pressure will be 3 Torr and the minimum distance should be the gap between the fins, $2g$. The maximum $pd$ was set to 27 Torr$*$100 cm because any distance up to a meter could be achievable between the excited fins and the grounded thrust stand or vacuum chamber.

Figure 4: Paschen curve for Nitrogen using the formula for breakdown voltage.

From Figure 4 it is clear that arcing will most likely occur in the conditions expected within the vacuum chamber.

Conclusions on Electrostatic Calibration

ESCs and ESFs are excellent for low force calibration because of the consistent, non-contact force that they can apply to a thrust stand. Using these devices to calibrate at higher forces requires either decreasing the separation gap or increasing the applied voltage, width of fins, or number of fins. To obtain forces in the single newton and higher range, one would have to design a very large electrostatic plate, about 100$cm^2$, or decrease the separation gap to hundreds of micrometers. The voltage can also be increased but is limited by the breakdown voltage.

Applying the principles discussed in this section to the AFRL contract, it was decided that ESCs and ESFs will not work. The vacuum chamber intended for use during this project will hold a pressure of about 3 Torr and fluctuate up to 27 Torr during testing. At these pressures, arcing will almost certainly occur because the distance between the charged plate and any point on the thrust stand can satisfy the requirement for voltage breakdown at a low voltage of 300 volts. Also, designing the electrostatic calibration system for only 300 volts will significantly increase the size to obtain 1 newton of force. Primarily due to the breakdown voltage, ESCs and ESFs will not work as calibration devices for the AFRL thrust stand.

Electromagnets

Electromagnets provide a possible alternative to electrostatic calibration because of the non-contact force and relationship between electric current and force. Use of electromagnets for calibration has already be shown by Tang et al. where they developed an electromagnetic calibration system (EMCS) using a permanent magnet structure and an electric coil for the millinewton force range ~\cite{Tang2011}. The EMCS uses the magnetic structure shown in Figure \ref{fig:EMCS} to generate a radially uniform magnetic field that interacts with the coil to provide a linear force along the center axis when current is run through the coil.

Figure 5: (a) Sketch of the magnetic structure with magnet orientations and (b) the actual magnetic structure Tang.

Theory

The force produced by the EMCS is a phenomenon known as the Lorentz Force. Specifically, the Lorentz Force is the force applied to a charged particle when traveling through an electric and magnetic field. It is also present in a current carrying wire immersed in a magnetic field which will produce a perpendicular force described with the following equation [p. 751] Walker.

\vec{F}=I \int{d\vec{s}} \times \vec{B}

This equation applied to a ring carrying current, I, in a non-uniform magnetic field, $\vec{B}$, as shown in Figure \ref{fig:MagnetRing} will produce the force, $\vec{F}$, experienced by the EMCS.

Figure 6: Current-carrying ring placed in a constant, non-uniform magnetic field produced by a permanent magnet MIT.

The following algebraic expression for the force applied to the ring was achieved by converting to cylindrical coordinates and integrating the Lorentz Equation along the current-carrying differential element, $d\vec{s}$.

\vec{F}=(2\pi rIB\sin \theta ) \hat{z}

This equation is valid for one ring but can be applied to a coil of $N$ rings by multiplying the expression by $N$ and assuming that the magnetic field is uniformly distributed across the entire coil.

Electromagnet Testing

Using the permanent magnet and a voice coil from a speaker, the force of a simple electromagnet could be physically tested. The tested magnet had a setup similar to the sketch shown in Figure \ref{Sketch} where a radially uniform magnetic field is created between the gap of the ring and core.

Figure 7: Schematic of the electromagnet to generate radially uniform magnetic field.

The electric coil was placed within this gap with the top of the coil aligned with the top surface of the iron core and ring. To test the consistency of the force against displacement, the axial position of the coil was changed. The initial position was defined as the zero engagement and positive engagement was set in the direction towards the base of the core. The physical tests only tested negative engagement because the core of the tested magnet had a ledge to stop the coil in the positive direction. The experimental results were compared with analytical results obtained using equation \ref{Lorentz} and numerical results acquired using ANSYS Maxwell. The results show very consistent data and remain linear as expected from equation \ref{Lorentz}. The experimental results were extremely consistent which allowed the error bars to be neglected in Figure ~\ref{Results}. As the coil is removed from the magnetic structure, the force reduces which is consistent with equation \ref{Lorentz} since the number of coils, $N$, within the magnetic field is reduced. The numerical results match the experimental results to within 8.7\%. This deviation could result from (1) not knowing the exact material properties of the magnet and (2) the current source used for the experiment. Material properties for NdFeB, a common magnet material, are well known and were used for the simulation but this material may not have been used in the magnet tested which most likely caused the systematic error between the experimental and numerical results. The current source used for the experiment was not designed to provide an accurate current supply, especially not below 0.1 amps.

Figure 8: Analytical, experimental, and numerical results for the force imposed on the coil due to the magnetic field.

Use in Vacuum

The electromagnet uses current through the coil to drive the force which could cause high temperatures, especially in vacuum. The temperature of the coil can be calculated analytically and compared to the maximum allowable temperature of the wire to ensure proper operation in vacuum. The analytical model neglects radiation and convection which are reasonable because the temperature of the wire is not hot enough to produce significant losses in radiation and convection is negligible in vacuum. The only mode of heat loss is through conduction at the end of the wire. It was assumed that the temperature distribution throughout the wire is uniform which is reasonable because the wire is copper which conducts heat very well. Figure \ref{Coil} shows the sketch of the model where heat is generated from Joule Heating and removed through conduction.

Figure 9: Sketch of electromagnet coil in a vacuum with Joule Heating.

The following equation was derived from the conservation of energy to provide the increase from room temperature as a function of time, $\theta(t)$.

\theta(t)=\frac{i^2R_ER_K}{2A}\left(1-e^{-2t/R_KLc\rho}\right)

The specific heat, $c$, and density, $\rho$ of copper were obtained from ~\cite{Kreith2011} as well as the contact resistance in vacuum, $R_K$. The current, $i$, electrical resistance, $R_E$, and length $L$, were measured. The cross sectional area, $A$, was determined from ~\cite{AWG} using realistic wire gauges. The results are shown in Figure \ref{WireTemp} where the American Wire Gauge was an input and the temperature was calculated at various times. A maximum current of $0.5 amps$ was used for a worse case scenario. A maximum operating temperature of 60\degree C is stated in ~\cite{WireConductor} which will be used as a standard to ensure proper operation. Since calibration will most likely not exceed 20 minutes at $0.5 amps$ 21 or possibly 22 Gauge wire should be safe to use in an electromagnet in vacuum.

Figure 10: Temperature increase of various gauge wire for specified times.

Conclusions and Recommendations

Electrostatic calibration is not feasible with the operating pressures expected at AFRL so another calibration system must be developed. Based on the findings in this work, an electromagnetic calibration system is recommended. The electromagnet will operate efficiently in the expected environment and meet the requirements. The numerical model developed in this work was verified experimentally which makes it a viable source for designing an electromagnet capable of several force ranges. Using the numerical model, the force can be increased by increasing the radial magnetic field generated between the red ring and green core shown in Figure \ref{NumericalModel}. The radial magnetic field strength can be easily increased by in creasing the height of the magnet, shown as the blue ring in Figure \ref{NumericalModel}. The height of the blue magnet can be increased by stacking standard ring magnets which makes this design very practical and easy to build. For the future it will be beneficial to develop an electromagnet with better known properties to further validate the numerical model and acquire a more sensitive current source for calibration.

Figure 11: Numerical model that can be used to develop future electromagnetic calibration systems.

Sources