2.0 Magnetorquer coil system

The magnetorquer coil system level block diagram with respect to on-board computer (OBC) is shown in Fig. 3. Here the block diagram shows all the subsystems related with power management, attitude determination and attitude control. All the electronic components related with various subsystems are embedded on the bottom layer (layer-10) of the PCB whereas the top layer (layer-1) has solar cells. The embedded magnetorquer coil is in the six inner layers (layer-2 ∼ layer-7) as shown in Fig. 2. In order to protect the electronic components mounted on layer-10 from the magnetic effects of the coil, two ground plans are added in layer-8 and layer-9. These ground planes work as a shield and protect the layer-10 electronic components from the magnetic effects of the magnetorquer coil. COTS component MSP430 (MSP430F5438A) is used as OBC which is an eight ports (modules) microcontroller. On each module a separate subsystem is attached. Magnetorquer coil is connected on MODULE_E through magnetorquer coil driver. The coil is operated in different modes through the coil driver depending upon the control signals from the OBC. The next subsections describe in detail magnetorquer coil driver and magnetorquer coil.

Figure 3. Magnetorquer coil system block diagram.

2.1. Magnetorquer coil driver

The coil driver of the magnetorquer unit monitors the amount and direction of current through the coil and ultimately decides the amount of torque generated and direction of satellite spin. It is composed of Allegro A3953 (A3953), differential voltage sensor and load switches as shown in Fig. 4. The driver pins OUT_A and OUT_B are connected to two ends of the magnetorquer coil. The pins notBRAKE, PHASE, MODE, NOT_ENABLE and EN_COIL are connected to MODULE_E digital output signals of the tile processor. These signals drive the magnetorquer coil in different modes such as standby, sleep, forward/reverse, brake and fast/slow and current-decay modes. The current flow through the magnetorquer coil is controlled by the differential voltage sensor and sends it to the microcontroller, which computes the magnetic field and corresponding torque accurately. Load switches terminate the magnetorquer coil power supply on enable signal from OBC.

Figures 4 and 5. (4) Schematic of magnetorquer coil system. (5) Coil driver (A3953) internal current paths and coil (L), (a) Forward current flow. (b) Reverse current flow.

The topology of the transistor bridge of the magnetorquer coil driver circuitry is illustrated in Fig. 5. The topology of bridge permits bi-directional flow of current through magnetorquer coil (L). Transistors switches Q1 and Q3 permit the flow of current from left to right (illustrated through a solid line from supply ‘V’ to GND in Fig. 5(a)) while transistors switches Q2 and Q4 are switched off at this stage. In reverse current flow case, Q2 and Q4 transistors are switched on and current flows through the coil from right to left (shown through a solid line from supply ‘V’ to GND in Fig. 5(b). During this stage Q1 and Q3 are switched off. The flyback diodes (D1, D2, D3 and D4) helps in the de-excitation of the copper traces during intermediate resting period as shown through dashed lines in Fig. 5.

3.0. Magnetorquer mathematical model

Electromagnetic PCB integrated set of magnetorquer coils operate on the principle of dipole moment induced resulting from the current passing through the windings and its coupling with the geo magnetic field giving a control torque. Magnetic dipole moment is then represented by

(1) \begin{align}\vec D = N.S.I.\vec n\end{align}

Where N denote turns number, S represent the area of single turn and I gives the current flow in the winding.

The Earth magnetic field fluctuates between 0.15G and 0.45G [25] at an altitude of 800km with inclination angle of 89°. The time varying Earth magnetic field interacts with the resulting dipole moment of integrated magnetorquers printed in the internal layers of PCB for torque impartment onto the spacecraft for attitude manoeuverability which is given by the equation below.

(2) \begin{align}\vec \tau = {D_{coils}} \times {B^b} = - {B^b} \times {D_{coils}}\end{align}

Where the magnetic dipole moment column matrix is represented by ${D_{coils}}$ produced due to every integrated magnetorquer and ${B^b}$ is the geo magnetic field given in the body frame. Considering the spacecrafts’ body fixed frame, the equation for x, y and z components of printed magnetorquer coil is represented as

(3) \begin{align}\left[ {\begin{array}{*{20}{c}}{{\tau _{\left( x \right)}}}\\[5pt] {{\tau _{\left( y \right)}}}\\[5pt] {{\tau _{\left( z \right)}}}\end{array}} \right] = \left[ {\begin{array}{*{20}{ccc}}0 & {{B_z}} & {{B_z}}\\[5pt] { - {B_z}} & 0 & {{B_x}}\\[5pt] {{B_y}} & { - {B_x}} & 0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{D_{\left( x \right)}}}\\[5pt] {{D_{\left( y \right)}}}\\[5pt] {{D_{\left( z \right)}}}\end{array}} \right]\end{align}

For the optimal design selection, a single arbitrary reference axis is taken in our future calculations to simplify the computational complexity. The effective torques generated on the satellite when the magnetic field lines are normal to the reference axis will be considered to simplify the mathematical complexity.

The benefit of the embedded design is the reconfiguration through MOSFET switches based on mission needs according to power ratings, generated magnetic moment, imparted torque, and temperature ratings. The asymmetric coils embedded in the internal layers can be reconfigured in (series, parallel and series-parallel hybrid combination) to maximise or minimise the torque and maximum heat dissipation. The coils can be configured and made compatible with the satellites dimensions by changing the arrangement through the onboard processor at any given time. Single layer design parameters are tabulated in Table 1.

Table 1. Single layer magnetorquer coil design parameters

Figure 6(a) depicts the printed asymmetric magnetorquers possible combinations (n-series, m-parallel and n × m hybrid). In case of m parallel connected coils, to flow a constant current (I ₀) the applied voltage is the same as (nV ₀). Similarly, for series arrangement, the voltage is (nV ₀) to draw a current (I ₀) through n-series connected magnetorquers. In case of series-parallel hybrid configuration, for the same applied voltage (nV ₀), the resulting current is (I ₀). Figure 6(b) illustrates the circuit topology of MOSFET switches which provides reconfiguration to the magnetorquer sub-coils. The magnetic moment $\vec D$ of M coils arranged in any possible configuration (n-series, m-parallel, n × m-hybrid) is

(4) \begin{align}\overrightarrow{D} = M\overrightarrow {{D_0}} \;\end{align}

Figure 6. Voltages required for possible configurations magnetorquer coils providing constant current to 4U (a) and 8U (b) multi-cube satellites.

Correspondingly, consumed power of a single coil is

\begin{align*}{P_0} = I_0^2{R_0}\,\, \because{R_0} = \rho {{{L_t}} \over A}\end{align*}

Dissipated power by M coils arranged in any possible configuration (m-parallel, n-series, $\;{\rm{n}} \times {\rm{m}}$ -hybrid) is

(5) \begin{align}P = M{P_{0\;}}\end{align}

\begin{align*}\because{I} = mI_{0},\;V = n{V_{0}},\; M = n \times m\end{align*}

The magnetic moment and power dissipation ratio also increase when the number of coils are increased, either in parallel, series or the hybrid combination.

4.0. Rotational attitude manoeuverability

The attitude of a spacecraft is the rearrangement of its spatial orientation relative to a reference system. High spin control rate is needed to maintain the nanosatellite attitude known as spin stabilisation. The satellite acts like a gyroscope while it is spun around its axis of symmetry to align with the Earth’s inertial reference frame for stabilisation [Reference Slavinskis26, Reference Ehrpais, Kütt, Sünter, Kulu, Slavinskis and Noorma27]. Spinning rate of nanosatellites can be affected due to disturbing torques like gravitational and aerodynamic, eddy currents from the earth’s magnetic field and the expansion and contraction of spacecrafts due to extreme temperature difference in orbital changes [Reference Lyubimov and Podkletnova28]. These disturbing torques can significantly affect the changes in the moment of inertia and thus the corresponding spin rate. There is also a possibility of an accident occurring due to space debris that could result in uncontrolled tumbling of the satellite at excessive angular rates. The prolonged tumbling at higher angular rates could affect the structural integrity of the nanosatellite and its components as well. Therefore, precise torque is needed for the recovery of satellites intended position in orbit after tumbling.

Imparted torque T and the generated inertia moment $J$ of vehicle is accountable for driving the angular speed ${\omega ^o}\;$ along the major axis of inertia while manoeuvering the spacecraft and utilising the magnetometers to predict the direction and magnitude of geo magnetic field for copper traces excitation. Specific controlled torque produced is required for time interval (0 → T/2) to increase the satellites angular speed for desired angular distance covered $\varphi$ . A reaction torque (−τ) of same magnitude proportional to the torqueing time (T/2 → T) of equal interval is required to stop the spacecraft at desired position.

The rotation times of magnetorquer coils are analysed for single axis attitude manoeuvers using Newton’s second law of rotational motion.

(6) \begin{align}\tau = J{\omega ^\circ }\end{align}

\begin{align*}\omega ^\circ = \int_0^T {{\tau \over J}dt = } \int_0^{T/2} {{{{\tau _{max }}} \over J}dt} + \int_{T/2}^T {{{ - {\tau _{max }}} \over J}dt} \end{align*}

\begin{align*}\omega ^\circ = \left( {{{{\tau _{max }}t} \over J}\left|_{_{_{_{_{_{_{_{_{_{0}}}}}}}}}}^{^{^{^{^{T{\rm{/2}}}}}}} {;{{{\tau _{max }}(T - t)} \over J}} \right|_{T/2}^T} \right)\end{align*}

The angular distance $\varphi$ is derived from the following equation.

(7) \begin{align}\varphi = \int_0^T {{\omega ^ \circ }} .\,dt = \left( {{{{\tau _{max }}{t^2}} \over {2J}}\left|_{_{_{_{_{_{_{_{_{_{0}}}}}}}}}}^{^{^{^{^{T{\rm{/2}}}}}}} {;{{{\tau _{max }}T} \over J}\left(t - {{{t^2}} \over {2T}} - {T \over 4}\right)} \right|_{T/2}^T} \right)\end{align}

\begin{align*}\varphi = {\omega ^\circ }{\left( {\frac{T}{2}} \right)^2} + \,{\omega ^\circ }{\left( {\frac{T}{2}} \right)^2} = \frac{\tau }{{2J}}{T^2}\end{align*}

The required time T for the integrated magnetorquer to revolve the satellite for desired angular distance $\varphi$ can be calculated from (8):

(8) \begin{align}T = \sqrt {\frac{{2J\varphi }}{\tau }} \end{align}

Equation (8) depicts that torqueing time and torque generation needed to revolve the satellite are inversely proportional. The results are plotted for various current inputs as a function of applied voltage to different possible reconfigurable arrangements as shown in Fig. 7. The value of J is set to 0.09 and 0.53kg/m² for a 4U and 8U nanosatellite, respectively.

Figure 7. Current vs generated torque; for the 4U (a) and 8U (b) satellites.

For various coil arrangements, the torque imparted on the satellite in presence of 0.5G Earth magnetic field is shown in Fig. 7. The torque generation and the respective time required to rotate the multi-cube nanosatellites by an angle of 90° are illustrated in Fig. 8. In case of single or series configuration, generated torque is 5.2µNm that needs 114.5s and 239s to rotate the spacecraft by an angle of 90° for the 4U and 8U multi-cube nanosatellites, respectively. In case of parallel configuration, the torque generation is 42.13 and 126.3µNm, which needs 84.3 and 114.5s to rotate the spacecraft by an angle of 90° for the 4U and 8U multi-cube nanosatellites respectively. The results show that as the cube units of nanosatellites are increased, with the increasing dimensions, the spinning rate proportionally increase. The maximum spinning rate results from the parallel arrangement, but at a cost of high temperature gradient and power consumption. Thermals and power dissipation will be discussed in the next section.

Figure 8. Satellite’s rotation time required through an angle of 90°; for the 4U (a) and 8U (b) satellite.

Figure 9. (a–e) 4-U and 8-U multi-cube satellites.

Figure 10. Plots of magnetic moment, power dissipated and temperature against applied voltage for various coil configurations.

5.0. Thermal analysis

Thermal modeling is the most significant step in the design of printed magnetorquers integrated in the internal layers of the multi layers PCB. When current flow through these coils, power is dissipated which results in PCB surface temperature increase. Thermal analysis is required to ensure that the PCB surface temperature rise should be within the required limits of the components mounted over the PCB. Here we are assuming the system (PCB with embedded magnetorquers) in thermal equilibrium. Which means that the power absorbed by the PCB from the surrounding $\left( {{P_I}} \right)$ plus the electrical power dissipated by the magnetorquer coils $\left( {{P_d} = {I^2}R} \right)$ is equal to the total radiated power from the PCB surface to the surrounding $\left( {{P_o}} \right)$ . According to Stefan-Boltzmann’s law (Mark Wellons, 2007), the heat radiated from the PCB surface at a specific wavelength is given by $\alpha \sigma {T_o}^4S$ whereas $\sigma $ is the Stefan-Boltzmann constant, ${T_o}$ is the PCB surface temperature, $S$ is the PCB surface area and $\alpha $ is the emissivity. In thermal equilibrium state when zero power is applied to the coils results in ${P_o} = {P_I} = \alpha \sigma {T_I}^4S$ where ${T_I}$ is the surrounding temperature. Comparative thermal analyses for multi-cube small satellites with dimensions 4U (33 × 33 × 16.5cm³) and 8U (33 × 33 × 33cm³) have been done by measuring the amount of current (I) that passes through the embedded magnetorquers and the corresponding resistance $(R)$ . The integrated coil’s power consumption $\left( {{P_d} = {I^2}R} \right)$ and the corresponding rise of temperature in the PCB is calculated from the derived equations in Refs [Reference Ali, Khan, Dildar, Ali and Ullah7, Reference Ali, Tong, Ali, Mughal and Reyneri21, Reference Ali, Ullah, Rehman, Bari and Reyneri22, Reference Mughal, Ali, Ali, Praks and Reyneri29], which are shown below.

(9) \begin{align}{T_o} = \sqrt[4]{{\frac{{{I^2}R + \alpha \sigma {T_I}^4S}}{{\alpha \sigma S}}}}\end{align}

Table 2. PCB surface temperature versus applied voltage

Table 3. Comparison of the proposed embedded magnetorquers with the commercial designs

whereas $\alpha $ is the emissivity value photovoltaic panels which is represented in (10)

(10) \begin{align}\alpha = \frac{{{P_d}}}{{\sigma S\left( {{T_o}^4 - {T_I}^4} \right)}}\end{align}

The magnetorquers are excited for a range of 2V∼18V inputs such as those used on multi-cube nanosatellites. The plots are depicted in Figs 9 and 10. Here emissivity ( $\alpha $ ) is assumed to be 0.9 which can be measured through an experimental setup (Ali et al. 2014). The plots show that the printed magnetorquers generate sufficient torque to provide attitude manoeuvers for both the cases of 4U and 8U multi-cube small satellites. For 4U spacecraft, the torque generated by the eight coils in series (8 × 1) is 5.2µNm with a temperature rise of 25.4°C while the eight coils in parallel (1 × 8) generates a torque of 42.1µNm with a temperature increase of 73.18°C. Similarly, in case of 8U spacecraft, the torque generated by the 24 (24 × 1) coils in series is 5.2µNm with a temperature rise of 26°C while the 24 (1 × 24) coils in parallel generates a torque of 126.3µNm with a temperature rise of 133°C. This shows that, as the dimensions and cube units of spacecraft increase, torque and magnetic moment generation increases while the associated power dissipation and temperature rise subsequently increase, which corresponds to the coil’s direct dependence on trace width, size and metal spacing in the module. The analysis shows that the maximum torque to power dissipation ratio is given by the series configuration, which is due to the fact that it draws extremely low current reducing the ${I^2}R$ losses. The torque-to-power dissipation ratio also increases with the increase in the cube-unit dimensions of small satellites. The maximum torque results in case of the parallel configuration but at the expense of higher power dissipation and temperature gradient which again corresponds to the higher ${I^2}R$ losses because of the high current flow.

In order to further clear the picture of PCB surface temperature (To) rise in case of various combinations of the coils for 4U and 8U satellite units, To values against applied voltage are given in Table 2. All these values are measured at room temperature (25ºC). Due to difference in resistance for various coils combinations, the current drawn and resultant power consumed are changing. The larger is the current drawn, the higher is the power consumed and the corresponding increase in To values. As the dimension of the satellite increases, the To values against the power dissipation also increase. These results reflect that the parallel combinations should be avoided at higher input power because it results in higher PCB temperature.

Based on the magnetic moment generation, torque imparted and power dissipation requirements, the magnetorquer with additional reconfigurability features (24 × 1, 1 × 24, 8 × 3, 3 × 8, 1 × 8 and 8 × 1 etc.) gives more adaptability to the system design by altering the configuration through the onboard processor in real time. The commands from the telemetry processor unit are sent to the onboard computer from ground station for transmission based on the demanded torque and power consumption of the coil. Table 3 tabulates the performance of magnetorquer coils in comparison with the commercial best-case designs.