# Summary

Magnetic torque rods, or magnetorquers, are a type of electromagnets composed of a long range of wire looped around a magnetic core. When a current is run through the wire, a magnetic dipole moment is generated which interacts with the external magnetic field of the earth.

The basic law of electromagnetism is defined in equation ( 1 ) where($\mu$) is the magnetic moment, $N$ is the number of wire turns, $i$ is the current and $A$ is the cross-section area of the core. The magnetic moment of the magnetorquer is directly proportional to the number of wire turns, cross section of the core and the current.

$\mu=NiA$( 1 )

By controlling the amount of current running through the torque rod, the magnetic moment may be increased or decreased. If the earth’s magnetic field is not parallel to the actuator, the amount of torque generated is defined by the cross product represented in equation ( 2 ) where $\tau$ is the torque generated and $B_E$ is the external magnetic field of the Earth. By changing the magnetic moment vector, the torque induced on the CubeSat may be controlled as well. 

$\tau=\mu \times B_E$ ( 2 )

# Types of Magnetorquers

There are three types of magnetic torquers: solid core, air-core and embedded. For solid and air-core magnetorquers, the long wire is wound around either a solid ferrous material or a vacuum, respectively.

## Embedded Magnetorquers

For embedded magnetorquers, the spiral trace is inside the PCB of the solar panels which generates the effect of the coil. Although the embedded option has the least impact on the weight, space and power constraints of the CubeSat, due to the physical limitations it is both difficult to implement and inefficient. Embedded torque rods were not researched further due to the physical limitations.

## Air Core Magnetorquers

Air core magnetorquers are a large bundle of wire looped around a cross-sectional area for a given number of turns to generate the torque on the CubeSat. From equation ( 1 ) , to generate a large magnetic moment, the number of turns, current and cross-sectional must increase. Without a solid core, the amount of resistance in the torque rod is minimal and hence the power consumption of the air-core torquers is relatively low; however, to achieve higher magnetic moments the cross-sectional area and number of wire turns must be increased to compensate. Since the CubeSats are constrained to a maximum of 1U, the only variable that can be adjusted is the number of wire turns. As the number of wire turns increases, so does the mass.

## Solid Core Magntorquers

Solid core magnetorquers are essentially the same as the air-core magnetorquers, but with a magnetic ferrous material that the wire coils around instead of the vacuum. By introducing a ferromagnetic core, the efficiency of the torque rod is increased depending on the type of material used for the core and hence the number of wire turns, and cross-sectional area of the core may be proportionally decreased.  

# Magnetorquer Design

Both the air-core and solid-core magnetorquers have their advantages and disadvantages and it is up to the ADCS team to weigh their trade-offs and decide which type of rod is necessary for the given pay-load. The design process we used involved setting payload constraints, running our basic calculations, increasing the complexity and revising our assumptions. By iterating through the calculations, it allowed us to continually optimize and redesign for the final selection.

This is a summary of the detailed analysis of the magnetorquer design discussed here.

## Constraints

Our initial maximum constraints are defined by the CubeSat Design Specification  and the rules set by the CSDC  and are summarized in the table below:

Parameter Value
Pointing Accuracy <1.5°
Mass 4kg
Dimensions 10cm x 10cm x 30cm
Required Roll Angle <25°
Operate LEO 400km-800km
Pass Time 8min

To achieve a pointing accuracy <1.5﮿, magnetorquers will not be enough and reaction wheels will be necessary. From these constraints, as well as for design simplicity, the magnetorquer design will be the same for all three axes. Additionally, “The attitude Control subsystem shall be designed so that it is tolerant to at least one attitude actuator failure and will maintain at least a degraded state of performance.”  Although we will be designing with the intention of introducing reaction wheels, our magnetorquer design will be made such that it could still operate alone.

The maximum satellite mass is defined as 4kg. For our Satellite Design Team, there are 5 sub-teams, and the overall mass divided equally amongst the team is 800gr. For our magnetorquer design, we decided we would design for a constraint of no more than 20% of our maximum sub-team mass budget.

$800g * 20\% = \frac{160g}{3magnetorquers}=\frac{53.333g}{magnetorquer}$

In a single plane, the maximum length a single magnetorquer could be is 10cm as that is the maximum length of the X or Y axis. In practice, the rod must be shorter than 10cm to consider mounting brackets fastened to the ends, but for our calculations we assumed a maximum of 10cm.

The required roll angle is the maximum rotation necessary to point the camera at the target. As this constraint is the minimum roll angle required, our design considered the maximum roll necessary to achieve the same goal. For our design we used a maximum roll angle of 180﮿.

The CubeSat will be expected to operate in Low Earth Orbit, which is approximately 400km-800km above the earth’s surface. Since Low Earth Orbit may still have some atmospheric drag, for our calculations, we must consider the point at which the atmospheric drag is highest: the perigee.  For our calculations, we assume the satellite will be operating at the minimum 400km above the earth.

During a Q&A session with the CSDC Management Society for the mission requirements, a single “pass” was undefined. After receiving clarification, a single pass was defined as receiving the Mission Control Centre uplink, rolling into position, taking the image and downlinking the image to the Amateur Radio Operator. Since the downlink time is further constrained to be <1.5min, the rotation has a maximum of 6.5min to complete. For our calculations, we decided to design for the rotation to complete in a maximum of 4 minutes, which allows for delays and error.

Lastly, our design has the assumption the power bus will be 3.3V and the maximum power consumption is 0.2W. These values are provided by the power sub-system.

The individual magnetorquer constraints discussed are summarized in the table below:

Parameter Value
Mass <53.33g
Length <10cm
Width <10cm
Maximum Roll <180°
Operate at perigee 400km
Slew time <4min
Voltage 3.3V
Power 0.2W

## Moment Calculation

A magnetorquer is defined by the core material, wire material and the number of turns the wire is wrapped around the core, as defined in equation (1). If we can calculate the minimum magnetic moment required to rotate the satellite defined by our constraints, we can extrapolate the necessary parameters to define the magnetorquer. To solve for the minimum magnetic moment, we can use the equation defined in (2).

To start, we must find the magnetic field generated by the earth, or geomagnetic field; however, the earth’s magnetic field fluctuates depending on the altitude, time of year and where the satellite is located above the earth.  Since the amount of torque generated is proportional to the strength of the magnetic field it is acting upon, the point at which the magnetic field is at its minimum strength will represent the least amount of torque the magnetorquers can generate.

The earth’s magnetic field may be simplified by comparing it to a dipole magnet, although it is much more complex and is represented by a spherical harmonic expansion calculated by the International Geomagnetic Reference Field (IGRF).  Just as a simple bar magnet generates its magnetic field, the earth’s magnetic field leaves the north pole and re-enters at the south pole; however, the magnetic poles are not located at the geographic poles.

The geomagnetic field is weakest when it is furthest from the magnetic poles, called the magnetic equator, specifically in the South Atlantic Anomaly region ; if the designed magnetorquers can actuate the satellite at the magnetic equator, it can actuate the satellite at any given point in low earth orbit. For our calculations, the IGRF model provided by the Canadian Government was used to generate a vector representing the geomagnetic field at the magnetic equator. 

Earth's Magnetic Field Magnitude [T]
$B_{Ex}$ 27538e-9
$B_{Ey}$ -2280e-9
$B_{Ez}$ -16223e-9

Next is the torque vector, which we can find from equation:

$\tau=I\alpha$ (3)

Since we have already worked out the moment of inertia, which was summarized in the Mass Moment of Inertia explanation, we can use formula (3) here by simply finding the angular acceleration necessary to calculate the minimum torque which we can get from the basic angular rotation formula.

$\theta=\omega_0t+\frac{1}{2}\alpha t^2$

$\alpha = \frac{2}{t^2}(\theta -\omega_0t)$ ( 10 )

If we assume the satellite has no initial velocity at this point, then then equation for angular acceleration may be further simplified to equation (11). Given this is a simplified model for designing the magnetorquer, this calculation may not be used for detumbling and we must revisit to introduce perturbations for the controller algorithm.

$\alpha_{y,180}=\frac{2}{t^2} \begin{bmatrix} \cos{\theta} & 0 & \sin{\theta} \\ 0 & 1 & 0 \\ -\sin{\theta} & 0 & \cos{\theta} \\ \end{bmatrix} =\frac{2}{t^2} \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \\ \end{bmatrix}$ ( 11 )

Given our constraints defined for slew time and angle of rotation, we can simply plug in the values and find the minimum desired angular rotation to be the following:

Angular Acceleration Magnitude [$rad/s^2$]
$\alpha_x$ -3.4722e-5
$\alpha_y$ 3.4722e-5
$\alpha_z$ -3.4722e-5

By using equation (3), we multiply the moment of inertia for our CubeSat with the desired angular acceleration and we find the minimum torque required to rotate the satellite to be summarized as:

Torque Magnitude [Nm]
$\tau_x$ -1.157e-6
$\tau_y$ 1.157e-6
$\tau_z$ -2.3148e-7

Lastly, be rearranging the equation in formula (2), we can solve for the magnetic moment. The formula are derived in Appendix A for the detailed write-up.

Torque Magnitude [Nm]
$\mu_x$ $[\tau_z + B_{Ex}(\tau_x + \mu_zB_{Ey})]\frac{1}{B_{Ey}}$
$\mu_y$ $(\tau_x + \mu_zB_{Ey})\frac{1}{B_{Ez}}$
$\mu_z$ $\frac{[\tau_y + \frac{\tau_z}{B_{Ey}} + \frac{B_{Ex}}{B_{Ez}B_{Ey}}\tau_x]}{B_{Ex} - \frac{B_{Ex}}{B_{Ez}}}$

Notice $\mu_x$ and $\mu_y$ are in terms of $\mu_z$, so to solve these equations we simply plug the previously calculated values into µz first and then back substitute. The calculations were executed by Matlab and the code is in Appendix B.

Based on a slew rate of less than 4 minutes and the minimum torque listed above, the minimum magnitude of the magnetic moment for our design is 0.1301 Nm/T.

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