Editor’s Note: This article deals with complex mathematics and requires background knowledge to understand fully. As prior reading, the following articles and videos are recommended:

• Khan Academy: Introduction to Reference Frames
• Wikipedia: Rotation of Axes in Two Dimensions
• Khan Academy: Dot Product
• Khan Academy: Dot vs. Cross Products
• Khan Academy: Dot and Cross Products Using Unit Vector Notation

Introduction

The objective of the Attitude Determination and Control System (ADCS) is to maintain the spacecraft’s orientation within the desired limit with respect to a reference frame. To design a control system for a satellite, we need to understand the satellite’s dynamics. This article will introduce Hughes’s Vectrix notation, which is widely used in dynamic systems design.

Vectrix notation was invented by Professor Peter Carlisle Hughes (1940-2023) at the University of Toronto Institute of Aerospace Studies (UTIAS). Initially published in 1986 in the reference book ”Spacecraft Attitude Dynamics” [1], vectrix notation has been continuously taught in university courses and used by many professionals in the field.

In astrodynamics, we have to deal with physical phenomena in multiple reference frames. For example, the sun vector can be located in the Earth-Centered Inertial frame (ECI), but for ADCS, the sun vector must be converted into the satellite body frame, which is a non-inertial frame. Dealing with complex dynamic systems across multiple reference frames can be challenging. Vectrix notations offer a superior approach to representing diverse dynamic phenomena in a specific reference frame, which can enable us to demonstrate how changes in satellite motion can be represented in multiple frames [2].

What is a vectrix?

A vector is a useful way to define a quantity having direction as well as magnitude, this is why vectors are used widely in many branches of physics and engineering. Let us consider a vector \underrightarrow{v} in three-dimensional space. In order to give this vector a magnitude and direction, we must define a reference frame.

We can denote by \mathcal{F}_a a reference frame having basis vector \{\underrightarrow{a_1},\underrightarrow{a_2},\underrightarrow{a_3}\} of unit length. Note that only dextral orthonormal sets of basis vectors – that is, right-handed, orthogonal and normalized as the Cartesian coordinate system – can be used. Using this notation, we can express \underrightarrow{v} as:

\begin{equation}
    \underrightarrow{v} = \underrightarrow{\mathcal{F}_a^T} \bold{v} =  v_{1} \underrightarrow{a_{1}} + v_{2} \underrightarrow{a_{2}} + v_{3} \underrightarrow{a_{3}}
\end{equation}

where

\begin{equation}
    \bold{v} \triangleq \begin{bmatrix}
    v_{1} \\ v_{2} \\ v_{3}
    \end{bmatrix}
\end{equation}

is the column matrix containing the components of vector \underrightarrow{v} as resolved in \mathcal{F}. Therefore, \underrightarrow{\mathcal{F}} is the vectrix associate with \mathcal{F} and defined as

\begin{equation}
    \underrightarrow{\mathcal{F}} \triangleq \begin{bmatrix}
    \underrightarrow{a_1} \\ \underrightarrow{a_2} \\ \underrightarrow{a_3}
    \end{bmatrix}
\end{equation}

Note that the result may seem like a 1 \times 1 matrix because of the multiplication of a 1 \times 3 matrix with a 3 \times 1 matrix, but the result is a vector because \{\underrightarrow{a_1}, \underrightarrow{a_2}, \underrightarrow{a_3}\} are basis vectors so a vectrix is a 3 \times 3 matrix.

Therefore, we can also see the following result using the Vectrix notation.

\begin{equation}
    \underrightarrow{v} = \underrightarrow{\mathcal{F}_a^T} \bold{v} = \bold{v^{T}}\underrightarrow{\mathcal{F}_a}
\end{equation}

We note that the representation of \underrightarrow{v} in a given reference frame makes the vector a 3 \times 1 form that we are all familiar with. Vectrix notation explicitly separates reference frame and vector representation.

Dot Products and Cross Products

This section includes an overview of how the dot and cross products are calculated using vectrix notation.

Dot Product

Let us consider two vectors \underrightarrow{u} and \underrightarrow{v} expressed in the same reference frame \mathcal{F}.

\begin{equation}
    \underrightarrow{u} = \underrightarrow{\mathcal{F}^T} \bold{u} = \bold{u^T} \underrightarrow{\mathcal{F}} 
\end{equation}

\begin{equation}
    \underrightarrow{v} = \underrightarrow{\mathcal{F}^T} \bold{v}
\end{equation}

The dot product (inner product) of \underrightarrow{u} and \underrightarrow{v} is denoted by

\begin{equation}
    \underrightarrow{u} \cdot \underrightarrow{v} =  \bold{u^T} \underrightarrow{\mathcal{F}} \cdot \underrightarrow{\mathcal{F}^T} \bold{v} 
\end{equation}

We can see that by matrix multiplication and distributing the dot product operator to the elements of \underrightarrow{\mathcal{F}} \cdot \underrightarrow{\mathcal{F}^T} and invoke the dot products for a pair of basis vectors, we get the identity matrix \bold{I}.

\begin{equation}
    \underrightarrow{\mathcal{F}} \cdot \underrightarrow{\mathcal{F}^T} = \bold{I}
\end{equation}

This also further serves as a formal statement of orthonormality. Finally, we can find the dot product for two vectors.

\begin{equation}
    \underrightarrow{u} \cdot \underrightarrow{v} = \bold{u^T}\bold{v} = u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}
\end{equation}

Therefore, we can see the result of the dot product is a scalar.

Cross Product

Let us also consider the cross product of two vectors \underrightarrow{u} and \underrightarrow{v}.

\begin{equation}  
\underrightarrow{u} \times \underrightarrow{v} =  \bold{u^T} \underrightarrow{\mathcal{F}} \times \underrightarrow{\mathcal{F}^T} \bold{v}
\end{equation}

To avoid going into lengthy proof, we will define the cross operator (\cdot)^{\times}. We can apply this operator on a 3 \times 1 \bold{u} as:

\begin{equation}
    \bold{u}^{\times} = \begin{bmatrix}
        0 & - u_{3} & u_{2} \\
        u_{3} & 0 & -u_{1} \\
        -u_{2} & u_{1} & 0
    \end{bmatrix}
\end{equation}

This gives us the cross product of \underrightarrow{u} and \underrightarrow{v} as follows:

\begin{equation}
\underrightarrow{u} \times \underrightarrow{v} = \underrightarrow{\mathcal{F^{T}}} \bold{u}^{\times} \bold{v} =\underrightarrow{\mathcal{F^{T}}} \begin{bmatrix}
        
     u_{2}v_{3}  - v_{2}u_3\\ 
    u_{3}v_{1} - v_{3} u_{1}\\ u_{1}v_{2} - v_{1}u_{2} \end{bmatrix} 
\end{equation}

The cross-product gives us a vector in the \underrightarrow{\mathcal{F}} reference frame.

Rotation Matrices

The vector \underrightarrow{v} can have many equivalent expressions in different reference frames.

\begin{equation}
\underrightarrow{v} = \underrightarrow{\mathcal{F}_a^T} \bold{v_a} = \underrightarrow{\mathcal{F}_b^T} \bold{v_b}
\end{equation}

We can find \bold{v_a}from \bold{v_b} by changing \bold{v_b} from frame \underrightarrow{\mathcal{F}_b} to frame \underrightarrow{\mathcal{F}_a}.

\begin{equation}
\bold{v_a} = \underrightarrow{\mathcal{F}_a} \cdot \underrightarrow{\mathcal{F}_b^{T}} \bold{v_b}
\end{equation}

We can define \bold{C_{ab}}, the rotation matrix from frame \underrightarrow{\mathcal{F}_b} to frame \underrightarrow{\mathcal{F}_a}, as:

\begin{equation}
\bold{C_{ab}} \equiv \underrightarrow{\mathcal{F}_a} \cdot \underrightarrow{\mathcal{F}_b^{T}}
\end{equation}

Therefore,

\begin{equation}
\bold{v_a} = \bold{C_{ab}} \bold{v_{b}}
\end{equation}

We can say that \bold{C_{ab}} describes the orientation or attitude of \mathcal{F}_a with respect to \mathcal{F}_b. We can expand equation 15 to obtain equation 17, in which \theta_{ij} is the angle between \underrightarrow{a_{i}} and \underrightarrow{b_{j}}.

\begin{equation}
    C_{ab} \equiv \begin{bmatrix} a_{1} \cdot b_{1} & a_{1} \cdot b_{2} & a_{1} \cdot b_{3} \\
    a_{2} \cdot b_{1} & a_{2} \cdot b_{2} & a_{2} \cdot b_{3} \\
    a_{3} \cdot b_{1} & a_{3} \cdot b_{2} & a_{3} \cdot b_{3}
    \end{bmatrix} \equiv \begin{bmatrix}
        \cos{\theta_{11}} & \cos{\theta_{12}} & \cos{\theta_{13}} \\
        \cos{\theta_{21}} & \cos{\theta_{22}} & \cos{\theta_{23}} \\
        \cos{\theta_{31}} & \cos{\theta_{32}} & \cos{\theta_{33}}
    \end{bmatrix}
\end{equation}

We can see that the rotation matrix is composed of the directional cosines between the two sets of basis vectors.

Principal Rotations

We can consider a few special cases of \underrightarrow{\mathcal{F}_a} being displaced from \underrightarrow{\mathcal{F}_b} by a rotation through an angle $\theta$ (using the right-hand rule) about one of the principal axes. This is referred to as a principal rotation.

First let’s consider the rotation about \underrightarrow{b_1}. This will keep \underrightarrow{b_1} parallel to \underrightarrow{a_1} and the rotation matrix reduces to:

\begin{equation}
    C_{ab} = \begin{bmatrix} 1 & 0 & 0 \\
    0 & \cos{\theta} & \cos{(\frac{\pi}{2}-\theta)} \\
    0 & \cos{(\frac{\pi}{2}+ \theta)} & \cos{(\theta)} 
    \end{bmatrix} \equiv \begin{bmatrix}
        1 & 0 & 0 \\
        0  & \cos{(\theta)} & \sin{(\theta)} \\ 0 & -\sin{(\theta)} & \cos{(\theta)}
    \end{bmatrix} \equiv C_1
\end{equation}

This defines a principal rotation and, since it is a rotation about the 1-axis, we can denote it by C_1. The other principal rotations about the 2-axis (\underrightarrow{b_2}) and 3-axis (\underrightarrow{b_3}) can likewise be written as:

\begin{equation}
C_{2} \equiv
\begin{bmatrix}
\cos{(\theta)} & 0 & -\sin{(\theta)} \\
        0  & 1 & 0 \\ \sin{(\theta)} & 0 & \cos{(\theta)} \end{bmatrix}, \: \: C_{3} \equiv
\begin{bmatrix}
\cos{(\theta)} & \sin{(\theta)} & 0 \\
        -\sin{(\theta)}  & \cos{(\theta)} & 0 \\ 0 & 0 & 1 \end{bmatrix}
\end{equation}

These are the three principal rotation matrices.

Euler Rotation

Euler’s Theorem provides a way to represent multiple rotational transformations as a single rotation about a unique axis passing through a fixed point, offering a concise and widely used framework for describing rotations in three-dimensional space. Specifically, it states that any general displacement of a rigid body with one point fixed can be represented as a rotation about an axis through that point.

Directional cosine matrices can be expressed as Euler-axis rotations about a unit-normal axis \underrightarrow{a} and a rotation angle \phi.

\begin{equation}
    C(a,\phi) = \cos\phi \bold{I} + (1-\cos \phi)aa^{T} - sin\phi \; a^{\times}
\end{equation}

The rotation axis \underrightarrow{a} can be expressed in either the initial or final frame of the transformation.

Other Properties of Rotation Matrices

The inverse transformation is physically a rotation through the same angle in the opposite direction. The inverse of a rotation matrix is also its transpose, therefore, we can transfer from \underrightarrow{\mathcal{F}_a} back to \underrightarrow{\mathcal{F}_b} by taking the transpose of the rotation matrix.

\begin{equation}
    C_{ab}^{-1} \equiv C_{ba}  \equiv C_{ab}^{T} 
\end{equation}

If we introduce another reference frame \underrightarrow{\mathcal{F}_c} obtained from \underrightarrow{\mathcal{F}_b} via another rotation, then the components of \underrightarrow{v} in this new frame in terms of its components in \underrightarrow{\mathcal{F}_a} are given by:

\begin{equation}
    \bold{v_{c}} = C_{cb}v_{b} = C_{cb}C_{ba}v_{a} = C_{ca}v_{a}
\end{equation}

Where C_{ca} = C_{cb}C_{ba} is the rotation from \underrightarrow{\mathcal{F}_a} to \underrightarrow{\mathcal{F}_c}. Note that the order of rotation is important.

In conclusion, vectrix notation provides a means of representing a vector within a specific reference frame, and through the application of linear algebra, we can define dot and cross products within that same reference frame. Moving forward, the application of vectrix notation is especially valuable when working with dynamic systems in multiple reference frames, such as satellites or spacecraft orbiting Earth or another celestial body.

References

1. P. C. Hughes, “Spacecraft Attitude Dynamics,” Wiley, New York, 1986.
2. G. M. T. D’Eleuterio and G. R. Heppler, Newton’s Second Law And All That. Toronto, Ontario: University of Toronto, 2019.

What is ADCS?

The Attitude Determination and Control System (ADCS) is a fundamental part of every satellite mission. The objective of the ADCS is to maintain the spacecraft’s orientation within the desired limit with respect to a reference frame. The capability to actively control a spacecraft’s attitude empowers engineers to utilize a wide range of payloads.

The technology for ADCS has undergone significant expansion since the launch of Sputnik 1 on October 4, 1957. ADCS has evolved from simple spin gyroscope stabilization methods to complex optimization control strategies utilizing powerful onboard computers. The tremendous growth in ADCS technology has enabled the development of CubeSats capable of performing complex tasks comparable to those of larger satellites.

Most satellites are equipped with some form of ADCS, which can be either passive or active. Active attitude control involves the use of sensors and internal actuators, while passive control relies on natural forces, such as the Earth’s gravity or magnetic field, to orient the satellite. Satellites that rely solely on passive control are typically used for technology demonstrations, while those with active control have a wider range of applications, including earth observation, communication, scientific research, and technology demonstration.

A simple onboard ADCS for active control is shown below in Figure 1.

The design of ADCS for active control satellites typically involves a range of components, including sensors, actuators, control laws, attitude estimators, and space environment models. Additionally, a means of determining the satellite’s position in orbit is necessary; an error in position estimation will translate to errors in attitude estimation. A mission evaluation must be conducted to determine the appropriate hardware and software. This evaluation should also include pointing budget analysis, actuator sizing and simulation in order to ensure optimal performance. Mission evaluation will be discussed further in its own section.

ADCS Sensor Hardware

A satellite’s ADCS relies on onboard sensors to determine its attitude. CubeSats commonly use magnetometers and sun sensors, but gyroscopes and star trackers can be added when higher accuracy is needed.

A three-axis magnetometer is usually used to measure the local magnetic field vector and compare it with an onboard International Geomagnetic Reference Field (IGRF) ephemeris for attitude estimation. Some satellites have their magnetometers on protruding booms to provide magnetic isolation from the spacecraft body, but most satellites can have magnetometers placed internally without any detrimental effects. Satellites rely on fine sun sensors to measure the sun vector and combine it with an onboard Astronomical Almanac Sun Vector (AASV) ephemeris to facilitate attitude estimation. A very small number of satellites utilize solar panels to measure the sun vector, but this approach is only feasible when a sufficient number of solar panels are present, and its accuracy is not as high as that of sun sensors. By combining sun sensor and magnetometer measurements, full three-axis attitude information can be determined.

Other sensors can also be used in place of or in addition to magnetometers and sun sensors, though they are less common. Star Trackers are optical sensors that take a star picture to estimate the satellite’s attitude in all three axes, but require a satellite to be largely stable (i.e. not spinning without control). Earth horizon sensors are available, but they are not utilized as frequently as star trackers due to their limited accuracy. Gyroscopes can be used to more accurately determine the angular rotation of a satellite than measuring the change in the magnetic field with a magnetometer. Adding these advanced sensors can enable more missions with more challenging requirements, and can also increase redundancy if some of them fail during the satellite’s lifetime.

For satellites that require a high degree of pointing accuracy, Global Navigation Satellite System (GNSS) technology is often utilized for determining position, while other satellites relied on ground tracking and orbital propagator software. Although the orbital state is not an attitude parameter per se, the addition of GNSS technology improves the ability to perform attitude determination since sensors such as sun sensors and magnetometers rely on space environment models that are dependent on the satellite’s current position. An error in position estimation can lead to an error in attitude estimation. While incorporating GNSS technology can enhance a satellite’s performance, there are regulations that exist concerning the integration of GNSS systems on board.

ADCS Actuator Hardware

To control the orientation of a spacecraft, the ADCS requires actuators. Magnetic control is a common approach used by many satellites, which involves generating a local magnetic field that interacts with Earth’s magnetic field to control the spacecraft’s attitude. Passive magnetic control using permanent magnets or active magnetic control using magnetorquers are the most common methods.

Reaction wheels are also frequently used in satellites that require precise control. They work by storing momentum and using the principle of conservation of angular momentum to stabilize the spacecraft. Control Moment Gyroscopes (CMGs) use the same principle as reaction wheels for attitude control. CMGs are more commonly used for larger satellites, but there has been a trend toward miniaturizing CMG technology for use in CubeSats. Momentum dumping is necessary for both reaction wheels and CMGs to prevent saturation, which happens when the wheels or gyroscope reach their maximum rotational speed. Momentum dumping is often handled using magnetorquers on CubeSats.

Other actuator technologies, such as propulsion systems, gravity gradient booms, and solar sails, are rarely used due to their complexity and cost.

ADCS Software

Onboard software governs the ADCS, making all decisions related to spacecraft attitude including mode transitions, executing external commands, conducting attitude estimation, and evaluating control laws. Typically, sensor measurements and ground station commands serve as input to the software, while output commands are directed toward actuators.

The flight code for ADCS must incorporate sensor processing software, which receives and filters measurements from sensors. The software then converts the filtered measurements into a measured ephemeris vector. The ephemeris vector is utilized by an attitude estimator to compare with an onboard astrodynamical model to get the spacecraft’s estimated attitude.

Attitude estimation algorithms such as the Attitude Extended Kalman Filter (AEKF), QUEST and TRIAD are commonly used for CubeSats. AEKF is one of the most commonly use attitude estimation algorithms because it can process non-linear errors with low computation cost. Attitude estimation methods such as the AEKF linearize the satellite equations of motion about a discrete trajectory and propagate it through time.

The main purpose of an attitude controller is to evaluate the appropriate control torque. The control’s torque then gets transmitted to the actuators to control the attitude and the body’s angular rates. The choice of control law depends on the mission objective but there is a trade-off between computation time (accuracy) and power consumption. Tuning the control gains (parameters) and implementing them in a discrete fashion is required to achieve optimal operation. The system bandwidth, inherent errors, and actuator saturation limits place constraints on overall performance, emphasizing the importance of selecting appropriate hardware and software for a given mission.

What to Expect From ADCS Articles

In the upcoming articles, I will be discussing the mathematics involved in dynamic system satellites, as well as exploring different reference frames and timekeeping methods. We will also delve into the space environment and operational conditions, including topics such as disturbance torque. Afterwards, I will cover critical aspects of mission planning including how to set ADCS requirements, mission analysis and modes of operation. I will also discuss the design of GNC software and the selection of appropriate hardware. Lastly, we’ll delve into the crucial process of ADCS system integration and testing, ensuring a comprehensive understanding of this complex topic.

Introduction

When designing an RF transmitter or receiver, choosing an amplifier (or amplifiers!) is one of the most critical tasks for ensuring that it functions optimally. If it’s your first time, though, it can be difficult to parse through all of the information provided on datasheets to try and find an optimal part, especially if you don’t really know what your looking for. This article is going to focus on RF amplifiers, and how you can ensure that you’re choosing a good one for your transmitter or receiver.

To help you choose an amplifier, I’m going to go step-by-step through several of the more important characteristics you’ll need to know. This is by no means an exhaustive list, but in broad strokes you can go through this list one by one, in order, to narrow down your search.

1. Frequency Range
2. Characteristic Impedance
3. Gain
4. Is it a LNA or a power amplifier?
   1. Noise Figure
   2. P1dB
   3. Psat
5. Power Added Efficiency & Heat dissipation
6. Maximum Input Power
7. Temperature Range/Stability
8. Input/Output VSWR
9. Operating Bias Voltage
10. Operating Current
11. Form Factor/Case Suitability

Prerequisite Reading

1. Overview of Key Terms – Antennas
2. Noise Figure and the Friis Formula for Noise
3. Noise Floor

Important Information

It’s important to note a few assumptions before getting into the weeds. In this article I’m focusing on amplifiers for the most common types of transmitters/receivers used on CubeSats today, single center frequency, with constant envelope modulation schemes. If you’re planning on building a multi-frequency system or one with a modulation scheme that isn’t constant envelope (like On-Off Keying or Amplitude Modulation), you’ll probably need to do a bit of extra legwork in order to make sure the parts you’re choosing will satisfy your requirements. That being said many of these terms and factors of merit are still relevant, so read on, just be aware that this article isn’t going to go as in-depth as you may need it to.

When explaining a few of these items on the list above, I’ll be referencing a Mini-Circuits component datasheet for ease of understanding, referenced in the “References” section at the bottom. It’s important to note that not all of these parameters will always be available in every part’s datasheet. Sometimes you’ll have to calculate them yourself based on other information that’s in the datasheet, and sometimes manufacturer’s just don’t include what you’re looking for. Here I want to emphasize, as with all things in the CubeSat world, the importance of testing. Always make sure you have time to test the device you’re building thoroughly, as the real world doesn’t always jive perfectly with what you find in datasheets!

1. Frequency Range

The frequency range is the most important and easiest way to narrow down your search for an amplifier, and it’s as simple as it sounds. Determine the operating frequency of your transmitter/receiver, and make sure that it’s within the operating frequency range of the radio you’re trying to build. On the example datasheet shown in Figure 1, note the frequency range circled in green and marked with a “1”.

2. Characteristic Impedance

The characteristic impedance (or Z₀) of an amplifier is the impedance which it’s designed to be fed by and output to. If you don’t remember what an impedance is, check this article for a refresher. Almost all high frequency electronic components these days are designed to have 50 ohms as their characteristic impedance. 75 ohms was once commonly used for things like broadcast television sets, but it’s becoming rarer. Because of their ubiquity and low cost, I’d recommended that you stick with 50 ohm components unless you have a very pressing reason to switch. The characteristic impedance is circled in Figure 1 and labeled with the number “2”.

3. Gain

An amplifier’s gain describes the ratio of the signal power output from the amplifier to the input power while the power is within the amplifier’s linear region (more on that in section 4).

G = \frac{P_o}{P_i}

The equation above is for linear gain G, where output power is P_o and input power is P_i. You can use any units for power, as long as input and output power use the same units. Gain, being a ratio, is a unitless quantity.

As with most quantities in communications, you’ll often see gain listed as a logarithmic quantity with units of decibels (dB). You can convert from linear gain to logarithmic gain using the equation

G_{dB} = 10*log_{10}(G_{linear})

Correspondingly, you can convert from logarithmic gain to linear gain using

G_{linear} = 10^{\frac{G_{dB}}{10}}

The gain is labelled in Figure 2 with the number “3”.

ADDENDUM ON DATASHEETS

In the specifications section (Figure 2), note that there are several columns. This section is a quick explanation on each.

Condition (GHz) – The frequency the parameter is measured at (in this case, in GHz).

Vd=5.0V/Vd=3.0V – Indicates that values were measured using a 5 Volt or 3 Volt bias voltage

Min./Typ./Max. – Minimum/Typical/Maximum measured values within acceptable production variation range. You can normally design using the typical value, but beware that each part may have some variance within the minimum/maximum range.

Units – The units of the measured value.

4. Use-Case Dependent Quantities

Amplifiers are used in many places in a CubeSat communication system. Two of the most common uses are Low-Noise Amplifiers (LNAs) in receivers and Power Amplifiers (PAs) in transmitters. The actual construction of amplifier circuits is beyond the scope of what we cover on Crashcube, but essentially you’ll need to know two things. LNAs are designed, as you might expect based on their name, to have a low noise figure. PAs, on the other hand, are designed to have their linear regions extend up to higher output power values.

4-1. Noise Figure

All amplifiers are inherently imperfect, and each will introduce some amount of noise into the signal it’s amplifying. The noise figure of an amplifier is a quantity that describes how much noise is in the signal at the output of the amplifier relative to the noise at the input. This is quantified as a signal to noise ratio (SNR). The noise figure of an amplifier is defined as the ratio of the input to output signal-to-noise ratios (a ratio of ratios, I know that’s a little bit confusing). Similarly to the way gain can be expressed as a logarithmic quantity, noise figure is also often expressed in decibels.

NF = \frac{SNR_i}{SNR_o}

NF_{dB} = 10*log_{10}(\frac{SNR_i}{SNR_o}) = 10*log_{10}(SNR_i) - 10*log_{10}(SNR_i)

If you look closely, you’ll notice that noise figure has the input and output values inverted compared to the equation for gain. This is because we want to have positive values for noise figure (due to both common convention and to make some of the equations we use it in easier to manage), and because an amplifier will ALWAYS have a lower SNR at its output compared to its input.

Why is this, though? It’s important to remember that, while we distinguish between the signal and the noise when doing analysis, to an amplifier they both appear together as a single input current and voltage. There’s no way for it to distinguish or separate the two. This means that when a signal (with some amount of noise) goes into an amplifier, it amplifies both signal and noise equally. Since in reality, all amplifiers will also add some additional amount of noise to the signal they amplify, this means that the noise value at the output will ALWAYS be higher than at the input.

The noise figure of an amplifier is more important in the case of LNAs than it is for PAs. This is because typically an LNA is being used in a receiver to amplify a very weak signal that is closer to the noise floor. As an example, let’s say your input power is -90 dBm and you have a noise floor power of -110 dBm. If you have an amplifier with a noise figure of 12 dB, your output SNR after amplification is only 8 dB. Depending on what modulation and error correction methods you’re using on your satellite link, this may not be enough to decode it properly! If you use an LNA with a lower noise figure, such as 3 dB, it may be possible to decode that same signal properly since its output SNR would then be much higher at 17 dB.

For a PA, this is less important because the input power to the amplifier is often far higher than the noise floor. As an example, a typical power amplifier might have 30 dB of gain and be fed with a signal that’s been generated by a transceiver at 0 dBm. If the noise floor power for this example system is -80 dBm before amplification, an amplifier with a noise figure of 12 dB would increase the signal power to 30 dBm and the noise power to -38 dBm. This increase will have very little impact on the quality of the transmitted signal. The SNR in this case would be reduced from 80 dB at the input to 68 dB at the output.

Once the signal travels all the way across space to a receiver on the ground, both the signal and the noise will be greatly reduced in power due to geometric spreading out of the power density. Let’s assume that the geometric spreading losses are 120 dB and that the receiver has a similar noise floor power to the one in the first half of this example, -110 dBm. Our hypothetical receiver now has a signal input power of -90 dBm, above the noise power floor, but the noise that was amplified by the PA at the beginning is now at -158 dBm, far below the level that is inherently within the receiver from external noise sources.

Table 1 below shows numerically what was described above.

4-2. P1dB Compression Point

Amplifiers are generally linear devices, but only up to a point. By a linear device, I mean that the gain of an amplifier is not affected by the input power of the signal. For a 20 dB amplifier whose linear range encompasses both, for example, a -50 dBm input signal and a -20 dBm input signal should both be amplified equally (to -30 dBm and 0 dBm output, respectively).

The linear range of an amplifier is defined as the range of output power values where the amplifier will operate as a linear device. Typically, an amplifier will operate linearly for all power values up to the point where it becomes compressed (i.e. there’s no relevant lower bound other than the noise floor). The P1dB compression point is used as the upper bounding point of the linear region, and refers to the output power when the amplifier loses 1 dB of linearity (also sometimes denoted as OP1dB).

Figure 3 below illustrates the linear region and where it breaks down near the P1dB compression point, becoming non-linear. You’ll notice that on this plot there are a few other points labelled, the second-order and third-order intercept points. These (particularly the third-order intercept, IP3) are used for measuring and quantifying the linearity of devices. They are more complicated to explain conceptually than the P1dB compression point, so I won’t do so in this article, however I highly recommend this video from Rohde & Schwarz [3] that I think explains them very well.

The P1dB compression point is more relevant to power amplifiers used in transmitters than it is for the amplifiers used in receivers. This is because in almost all cases you’ll want to have a high transmit power to make it easier to close your link budget. It’s important to choose an amplifier whose P1dB compression point is at or above your desired output power so it won’t be stressed trying to amplify an already too-strong signal. Stressing parts more than they need to be is a quick and easy way to cause them to fail, and for something as critical as your transmitter, having an amplifier fail could mean your whole satellite fails!

4-3. Psat

Sometimes datasheets for amplifiers will also include a value P_{sat} that refers to the saturated power point. This is the absolute maximum output power level of an amplifier, where the slope of the input-output curve becomes horizontal. It is always above (at a higher power level than) the P1dB compression point.

While it’s generally preferable to operate an amplifier below the P1dB point, if using a modulation with a constant power envelope (such as BPSK or QPSK, among others), it is possible to operate closer to the saturation region with careful design. Generally though, it’s better to avoid it when designing smallsat systems. The extra margin of error you get from being a bit more conservative is almost always worth it when it comes time to test your devices.

5. Power Added Efficiency (PAE) and Heat Dissipation

Power Added Efficiency refers to how efficient the amplifier is while taking into account the gain of the amplifier, and can be described by the below equation, where P^{RF}_{out} is the output RF power, P^{RF}_{in} is the input RF power, and P^{DC}_{in} is the DC power consumption/dissipation. It’s important here to note that P^{RF}_{out} - P^{RF}_{in} will NOT equal the gain of the amplifier! This is because you need to use linear power units (such as milliwatts, Watts, etc.) instead of logarithmic ones for this equation to be valid.

PAE = 100 \cdot \frac{P^{RF}_{out} - P^{RF}_{in}}{P^{DC}_{in}}

In the datasheet in Figure 2 I’ve circled the absolute maximum power dissipation and labelled it with a “5”, which you can use as a conservative estimate for the DC power consumption when calculating PAE. PAE is important primarily because a cubesat will have a limited ability to generate power using solar cells and it’s important to use that power efficiently, but it is also important for another reason. Every milliwatt of heat that’s generated by using power has to be managed by your cubesat’s thermal control system. It could be a good or bad thing to generate more heat, depending on your design, but either way it’s an important systems engineering consideration.

Generally, PAE and heat dissipation are more relevant to the choice of PAs for transmitters versus LNAs for receivers. It is important in both cases to be efficient, but LNAs will often be consuming much less power overall, so an inefficient ampifier using 100 mW of DC power will be much less of a concern than one using 5 W of DC power.

As a good rule of thumb, a power amplifier with reasonable PAE for a cubesat transmitter will be somewhere in the 20% to 40% range. Amplifiers will also generally be more efficient at lower frequencies vs. higher ones. You may be able to find an inexpensive amplifier in the VHF band with a 40% PAE relatively easily, but amplifiers for S-band above 25% may be much more expensive or difficult to locate.

6. Maximum Input Power

In Figure 2, the maximum input power is circled and labelled with a “6”. Maximum Input Power (CW) refers to the maximum power input that an amplifier can sustain without damage. This is a value that only indicates whether or not an amplifier will be damaged by an input signal at that level. It does not indicate whether or not an input signal will be effectively amplified without distortion or intermodulation problems, so using this value together with the gain can’t be used as a shortcut to properly determining whether or not your signal will be compressed.

The (CW) in the label refers to power using a Continuous Wave modulation scheme. This means that the power is valid for modulation schemes have a consistent amplitude, such as phase modulation and frequency modulation. Modulation schemes that vary the amplitude of the signal are referred to as pulsed wave or pulsed modulation schemes (like on-off keying or the amplitude modulation used for AM radio). Most modulation schemes used for satellite communication are continuous wave, though, since they tend to be more spectrally efficient. Pulsed modulation likely isn’t something you’ll need to worry too much about unless you’re tracking aircraft using ADS-B [4].

7. Temperature Range

As mentioned in Section 5, thermal considerations are important for cubesat systems engineering, and therefore must be taken into account when choosing parts. Both of the temperatures labelled as “7” in Figure 2 refer to the temperature at the mounting points or leads of the package, not the temperature of the package itself. There are two temperatures listed.

The Operating Temperature is the range of temperatures that the amplifier can safely operate at while powered on and in use.

The Storage Temperature is the range of temperatures that the amplifier can be exposed to while powered off with no current flowing through it.

8. Input/Output VSWR

The VSWR, or Voltage Standing Wave Ratio, is an important characteristic for determining how much of the power of a signal is transmitted through an amplifier, and how much is reflected at its input and output ports. In figure 2, it is labelled with an “8”. For a more detailed explanation on VSWR, see this article.

Essentially, for more optimal matching and therefore power transfer, we want the VSWR to be as close to 1:1 as possible. A value of 1.5:1 corresponds to 96% of the power being transferred to the amplifier, and 4% being reflected. A value of 2:1 corresponds to approximately 90% throughput, 10% reflection. A value of 3:1 corresponds to approximately 75% throughput, 25% reflection. To calculate this value, you can use this handy online calculator from All About Circuits, and then convert the Mismatch Loss from dB to a linear value [5].

This is an especially important value if you have multiple amplifiers and are using them together. If the reflected portion of the signal bounces between the output of the first amplifier and the input of the second, you can quickly end up with an unstable system that leads to overload and failure. This is why engineers will often put a small value attenuator (1 or 2 dB) of the same characteristic impedance as the amplifiers between them. While it does slightly decrease efficiency, it’s an easy way to impedance match the two amplifiers better over a wide range of frequencies and often doesn’t waste much power or decrease link budget significantly, due to the way that system noise temperature is cascaded [a more detailed article on this subject is in progress].

9. Operating Bias Voltage

The operating bias voltage is the DC voltage that will be applied to the amplifier in order to power it. Most of the time, an amplifier will expect a very particular voltage, and so the supplied bias voltage must be regulated to that value.

Power subsystems can be designed in several ways, either providing regulated or unregulated voltages to the various units/boards that make up the subsystem. If you have certain regulated voltages available, it is often preferable to choose an amplifier that uses one of those voltages, as otherwise you’ll have to convert the voltage a second time. Each voltage conversion stage introduces inefficiencies, and if you are using a switched mode converter like a buck, boost, or buck-boost, you can also introduce noise into your signal that will have to be filtered out.

The operating bias voltage is labelled in Figure 2 with a “9”.

10. Operating Current

Operating current is the current that the amplifier draws when operated at a bias voltage. This is an important consideration, since the power subsystem will have to be able to supply an appropriate amount of current, and the PCB you design for the radio will likewise need to handle it appropriately. In most cases, you won’t require more than 1 or 2 Amps for power amplifiers and even less for LNAs. This is marked in Figure 2 with a “10”.

11. Form Factor/Case Suitability

The form factor of the amplifier tells you what size it is, how many electrical connections it requires, and what the layout and dimensions of those connection points to a PCB are, also called a footprint.

The device footprint is labelled as “11” in Figure 1 at the top of the page. Many components will use industry standard packaging, and be described simply as such. This component, for example uses the SOT-343 package. The benefit of using these standardized packages is that many devices that have a similar size and number of electrical contacts, even if they have different functions, can use the same footprint. This makes life much easier for PCB designers using the components, as they don’t have to draw a custom footprint for each and every part.

More information on the footprint of most standard packages can be found online. The SOT-343 package has information available from Infineon Technologies, for instance [6].

Conclusion

I hope this article was helpful for any aspiring cubesat communication engineers. If you find any errors or important omissions in the information presented here, we encourage you to submit feedback via the comments and we will endeavour to correct anything that needs correction or clarification. Happy orbiting!

References

1. Mini-Circuits PSA4-5043+ Low Noise Amplifier Datasheet. Retrieved on Feb 3, 2020. URL: https://www.minicircuits.com/pdfs/PSA4-5043+.pdf
2. RF Wireless World, “IIP3 vs. OIP3”. https://www.rfwireless-world.com/Terminology/IIP3-vs-OIP3.html
3. Rohde & Schwarz, “Understanding Third Order Intercept”. https://youtu.be/m-2H8ddSwTI
4. https://www.mathworks.com/help//comm/ug/airplane-tracking-using-ads-b-signals.html
5. All About Circuits, “VSWR/Return Loss Calculator”. https://www.allaboutcircuits.com/tools/vswr-return-loss-calculator/
6. Infineon Technologies, “SOT343 Package Overview”. https://www.infineon.com/dgdl/SOT343-Package_Overview.pdf?fileId=5546d462580663ef015806a5338d04ef

If you’re here, it’s likely you’ve heard of cube satellites, or CubeSats for short. But what goes into a CubeSat? What distinguishes it from larger satellites that you see, like the James Webb telescope or the ones that make up the GPS system?

Essentially, it all comes down to size. A CubeSat is basically a baby satellite, usually between 1 and 5 kilograms (though sometimes larger), designed and built to conform to the CubeSat standard. First published in 1999 by a group of researchers at CalTech, the standard defines a whole host of things from the size and shape of a CubeSat to the mass, electrical characteristics, and even provides standards for included propulsion systems and materials that should make up the structure (the complete, most recent version can be found here).

The reason for defining all these things is to encourage compatibility and standardization. When more satellites have similar requirements, it is easier to build components such as computers, radios, and batteries that can be used by multiple different satellites. This leads to more prolific mass-manufacturing of components, bringing costs down for organizations that may want to build and launch a satellite but are unable to afford the non-recurring engineering costs required to develop a system from scratch. In addition, it simplifies launching the satellites, since they all have similar size and mass. This allows for standardized deployment mechanisms and loading configurations on rockets, further bringing costs down.

What Makes Up a CubeSat?

There are 5 standard sizes for CubeSats set by the CalTech standard, each defined in terms of the amount of volume and mass they take up. Choosing the correct size is one of the first and most important things your team will need to do as CubeSat engineers. It will likely come down to 2 things, the engineering requirements of the payload(s) you are planning to fly and your team’s budget, though other factors may come into play as well. The table below shows the typical CubeSat configurations covered by the CubeSat standard.

CubeSat Type	Volume	Mass	Configuration
1U	10 cm x 10 cm x 10 cm	1.33 kg
1.5U	15 cm x 10 cm x 10 cm	2 kg	–
2U	20 cm x 10 cm x 10 cm	2.66 kg
3U	30 cm x 10 cm x 10 cm	4 kg
6U	30 cm x 20 cm x 10 cm	8 kg	–

Sometimes, you’ll hear of even larger configurations like 12U or 18U. These typically involve the stacking of multiple 6U volumes together. Smaller configurations exist that are a fraction of a unit, like 1/2U or 1/3U. We won’t focus too much on these in this article, though many of the principles you learn on Crash Cube can be applied directly to larger and smaller satellites as well.

Like all satellites, a CubeSat is a system that can be broken down into many smaller subsystems, each with a specific purpose. When discussing subsystems, it’s useful to use a hierarchical structure to examine what each is describing. This structure doesn’t follow any hard and fast rules and may differ depending on where you learn it. For the sake of consistency on this site, however, we’ll try to stick to it as often as possible.

\small
System \rightarrow Subsystem \rightarrow Unit \rightarrow Subassembly \rightarrow Part/Component
\\
CubeSat \rightarrow ADCS \rightarrow Reaction \, Wheel \rightarrow Electronics \, Board \rightarrow Capacitor

A part (or component) is the smallest piece of a CubeSat. These will typically be the raw materials you purchase from a manufacturer, such as a capacitor that goes on a board, a piece of the metal structure, an integrated circuit chip, or a screw that’s used to attach pieces of structure together.

A subassembly is the next level up. It is an agglomeration of parts that forms a recognizable, self-contained part of the satellite. A subassembly will typically serve a specific function as part of a larger unit, and often can be built and tested separately from the rest of the unit. The electronics board is a subassembly of a reaction wheel, along with the wheel itself, the motor, and the mounting bracket.

A unit is one stage up from a subassembly. It forms a recognizable part of a subsystem that can be built and tested independently of the subsystem itself. An example of a unit would be a computer, reaction wheel, or radio receiver. Sometimes a unit will not have subassemblies. For example, while a reaction wheel may have several subassemblies, a radio receiver may just be a single PCB populated with components.

A subsystem is a collection of different units that serve a role within the satellite. The various subsystems are defined below in the next section. Each of them will have different requirements placed upon them (and therefore be designed differently and composed of different units) based on the mission. An example of a subsystem is the Attitude Determination and Control Subsystem (ADCS), which can be made up of reaction wheels, magnetometers, magnetorquers, star trackers, sun sensors, and more (or less!).

The system is the highest level subdesignation that’s part of a satellite mission. This would be the CubeSat itself in most cases. The ground station is an example of another system that’s part of a satellite mission.

CubeSat Subsystems

Despite its small size, a CubeSat contains many different subsystems that each do something important to ensure that its mission can be completed, and often it’s impossible for one person to be completely familiar with the inner workings of all of them. A team of engineers is therefore needed, but to make sure that all the subsystems function together properly, one (or more) people typically take on the role of a systems engineer. The systems engineer’s role is to define concrete requirements for each subsystem and the system as a whole to ensure that they work seamlessly together to accomplish the mission.

Each subsystem will also typicaly have a lead engineer, who reports to the systems engineering team. Below is a list of the most common subsystems that make up most CubeSats.

ADCS, the Attitude Determination and Control System

For a lot of satellites, the objective is to point a device (like a telescope or other sensor) at something worth observing. The Attitude Determination and Control Subsystem is where this happens. Attitude is the orientation of the spacecraft, and therefore which way it is pointing (but not concerned about the translation of the spacecraft). For this subsystem, we need to measure (determination) how the spacecraft is oriented using a variety of different sensors and feed these results into actuators that change (control) the orientation. The complexity of the ADCS hardware and software varies greatly by mission; some require redundant parts and high precision sensors, while others might forego an ADCS entirely. It depends on what the payload and mission objectives are for the spacecraft.

Communication

For any satellite, it’s important to be able to tell it what to do, and to be able to send any data it collects back to the Earth. The Communication subsystem is all the components directly related to this goal, including receivers, transmitters, transceivers, cables and waveguides, and antennas. Some CubeSats may even use optical communication systems, which would include lasers, receiving diodes, and optics. It’s important to note that if your payload is a communication system, that will typically be classified as the payload subsystem.

Computer

The brain at the centre of your satellite is a computer. Data collection and processing from the payload can occur here, which is then stored in onboard memory or passed to the communication system for transmission. Housekeeping telemetry, which lets satellite operators know the health of the satellite, is also collected by the computer from each subsystem and their components. The computer must also act on any commands sent by operators, such as to take a picture or restart all systems.

Payload

Payloads come in several forms, but they are there to serve the main purpose of the satellite. Whether the objective is to measure greenhouse gases in the Earth’s atmosphere or relay communications to a remote receiver, the payload defines whatever hardware and software is needed to meet these objectives. Examples of payloads include telescopes for Earth observation missions, communication systems for satellite-based internet and communications, platforms for onboard biological and chemical experiments, and possibly any other device that needs flight testing.

Power

The power subsystem is all the components related to electricity generation, conversion, storage, and distribution. Most CubeSats generate power using photovoltaic cells (solar panels) attached to the body, though some will also use larger deployable panels to generate more power. Power is converted between voltages by using voltage converters and stored in a battery. Switches are used to turn on and off various subsystems, units, and components based on the needs of the mission, and wires carry electrical current from the power subsystem to the other subsystems.

Propulsion

Many CubeSats will not use propulsion systems, but for some specialized missions they are necessary. Propulsion is used to adjust the orbit of the CubeSat. It comes in several varieties including solid, liquid bipropellant, liquid monopropellant, cold gas, and electrical. Cold gas and, to a lesser extent, electrical propulsion systems are far and away the most common on CubeSats due to simplicity as well as safety and regulatory concerns.

Software

The software subsystem includes all the code that is written to control computers and microcontrollers located throughout the satellite. Typically this will primarily consist of the code for the main onboard computer, but depending on how your team is distributed may also include microcontroller code for other subsystems and/or firmware for FPGAs (Field-Programmable Gate Arrays) if you use them.

Structure

Holding the spacecraft together is the structure. This structure protects all other subsystems from the vibrations of launch and provides the precise positioning to ensure everything fits and is oriented properly. The structure also serves as a primary conduit for heat transfer and as the common ground for all electrical systems. Mechanical (i.e. moving) components also fall within the structure subsystem, and can include deployable antennas, solar arrays, and payloads.

Thermal

The Earth orbit environment can be challenging for temperature control. In the vacuum of space, no convection heat transfer is available to cool the spacecraft’s electronics, so engineers must manipulate conduction and radiation to ensure the system survives. By understanding and selecting the conductivity, absorptivity, and emissivity of the spacecraft, thermal engineers can regulate the spacecraft temperature. In some cases, passive and active thermal control systems are required, however this is typically reserved for specialty payloads and extreme orbits.

References

1. The CubeSat Program, CalPoly SLO. CubeSat Design Specification Rev. 13, California Polytechnic State University, San Luis Obispo, CA, USA. Feb 20, 2014. Accessed on: Feb 03, 2020. [Online]. Available: https://www.cubesat.org/s/cds_rev13_final2.pdf
2. The CubeSat Program, CalPoly SLO. 6U CubeSat Design Specification Rev. 1.0, California Polytechnic State University, San Luis Obispo, CA, USA. June 7, 2018. Accessed on: Feb 03, 2020. [Online]. Available: https://www.cubesat.org/s/6U_CDS_2018-06-07_rev_10.pdf
3. Pelton J., Finkleman D. (2020) Overview of Small Satellite Technology and Systems Design. In: Pelton J. (eds) Handbook of Small Satellites. Springer, Cham. https://doi-org.myaccess.library.utoronto.ca/10.1007/978-3-030-20707-6_7-1

About the Authors

Introduction

This article will give you an introduction to some of the key terms used by spacecraft engineers when designing and analyzing antennas. While this article doesn’t require any prior knowledge of spacecraft engineering, some knowledge of circuit theory as well as mathematical concepts including logarithms and complex numbers are used.

This article is a bit of a work in progress that we’ll be adding to continuously, so there are a few sections that aren’t yet completed. These are marked with a [TBC] for “To Be Completed”.

Prerequisite Knowledge

1. Frequency domain, (video)

Background

This section will introduce many of the theoretical and conceptual terms you will come across on Crash Cube on pages related to antennas and electromagnetics.

Electric Field

An electric field, commonly abbreviated as an E-field, is a field generated by electrical charges (such as electrons) or by a magnetic field that changes with time. It is one half of the force of electromagnetism, one of the four fundamental forces of nature. The strength of an electric field (E) is given as the force (F) exerted per unit charge (q) and is often measured in units of volts per meter, or electric potential per distance. Electric field is a vector quantity, meaning that direction is important in determining its effect.

\overrightarrow{E} = \frac{\overrightarrow{F}}{q}

Dimensional Analysis:

\frac{Newton}{Coulomb} = \frac{N*m}{C*m} = \frac{J}{C*m} = \frac{V}{m}

Convention is to define electric field lines generated by a positive charge as emanating from the charge, outward.

Magnetic Field

A magnetic field, commonly abbreviated as an B-field, is one generated by a moving electric charges (such as an electric current) or by an electric field that changes with time. It is the other half of the force of electromagnetism along with the electric field.

It is convention to define the direction of the field as perpendicular to the direction of the current generating it, following the right-hand rule.

Electromagnetic Wave

An electromagnetic wave is a paired, sinusoidally oscillating electric field and magnetic field that travels through space, either through vacuum, air, or some other non-conducting material. In almost all typically used materials, including PCB substrates and vacuum, the electric and magnetic fields are perpendicular to both each other and the direction of travel of the wave. They follow the right hand rule such that the cross product of the E field and the B field is in the direction of propagation.

The direction of Figure 1 shows an illustration of an electromagnetic wave, with the electric field (E) in blue oscillating in the x-axis, the magnetic field (B) in red oscillating in the y-axis. The z-axis is the direction the wave is travelling in.

Polarization

An electromagnetic wave can oscillate in different axes, or multiple axes simultaneously. This is called the polarization, essentially how the wave oscillates in the plane perpendicular to its axis of propagation.

In Figure 1 for example, the electric field is oscillating in the x-axis. If the wave was projected into the x-y plane, the tip of the arrow would trace out a line over time, only moving in the x-axis. This is an example of linear polarization. If the ground is in the y-z plane in that image, this could also be called vertical polarization. If the wave was rotated 90 degrees to either side, it would be an example of horizontal polarization. It is also possible for a linearly polarized wave to be rotated at an arbitrary angle around the axis of propagation (z in this example).

A wave can also be polarized in more than one axis simultaneously. A common form of this in the space industry is circular polarization, in which the electric field oscillates in both the vertical and horizontal axis with the same magnitude, 90 degrees out of phase with each other. If the oscillations are of differing magnitudes or have different phasing, the polarization is called elliptical. Figure 2 shows an example of linear, circular, and elliptical polarization.

Frequency

The frequency of an electromagnetic wave is the rate at which the phase of the wave changes per time. All electromagnetic waves oscillate sinusoidally. Frequency is commonly measured in Hertz (abbreviated Hz), or cycles per second. In antenna engineering, however, it is also common to measure frequency using angular measurements, radians per second. Recall that for each 360° phase cycle of a sine wave, there are 2π radians. Frequency is often abbreviated using the symbol f.

Wavelength

The wavelength of an electromagnetic wave is the distance between the peaks of the wave, represented by the symbol λ. In vacuum, this is equal to the speed of light divided by the frequency.

\lambda_0 = \frac{c_0}{f}

The subscript 0 (often pronounced “naught”) is used to indicate that the wavelength is that measured in vacuum. In materials other than vacuum, the frequency of a wave remains the same. The wavelength, however, will change based on the material’s properties since the speed of light in a material other than vacuum is lower than the speed of light in vacuum.

Modulation

An electromagnetic wave is sinusoidal, but it is impossible to encode any information onto a pure sinusoid without changing it over time. By changing the sinusoid in some way that is detectable, we can encode information. Changes to the wave over time to encode information can take many forms, from changing the phase at specific intervals, to changing the frequency, to shutting the wave on and off. These changes, which can be decoded by a receiver as 0’s and 1’s, are called modulation.

Bandwidth

An unmodulated signal with one frequency appears as a single value on a frequency spectrum plot (see prerequisite reading #1). When a signal is modulated it is no longer a pure sinusoid, instead it’s power spreads out wider in the frequency spectrum in a manner that is dependent on the type of modulation that’s being used and the rate at which the signal is being modulated.

Bandwidth is the range of frequencies under which most of the power of the signal resides. It can be defined in a couple of ways most commonly 98% power (for signal bandwidths), -3 dB off-peak insertion loss (for filter bandwidths), or -10 dB return loss (for antenna bandwidths), but it’s possible to define it however is relevant to your specific application.

Essentially, it boils down to the difference between the minimum significant frequency component and the maximum frequency component of your signal. A modulated signal that has a minimum frequency component of 10 MHz and a maximum frequency component of 15 MHz has a bandwidth of 5 MHz.

Rather than defining a signal by the minimum and maximum frequency components, its common to describe it using the point in between them, called the center frequency, and the bandwidth (in this example 12.5 MHz and 5 MHz, respectively).

Decibels

Decibels, often abbreviated as dB, are a logarithmic unit of measurement for measuring relative quantities. Most often you’ll see them used to describe quantities like gain (dBi, decibels relative to an isotropic antenna), power (dBm, decibels relative to one milliwatt), or voltage (dBV, decibels relative to one volt), but it can be used to describe any measurement. The purpose of using decibels for many quantities in communications is twofold. First, they allow representations of both extremely large and extremely small values in a manner that’s easy to represent and perform mathematical operations with. Second,

For example, 50 mW can be represented as 17 dBm. If a 1 GHz signal is transmitted over 50 km, the amount of power at a receiver is 2.3*10^-14 of what was originally transmitted, or -106.4 dB. Rather than multiply these two numbers together to find the amount of power at the receiver (1.15*10^-9 mW), by using decibels the two values can simply be added together. This becomes far more convenient, especially when many different gains and losses need to be taken into account in an equation.

To convert a value x to decibels, for linear quantities such as voltage, use the following formula.

20\log_{10}x

For nonlinear quantities such as power or gain (which is itself a measure of relative power), use the following formula.

10\log_{10} x

Impedance

The impedance of an electromagnetic wave is the ratio between the amplitudes of its electric and magnetic fields. In a circuit, the impedance describes the ratio between the voltage and current of a wave travelling in the circuit. It is measured in units of ohms, similarly to resistance, and can be a complex value. Impedance is important because, as a wave reaches the boundary between two mediums or transmission lines (such as transitioning between a coaxial cable and a an antenna), if the impedances aren’t matched, some of the energy gets reflected resulting in inefficiencies.

The two relationships that impedance describes are related. If you wish to learn about them in more detail, check out Chapter 2 in [3], an excellent textbook on the fundamentals of electromagnetics.

VSWR and Reflection Coefficient

[TBC] – While we continue work on this article, here is an external link covering the concept we haven’t gotten around to yet.

https://www.electronics-notes.com/articles/antennas-propagation/vswr-return-loss/what-is-vswr.php

Directivity, Gain, Realized Gain

[TBC]

Radiation Pattern

When an antenna radiates, it does not do so equally in all directions. Based on the shape, size, and material composition of an antenna, the amplitudes of the fields radiated in different directions will be different. By plotting the strength of the fields relative to the direction in which they emanate, a radiation pattern can be generated. The radiation pattern is one of the most useful tools for an antenna engineer.

A radiation pattern can be generated for several different quantities, most of which are what is called far-field quantities, i.e. they are the values that would be measured if an observer was very far away from an antenna, such that the entire antenna essentially appears as a point source. Typical quantities measured using a radiation pattern are directivity, gain, realized gain, and axial ratio. An example of a 2D radiation pattern (normalized to the peak value, 9 dBi) is shown below in Figure 3.

Further Reading

1. [TBC]

References

[1] – By SuperManu – Self, based on Image:Onde electromagnetique.png, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2107870

[2] – R. Nave, Classification of Polarization. [Online]. Available: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/polclas.html. [Accessed: 12-Jan-2021].

[3] – Ulaby, Fundamentals of Applied Electromagnetics

[4] – https://commons.wikimedia.org/wiki/File:Patch_antenna_pattern.gif

To help us better understand those needs as we assemble our site’s content, we’ve put together a short survey. The survey is located here and takes about 5-7 minutes to fill out. We’re interested in the opinions of everyone, from your team’s chief engineer to the first-year who joined the team last week.

Happy Orbiting!

– Ian

Even though the cost has been greatly reduced since the first CubeSat was launched more than 20 years ago, getting a student-run satellite project off the ground (both metaphorically and literally) can still be a huge challenge. It can be difficult to jump in without having an experienced hand to help guide your way, and even then, much of the information you need to get your project going isn’t taught in university courses or available on Wikipedia.

We’ve found that information on many of the topics needed to build an amateur satellite can be difficult to find, and even when it can be found it’s often hidden behind paywalls in academic journals and expensive textbooks full of confusing terminology that isn’t explained well.

Crash Cube is a centrally maintained repository of technical information related to all aspects of amateur satellite analysis, from structures and thermal to power systems to communication, and everything in between. We’re doing the work of collecting everything you’ll need to get your satellite into space into one place, verifying its accuracy, and explaining it in plain language so you understand not just how to do it, but why.

Most importantly, this site is and always will be non-profit and available to all, free of charge.

– Ian