Transverse Electric and transverse Magnetic modes

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Theory of operation

Standing wave [3]

As the incident wave inside the RC reflects from the conducting surfaces inside the chamber (walls, ceiling, floor, stirrer blades), the reflected wave interferes with the incident wave constructing a standing wave.
Standing wave pattern is determined by the mathematical summation of both the incident and reflected waves. So that the maximum value of the standing wave pattern corresponds to the position at which both the incident and the reflected wave are in phase (having 2nπ phase shift; n is an integer) and therefore add constructively figure(1.1). The minimum value of the standing wave pattern corresponds to the position at which both the incident and the reflected wave are in phase-opposition (having [2n+1] π phase shift; n is an integer) and therefore add destructively figure (1.2). If the amplitude of the reflected wave is less than that in the incident wave, the resulting standing wave constructs an envelope with a repetition period of λ/2. Considering the incident and the reflected wave repetition period individually is λ (wavelength in meter), the repetition period of the standing wave is λ/2 this means that the maxima of the standing wave appear every half wavelength of the incident wave figure (1.3)
Figure (1.1) constructive interference Figure (1.2) destructive interference figure (1.3) standing wave

Transverse Electric and transverse Magnetic modes [3]

Transverse Electric Mode (TE): Considering the three orthogonal directions x, y, and z, when the electromagnetic wave (EM) propagates in the z-direction and the (z) component of the electric field vector is equal to zero. So that the electric field vector is always perpendicular to the direction of propagation (z). This mode of propagation is called the Transverse Electric mode (TE).
Transverse Magnetic Mode (TM): If the (z) component of the magnetic field vector is equal to zero, the magnetic field vector is always perpendicular to the direction of propagation (z). This mode of propagation is called the Transverse Magnetic mode (TE).
Transverse Electromagnetic Mode (TEM): In TEM mode, the z-component of both the electric field and the magnetic field vectors are equal to zero. The electric field and magnetic field are propagating orthogonally to the direction of propagation. This mode needs a current source to exist; that is why it exists in coaxial cable carrying current but does not exist in a rectangular waveguide and RC. For a rectangular waveguide, if the direction of propagation is (z) while (x) and (y) are the width and the height respectively, the lowest mode of propagation TE10 means that there is one half of the wavelength λ in (x). TE20 means there are two halves of the wavelength λ in (x), whereas TE01 means there is one half of the wavelength λ in (y), and TE20 means there are two halves of the wavelength λ in (y). Figure (1.4) presents TE10&TE20&TE30 in x-y plane. Different TE and TM modes are illustrated in figure (1.5)

Operating parameters

Resonance frequency and number of modes

RC operates on the phenomenon of the cavity resonator in which several modes are created according to the RC dimensions and the exciting frequency inside the chamber. As the electromagnetic radiation inside the RC reflects from the ceiling, walls, and floor, moreover stirred by the mechanical stirrer, it causes the incident wave to interfere constructively with the reflected wave resulting in standing waves. Nevertheless, there is destructive interference which quenches the wave.
The resonance frequencies of the chamber can be calculated by the following equation (1.1) which is derived from solving Maxwell’s equations[1]
Where m, n, p are the number of modes in x, y, z directions respectively m=0,1,2,3,… & n=0,1,2,3,… & p=0,1,2,3,… while a, b, d are the chamber dimensions in x, y, z in meter, and fmnp is the resonance frequency in Hz, where μ, ε are the permeability and the permittivity of the medium inside the cavity, respectively.
Number of modes is given by Weyl’s formula (1.2) for rectangular cavities[1] = 8 ( . . ℎ) 3 − ( + +ℎ) + 1 (1.2)
Where ‘c’ is the speed of the electromagnetic wave inside the cavity, which equals to 3*108 m/s. and w,l,h are width, length, and height, respectively. Mode density also is an important parameter to be considered. It reflects how many modes available in a narrow bandwidth of a specific center frequency. It is given by equation (1.3)[1] as the number of modes per frequency unit.
It is noted that RC dimensions constitute a significant factor in determining many important operating parameters like resonance frequencies and the number of modes. Lowest Usable Frequency (LUF) is one of these parameters. It is the minimum frequency at which the RC can achieve field uniformity in the working volume of the RC, and it occurs at a frequency slightly above three times the first chamber resonance frequency. LUF can be determined alternatively by substituting the number of modes (N) in equation (1.2) with 60 modes; according to (IEC 61000-4-21) this is the minimum number of modes that can achieve the field uniformity inside the RC [19]. The working volume is one of the RC essential parameters. It is the cube inside the RC whose eight vertices are separated by the chamber walls, ceiling, floor, stirrer, transmitting or receiving antennas, and any conducting object inside the chamber by 0.25 wavelength of the LUF [19].

Chamber Quality factor

The quality factor (Q-factor) of an RC gives information about how much electromagnetic energy that RC can store concerning losses[5]. This energy is determined by the volume of RC and the excitation frequency while losses due to resistive loss of the wall’s conducting material, leakage through apertures, loading RC with Equipment-Under-Test (EUT), holders, and antenna loss. The higher the losses, the lower the Q-factor, which leads to lower field strength at certain input power. So during empty chamber validation, the existing power inside the RC is supposed to be greater than that if it is loaded with EUT, this should be considered to increase the input power during loading to compensate losses hence getting considerable average testing power Pav = Pin – Ploss (1.4)
Where Pav is the average power, Pin is the input power, and Ploss is the power losses. A remarkable enhancement of the quality factor and the RC performance can be achieved by installing metallic hemispheres or caps on the inner walls of RC. A significant improvement of the field statistical properties (homogeneity and isotropy) can be attained when the number of hemispheres or caps increases.[5]

RC physical description [1]

RC is a shielded enclosure Figure (1.7) with high electrical conductivity walls, ceiling, and floor to eliminate absorption of the Electromagnetic wave (EM) from one side and to accomplish as a complete reflection as possible from the other side. RC operates under the cavity resonator phenomenon. The three dimensions of the chamber are a function of the cavity resonance frequency and the number of modes inside the chamber at a specific frequency. Inside RC, there is a transmitting antenna to provide electromagnetic power. One or more stirrers (the dimension of the smallest side of the stirrer sides is 0.25 of the lowest used frequency of RC) are installed to achieve a statistical uniform and isotropic EM field distribution as well as random polarization on the EUT. Stirrers work in two modes mode-stirred when the stirrer rotates continuously during the measurement procedure, mode-tuned when the stirrer rotates in discreet angles and stops at each angle until measurement has been taken.
Moreover, in contrast with its counterpart AC, RC not only provides shorter testing time but also high field strength for moderate input power, EUT doesn’t need to rotate inside the RC, and no need to change the polarization of the transmitting antenna.

RC applications

RC provides a remarkably ideal environment for Electromagnetic compatibility tests to ensure that an electrical device, from one side, can operate correctly in an environment without negatively affecting other electrical devices which are sharing the same environment by its own electromagnetic radiation. On the other hand, its performance is not degraded by the effect of other neighboring devices’ radiation. All EMC test environments, limitations, and related standards are conducted in the IEC61000-4-21 standard[19]
An electrical device needs to pass different tests on complying for the EMC. Radiated Emission [1] test measures how much EMI is caused by EUT during operation and its effect on the neighboring electrical devices sharing the same environment. Radiated immunity test measures to what extent the EUT can resist the existence of EMI, and its performance doesn’t deteriorate accordingly. Shield Effectiveness of different gaskets and materials test measures to what extent shielding gaskets can protect the shielded EUT from external EMI and how much leakage from EUT’s radiation to the outer environment. Using RC provides a better testing environment than other measuring techniques like AC or Open Area Test Space (OATS) as the EUT is exposed to EM radiation from all directions with all possible polarizations without even the need to rotate or move the EUT. This aspect provides a high degree of flexibility for large size EUT as the case of the automotive industry.
An effective test setup developed by Helge Fielitz [7]that combines discrete reflections generated by a fading simulator with the continuous distribution of reflections created in a reverberation chamber provides a cost-effective test environment for outdoor urban wireless propagation studies. Characterization of antennas for mobile and wireless terminals can now be done by RC[2].
All the previous electromagnetic applications are used along with the acoustical applications like microphone calibration and measuring sound power of an acoustical source that RC can be an ideal testing environment.[8]

Reverberation chamber versus Anechoic Chamber (AC)

On the contrary of the RC operation, which is based on EM wave reflections from all inner sides, the AC figure (1.8) contains absorbers to eliminate the field reflections and provide a reflection-free test environment. The great advantage of using the RC is the reduced power [9]required to establish the field test level due to the high-quality factor, typically some watts versus some hundreds of watts. Another advantage is that there is no need to change the EUT orientation or the transmitting antenna polarization during the test in RC as the stirring of the field and the random reflections inside the RC achieves field isotropy. Concerning cost, RC is remarkably differentiated from AC; in fact, the ratio ranges from one-tenth to one-seventh, according to the way used to realize the AC (absorbing material only, absorbing materials and ferrites, and so on). Not only the saving for the chamber cost but also the saving in the power amplifier cost should also be considered because the RC test requires a very low power compared to AC[9]. Antenna pattern is obtained using AC because it provides an ideal test environment for antenna pattern determination as the received power at the Rx antenna can be measured at particular polarization as well as at a specific angle of arrival (AoA). This can’t be achieved by the RC in which polarization is completely destroyed, and the received power at the Rx antenna coming from arbitrary directions.[9].

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Antenna principles

An antenna is an electromagnetic transducer which converts guided waves within transmission lines to radiated free-space waves (in a transmitting mode) or to convert free-space waves to guided waves (in a receiving mode). Antennas are reciprocal, which means they have the same properties (radiation pattern, gain, directivity, losses, polarization), whether it is used for transmitting or receiving.[10]
To describe the mechanism by which the electromagnetic waves propagate from an antenna, a center-fed dipole shown in figure (1.9) is considered to be fed by a sinusoidal voltage source. In the first quarter of the period during which the charge has reached its maximum value, the electric field lines have traveled outwardly a radial distance λ/ 4. During the next quarter of the period, the original lines travel an additional λ/ 4 (a total of λ/ 2 from the initial point), and the charge density on the conductors start decaying. This decaying can be assumed as opposite charges being introduced during the decay period of the second quarter of the first half cycle.
These opposite charges cause electric field lines in the opposite direction of the original lines and travel a distance λ/ 4 during the second quarter of the first half. The result is that there are lines of force pointed upward in the first λ/ 4 distance, and the same number of lines directed downward in the second λ/ 4. At the moment of the end of the first half cycle, the two opposite lines groups are being touched at the center point of the dipole. Regarding the fact that there is no net charge on the dipole, the lines of force must have been forced to detach themselves from the conductors and to unite together to form closed loops. The same process is followed in the remaining second half cycle in the opposite direction so that the process continues and creates the propagation of electromagnetic waves.

Radiation Pattern

An antenna radiation pattern is a 2-D or 3-D illustration in a polar plot that illustrates either radiated power versus angle of measurement figure (1.10a) as well as it may demonstrate the electric (E) or magnetic (H) field strength versus the angle figure (10.b).
Figure (1.10b) R&S HL562E radiation pattern at 80 MHz[11] Figure (1.10a) 3-D antenna pattern [12]

Antenna Directivity, Gain, and Efficiency

Antenna directivity is the ratio of the power density that the antenna radiates in the direction of its strongest emission to the power density radiated by an ideal isotropic antenna (an antenna that radiates power equally in all directions). Directivity is measured in dBi[10], [12]
Antenna directivity and antenna gain are basically the same concepts, except the gain takes the antenna efficiency into account while the directivity does not. Gain is the directivity multiplied by the radiation efficiency (η) (1>η>0) equation (1.5). Gain measures how much power is transmitted in the direction of peak radiation to that of an isotropic source. An antenna of 3dB gain can deliver power in the direction of maximum power radiation twice more than if it would be transmitted by an isotropic antenna fed by the same input power. G = ηr D (1.5)
Where G is gain, ηr is radiation efficiency, D is directivity. Antenna efficiency measures the efficiency of an antenna of converting the electrical power into EM power and vice versa. It can be calculated by dividing the antenna radiated power density at a distant point (Pr) to the total antenna input power (Pin) equation (1.6)[12]
Where Pr is the received power, and Pin is the input power.

Antenna polarization

The electric field orientation of the far-field plane wave determines the wave polarization. The common antennas polarizations, considering the EM wave comes out of the page are vertical, horizontal and helical polarization shown in figure (1.11)[12]

Spectrum analyzer theory of operation[13][14]

At the most basic level, a spectrum analyzer can be described as a frequency-selective, peak-responding voltmeter calibrated to display the root mean square (rms) value of a sine wave. But it is not considered as a power meter even though it can measure the power of the signal under measurement. Phase measurements also can be performed. Modulated signals of different types of modulations (AM, FM, PM), as well as wireless communication signals (GSM, WCDMA, LTE), can be analyzed.
It is well known from Fourier theory[3] that any electrical signal of any arbitrary shape can be analyzed into one or more sine waves of appropriate frequency, amplitude, and phase. Hence it can be transferred from a time-domain signal into its frequency domain equivalent. Measurements in frequency-domain give information about how much energy is present at each particular frequency (i.e., the harmonic contents of the signal). The waveform shown in figure (1.12) can be analyzed into two separate sinusoidal waves (two spectral components) so that they can be evaluated independently. The two time-domain signals shown in figure (1.12) have two different frequencies, amplitudes and they have a difference in phase as well. They are represented in frequency-domain as two peaks of different amplitudes; each amplitude reflects the (rms) of the corresponding signal. They are separated in frequency-domain according to the difference of their frequencies. This type of signal analysis is called spectrum analysis.[13]
Figure (1.12) Frequency domain versus time domain [13]

Applications of a spectrum analyzer

There is a wide variety of applications that need determining the harmonic content of a signal and how much power is contained by each frequency.
– Government regulatory agencies allocate different frequencies for various radio services. For instance, television and radio broadcasting, cellular networks, police and emergency wireless communications, and a wide range of other wireless services. Every service provider must be committed to the allocated frequency band by the regulatory agencies to avoid causing EMI with other services.
– Signal-to-noise ratio (SNR) and noise figure characterize the performance of an electronic device and measure to what extent it affects the overall system performance.
– Measurement of the third-order intermodulation.
– Investigation of electromagnetic interference.

Spectrum analyzer function block diagram description

Radio frequency (RF) attenuator figure (1.13)
RF input attenuator is the first stage. It plays an essential role in instrument overload protection. It ensures the input signal enters the mixer at the optimum level. It has a blocking capacitor to block the direct current (DC) signal or a DC offset of the signal under measurement.
Pre-selector figure (1.13)
It is a tunable low-pass filter that prevents high-frequency signals from reaching the mixer and passes only the frequencies in the measuring range of the spectrum analyzer.
Mixer, Local Oscillator (LO) and intermediate frequency filter (IF) figure (1.13)
Tuning the analyzer[13], [14]
The LO frequency (fLO) is tuned to the sum of the signal frequency (fsig) and the IF frequency (fif). The mixer output contains fLO + fsig , fLO – fsig and higher-order harmonics of fsig. The IF band-pass filter passes the mixing product fLO – fsig and block the mixing product fLO + fsig . If the fLO is not quite high enough to cause fLO – fsig to fall in the IF passband, so there is no response on the display so the ramp generator needs tune the LO higher. However, this mixing product will fall in the IF passband. The ramp generator controls both the horizontal position of the trace on display and the LO frequency. Hence the horizontal axis of the display can be calibrated in terms of the input signal frequency. fif is chosen to be above the highest frequency of the spectrum analyzer tuning range.[13]
Figure (1.14 ) Mixer operation [13]
To distinguish two signals having close frequencies, the IF band-pass filter should be as narrow as possible. But at high frequencies (GHz range), it is challenging to implement an extremely narrow band-pass filter. A practical solution to overcome this problem is to add additional mixing stages, typically two to four stages are added to down-convert the fif to the optimum frequency can be sharply filtered by the IF band-pass filter as shown in figure (1.15)

Table of contents :

1 introduction
1.1 Historical view
1.2 Theory of operation
1.2.1 Standing wave
1.2.2 Transverse Electric and transverse Magnetic modes
1.3 Operating parameters
1.3.1 Resonance frequency and number of modes
1.3.2 Chamber Quality factor (Q)
1.4 RC physical description
1.5 RC applications
1.6 Reverberation chamber versus Anechoic Chamber
1.7 Antenna principles
1.7.1 Radiation Pattern
1.7.2 Antenna Directivity, Gain, and Efficiency
1.7.3 Antenna polarization
1.8 Spectrum analyzer theory of operation
1.8.1 Applications of a spectrum analyzer
1.8.2 Spectrum analyzer function block diagram description
1.8.3 Resolution bandwidth and phase noise
1.9 Transmission Line
2 Methodology
2.1 Instruments and software
2.1.1 Signal generator (Ana-Pico APGEN 3000)
2.1.2 Spectrum analyzer Rohde & Schwarz R&S FPC 1000
2.1.3 Transmitting (Tx) antenna
2.1.4 Receiving (Rx) antenna
2.1.5 Stirrer system
2.1.6 Software
2.1.6.1 Python-Visa
2.1.6.2 SCPI
2.2 calculations
2.2.1 Lowest Usable Frequency (LUF)
2.2.2 Working volume
2.3 Field uniformity validation
2.3.1 Variance and Standard Deviation
2.3.2 Test set up
2.3.3 Measurements results
2.3.4 Sweeping measurements
2.3.5 Quality factor degradation test
3 Discussion
3.1 Field uniformity and isotropy
3.2 Sweeping measurements
3.3 Quality factor degradation test
4 conclusion
5 Recommendations for continued work
References
Acronyms
Appendix
Python code

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