EQUATIONS

Experience and Familiarity with S-Parameters

In common with most engineers, Chris Angove has a good appreciation of the mathematics behind the everyday calculations performed, whether those are done ultimately by a computer or 'manually'. With all new analytical tasks, when time allows, he makes a point of understanding the underlying theory behind the task in hand and how the results are arrived at. He can often contribute to the fundamental approaches used in obtaining the results and the way they are presented.

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  1. Transmission Line Equation
  2. Rectangular Waveguide Equation
  3. Fourier Transform
  4. Microwave Magnetic Properties
  5. S and T Parameters (2 Port)
  6. Cascaded Noise Figure
  7. Laplace Transform
  8. Skin Depth
  9. Voltage Standard Wave Ratio
  10. Intrinsic Impedance of Free Space
  11. Dual Modulus Prescaler Synthesizer
  12. Poynting Vector
  13. Maxwell 1
  14. Maxwell 2
  15. Maxwell 3
  16. Maxwell 4
  17. Optical Coupler
  18. Propagation Coefficient
  19. Differential Amplifier
  20. Intrinsic Impedance
  21. Radar Equation
  1. Transmission Line Equation

    Transmission Line Equation

    This is the transmission line equation for the general (lossy) case. It is frequently simplified to the loss-free case for which the hyperbolic tangent becomes a normal tangent and g becomes b. g is the propagation constant and b is the phase constant. For a transmission line of characteristic impedance Z0 W, terminated in an impedance ZL W,ZT is the impedance seen at a distance l metres, measured back from the termination. g, b and the attenuation constant a are related by g=a + jb. See also propagation coefficient.

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  2. Rectangular Waveguide Equation

    Waveguide Equation

    For a rectangular waveguide excited in the fundamental (TE10) mode, this equation relates the free space wavelength l0, to the guide wavelength (lg) and the cutoff wavelength (lc).

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  3. Fourier Transform

    Fourier Transform

    f(x) is a periodic function of x. The basis of Fourier's series is that any periodic waveform can be represented as a constant term (a0) plus the sum of a series of sinusoidal and cosinusoidal terms and their harmonics to infinity. The Fourier coefficients are an and bn.

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  4. Microwave Magnetic Properties

    Tensor Permeability

    Ferrimagnetic materials such as ferrites are designed to have useful properties when subjected to high frequency fields, possibly whilst a constant (DC) magnetic field is superimposed.

    The steady state relationship between the magnetising intensity H, the resulting magnetic flux density B and the permeability m is B=mH. The permeability is often expressed in terms of the absolute permeability of free space m0 and the relative permeability mr as m=m0 mr. However mr varies very wildly with just about everything and a completely different consideration is necessary which is dealt with elsewhere. At a constant (usually microwave) frequency in the presence of a static magnetic field, the above tensor relationship is more appropriate in which the tensor permeability is given by:

    where k is called the cross diagonal component of tensor permeability.

    Typically this might be used in isolator design work, where a steady magnetic field is applied usually by a permanent magnet and the microwave field is that due to the RF applied.

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  5. S and T Parameters (2 Port)

    S Parameters

    S Parameters

    'S' parameters stands for scattering parameters. They are just another form of parameter used for describing the behavoir of a network like for example 'H', 'Y' or 'Z' parameters. S parameters are used extensively in RF and microwave engineering to characterise electrical networks at the higher frequencies.The 'S' parameters of an 'N' port network at the frequency of interest can be expressed by a square S parameter matrix containing N rows and N columns, so the total number of S parameters would be N2. The example shown refers to a two port network with a total of 4 separate S parameters S11, S12, S21, and S22. The test frequencies do not have to be high by definition, but they are generally more useful at higher frequencies as all parameters are expressed in both magnitude and phase. Note that by phase, I mean just the spacial part, ignoring the time-varying part. That is, at a specified frequency, to describe the electrical characteristics of the network in terms of the incident and reflected voltage waves at each port whilst under specified conditions at each of the other ports.

    In this example the individual ports are designated 1 and 2. The incident waves at port 1 and port 2, normalised by dividing each by the quare root of the system impedance, are respectively a1 and a2. The reflected waves are respectively b1 and b2. The matrix form is simply a compact and convenient way of representing the elements. Expanding out the matrix gives:

    b1=S11a1+S12a2

    and

    b2=S21a1+S22a2

    An electrical network can have one or more ports. I have rarely come across more than 4 ports but they certainly do exist, examples may be multi-way splitters and combiners. By far the most common S parameters are defined for a system impedance of 50 W. That is, the source and load impedances are both 50 W, and every port apart from those being measured are terminated in 50 W.

    S parameters can be measured with an instrument that can both generate a CW signal and can measure the same signal in spatial amplitude and phase. These are the basics of a network analyzer (NA). Over the last 30 or so years, NA's have become very advanced. Even a basic model would typically include several of the following faciltiies:

    • Simultaneous measurement of S parameters for 2 ports.
    • Accurate and comprehensive frequency control.
    • Full bi-directional data bus interfacing.
    • Accurate internal calibration capability using error correction models.
    • Multi-output formats.

    T Parameters

    T Parameters

    Like with S parameters, every T parameter is also expressed in amplitude and phase, but the definition in terms of the incident and reflected waves a and b respectively differs as shown by the matrix equation above.

    The main advantage of T parameters is the simplicity with which they can be used to determine the properties of cascaded 2 port networks.

    Suppose two 2-port networks 'A' and 'B' have T parameters represented by the 2x2 matrices (TA) and (TB) respectively. When they are cascaded, by connecting port 2 of network A to port 1 of network B, the resulting 2-port network will have the new set of T parameters (TAB), where (TAB)=(TA)(TB). The T parameter matrices are simply multiplied, using the normal rules of matrix multiplication.

    Conversion Between S and T Parameters

    Some matrix arithmetic and a great deal of time and patience yields the following formulas for converting between S and T parameters for 2-port networks, where det(S) and det(T) refer to the determinants of the S and T matrices respectively.

    For converting from S to T parameters:

    S to T Parameters

    For converting from T to S parameters:

    T to S Parameters

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  6. Cascaded Noise Figure

    Fris's Formula

    This equation, also known as Fris's formula, is an expression for the overall noise figure for two cascaded 2-port devices. The input and output impedances of each stage must be the same and there must be no frequency conversion. FT is the overall noise figure of both stages. F1 and F2 are the noise figures of stages 1 and 2 respectively, and G1 is the gain of stage 1. Noise figures and gains are both expressed as linear power ratios, not in dB which is of course a logarithmic ratio.

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  7. Laplace Transform

    Laplace Transform

    The Laplace transform is a very useful mathematical tool, just like the Fourier transform, for converting a function in the time plane to the complex frequency plane 's', where s=jw, and w is the angular frequency in rad/s. The inverse Laplace transform does the opposite conversion. Complex frequency is a little more than just a shorter way of writing jw, it is widely used in circuit network theory where any type of time varying waveform is concerned.

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  8. Skin Depth

    Skin Depth

    An alternating current induced in a real conductor from an alternating electric field on the outside will decay exponentially into the surface of the conductor. This equation expresses the normal depth at which the magnitude of the electric field is 1/e times its value at the surface. The conductor has conductivity s and permeability m and f is the frequency in Hertz. In a theoretically perfect conductor, the current does not penetrate at all and resides just on the surface so it can't go anywhere because it doesn't have any thickness, which is quite interesting.

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  9. Voltage Standard Wave Ratio

    VSWR

    A voltage standing wave occurs on a transmission line that is not matched, when a voltage is applied at one end. Supposing the forward wave is VF, then a reflected wave VR will be generated by the mismatch in the opposite direction. Under steady state conditions, the magnitude of the voltage reflection coefficient, |rV| is given by |VF|/|VR|. This equation relates the voltage standing wave ratio (VSWR) to the magnitude of the voltage reflection coefficient.

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  10. Intrinsic Impedance of Free Space

    Intrinsic Impedance

    A parameter of impedance can be applied to free space, and this is it! It works out to about 377 W. Whereas in a circuit, an impedance is the ratio of voltage to current, in free space, the intrinsic impedance is the ratio of the electric field, E to the magnetic field, H. It should be applied to a plane wave in which E, H and the direction of propagation are mutially at right angles.

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  11. Dual Modulus Prescaler Synthesizer

    Dual Modulus Prescaler

    Dual modulus refers to the first divider in the feedback loop of which the division ratio can be P or P+1, controlled by the contents of the A and N counters. The value programmed into the A counter must be less than that programmed into the N counter. The contents of P, A and N are chosen according to this formula to give the required division ratio from the VCO output frequency to the phase detector. Modern synthesizer ICs normally include all counters in one package.

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  12. Poynting Vector

    Poynting Vector

    The Poynting vector is the vector cross product of an electric field and a magnetic field. When evaluated, the result of this comprises some interesting components, including the power flux density in the direction of propagation which is probably the most useful. It is often used in the study of how plane radio waves are propagated through free space and various other media.

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    Maxwell's Equations

    For some interesting background on Maxwell's equations, see Halliday and Resnick. James Clerk Maxwell (1831-1879) was the Scottish physicist who was responsible for the unification of electricity, magnetism and light into one set of equations, the famous Maxwell's Equations. He was the first to introduce the concept of an electric field and to provide a mathematical treatment for Faraday's 'lines of force'. He also identified light as a form of electromagnetic radiation. It was not until many years after Maxwell died that Hertz generated and detected the electromagnetic radiation that Maxwell had predicted.

  13. Maxwell 1

    Maxwell 1

    This was derived from Gauss's law for electricity an relates charge and electric field. D is the surface charge density and r is the total charge:

    • The charge on an insulated conductor resides on its surface.
    • Like charges repel and unlike ones attract as the inverse square of their separation.

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  14. Maxwell 2

    Maxwell 2

    Also known as Gauss's law for magnetism, this equation results from the fact that, unlike with charges, magnetic monopoles have not been observed. B is the magnetic flux density.

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  15. Maxwell 3

    Maxwell 3

    This is Gauss's version of Faraday's law for induction and associates the induced voltage with the speed at which the conductor is moved, or the coupled field is changed, within the magnetic field. The negative sign is attributable to Lenz's law which states that the induced voltage is generated in a direction that opposes the force causing it. E is the electric field.

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  16. Maxwell 4

    Maxwell 4

    This is Gauss's version of Ampere's Law and describes the magnetic effect of a changing electrical field or of a current. It covers the universal range of media from perfect conductors (D=0) to perfect insulators (J=0). H is the magnetic field and J is the current density.

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  17. Optical Coupler

    Optical Coupler

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  18. Propagation Coefficient

    Propagation Coefficient

    Take the circuit model for a short element of practical (lossy) transmission line, apply the circuit network equations, derive the differential equations. Then, if you solve them and express the result in exponential form you get an equation that describes the way the voltage amplitude and phase varies at different positions down the line. There will be both a phase and an amplitude contribution.

    a is the attenuation constant in nepers per metre, assuming use of SI units.

    A neper is a logarithmic unit used to express power ratios, very like decibels, but with a slightly different definition. The attenuation constant in nepers is calculated simply from from the natural logarithim of the power ratio.

    b is the phase constant in radians per metre.

    g is the propagation constant, a complex quantity which combines together the attenuation constant, the real coefficient and the phase constant, the imaginary coefficient.

    If the transmission line is loss-free, a is zero. Quite often in practical set-ups, particularly at low frequencies, putting a equal to zero drastically simplifies the calculations, so it is well worth finding an excuse to do this.

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  19. Differential Amplifier

    Differential Amplifier

    An ideal differential amplifier has an inverting and a non-inverting input terminal (- and + respectively) and an output terminal. The voltage at the output terminal VOUT, is related to the voltages at the input terminals (V- and V+) by this equation, where A is usually known as the open loop voltage gain, a very large number. Most configurations using these devices utilise feedback and the closed loop equations are greatly by the assumption that 1/A is approximately equal to zero.

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  20. Intrinsic Impedance

    -

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  21. Radar Equation

    Radar Equation

    This is the famous radar equation.

    It applies to primary radar, that is, where the target is a completely electrically passive object and does not deliberately transmit anything back to the radar receiver. PT is the transmitted power arriving at the transmission antenna. GT and GR are the gains of the transmit and receive antennas respectively. Often a common antenna is used so GT = GR. s is the radar cross section, that is, the equivalent electrical area of the target determining how much of the incident power is reflected. l is the wavelength and S/N is the signal to noise ratio necessary at the radar receiver to give the specified maximum sensitivity. T is the standard noise temperature of the receiver, normally 290 K assuming it is uncooled. B is its noise bandwidth and k is Boltzmann's constant. The maximum target distance under these conditions is d.

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