Know all about Microwave Networks, theory, and the analysis. We have shared the complete notes on the Microwave Networks that you must go through.
At low frequencies for electrically small circuits, lumped active and passive circuit elements are enough for analyzing the circuit leading a type of a Quasi-Static solution (assumption of negligible phase change in anywhere of the circuit) of Maxwell's equations and to Kirchoff Current and Voltage Laws with impedance concept. Moreover, fields are considered a TEM type. But this way is not possible to analyze microwave circuits. The circuit concept should modify and apply to microwave network theory developed at MIT in 1940. The reasons for using it are as follow:
- Much easier than field theory.
- Calculations are performed at terminals, not everywhere.
- Easy to modify and combine different problems.
- The field solution of Maxwell's equation gives more information at the every time and place of the network, but difficult. At microwave frequencies, although the definition of the terminal pairs for TEM line is relatively easy, the terminal pair for non - TEM line does not strictly exist.
Voltage, Current and Impedance
- The measurements of V and I at microwave frequencies are difficult due to not easily defined terminals for non-TEM waves. Because of that the fields are measured and used as
- then, the impedance can be defined as Z = V/I. Because the fields depend on the coordinates (like in waveguide), special attenuation should be given for extraction of V and I. The way is to do that
- It can be shown that the real parts of and are even in , but the imaginary parts of them are odd in . and are the even function in
Impedance & Admittance Matrices
- The v and of an N-port microwave network having n'th terminal
- The impedance matrix is in the form of [v] = [Z][i]
Similarly the admittance matrix is in the form of [i] = [Y][v]
Clear that [Y] = [Z]–1. It can be shown that
Then Zii and Zij are known as Input Impedance and Transfer Impedance, respectively.
- If the network is reciprocal (no ferrites, plasmas and active devices inside), Zij = Zii and Yij = Yii are the right relations.
- If the network is lossless, Zij and Yij are purely imaginary.
- The form [V–] = [S][V+] of the scattering matrix gives the complete description of the N port networks with the incident and reflected waves as
- Any element of the scattering matrix
- In the [S], all the Vi and Ii are defined as a reference point at the end of every lines. If the reference point is shifted, then
Where kili is the electrical length of the outward shift.
Generalized Scattering Matrix
- In the previous chapter, [S] is defined for networks with same characteristic impedance for all ports. Generally for not same impedance for all ports, a new set of wave amplitudes as
Then, generalized S-matrix
- In reciprocal networks, [S] = [S]t is symmetric,
- In lossless networks, [S] is unitary and satisfies the equation
Transmission (ABCD) Matrix
- Many microwave networks consisting cascade connection and need building block fashion in practice. ABCD matrix is defined to satisfy this as
- where port 1 and port 2 are completely isolated in the equation means that cascade multiplication is possible. The current direction is also specially designed for ABCD matrix.
- The relation between Z and ABCD matrix parameters are as
- If the network is reciprocal (Z12 = Z21), then AD – BC =1.
Equivalent Circuits for 2 Port Networks
- Transition between a coaxial line and microstrip line can be chosen as an example of two port networks. Because of discontinuity in the transition region, EM energy is stored in the vicinity of the transition leading to reactive effects mean that the transition region should be modeled as black box (There is an unlimited way of equivalent circuits, but choosing S matrix equivalence), then
- A nonreciprocal network can not be represented by passive equivalent circuit using reciprocal elements. If the network is reciprocal (there are six degrees of freedom, six independent parameters), the presentations lead naturally to T and equivalent circuits as
- If the network is lossless, the impedance and admittance matrix elements are purely imaginary, the degrees of freedom reduces to three and the elements of T and equivalent circuits should be constructed from purely reactive elements.
Signal Flow Graphs
- Signal flow graph is an additional technique to analyse microwave networks in terms of reflected and transmitted waves. Three different forms of it are given below with nodes and branches.
- The decomposition rules are also given below
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