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and allow an accurate modeling of the circuits within a finite frequency range.
“Compact models of broadband distributed microwave circuit are a valuable tool for
efficient modeling and design of complex circuits. Lumped element circuit models are not
only compact but also describe the physical quantities, like energy and power of the
original structure”[1, 2]. The advantage of this project is that in case of a complicated
structure, we can get the equivalent simple circuit easily without focusing on every element
on the structure. This new lumped element circuit will have the same I-V behavior at the
ports as that of the original one.
1.2. Tools
1.2.1 SONNET ®
SONNET®’s 3D Planar Electromagnetic software suites of high-frequency
electromagnetic (EM). It can simulate microstrip, stripline, coplanar waveguide and similar
structures. Complicated geometries in which variation of material occurs frequently in all
these dimensions can be simulated in SONNET with some limitation but it poses high cost
in terms of memory and time [3].
1.2.2 Spice 3
Spice 3 is a simulation tool for lumped element circuit, and it is on industry standard.
Spice 3 can analyze linear and nonlinear circuits. It can also perform DC, nonlinear transient, and small-signal AC analysis. Spice was developed at the University of
California at Berkeley [4].
1.2.3 Matlab® R2012a
Matlab is 4th generation programming language and powerful computational
software. Vector Fitting [5, 6, 7] procedure is already implemented in Matlab. Due to ease
of use, circuit synthesis procedures were also implemented in Matlab.
1.3. Preliminaries
1.3.1 System
System is an interconnection of devices, subsystems and components [8]. For the
system to be useful or controllable, it must have at least one input and one output.
Importantly, the system can be represented by its behaviors at its ports. The system
response can be either described in time domain or in frequency domain using laplace
transform. The properties of the system include causality, time invariance, linearity,
stability and memory.
Figure 1: System
(1) Causality
A system is causal if the output at any time depends only on values of the input at the
present time and in the past. The region of convergence (ROC) associated with the system
function for a causal system is at the right side of right-most pole on complex frequency
plane. corresponding to the input x(t), a time-invariant system will have y(t-ݐ) as the out put
when x(t-ݐ)is the input.
(3) Linearity
If an input consists of the weighted sum of several signals, then the output is the
superposition-that is, the weighted sum-of the responses of the system to each of those
signals. For example, let ݕଵ(t) be the response of a continuous time system to an input ݔଵ(t),
and let ݕଶ(t) be the output corresponding to the input ݔଶ(t). then the system is linear if :
The response to ݔଵ(t)+ ݔଶ (t) is ݕଵ(t)+ ݕଶ(t).
The response to aݔଵ(t) is aݕଵ(t),where a is any complex constant.
(4) Stability
The stability of a system means that if the system has bounded input, it will have
bounded output (BIBO). An LTI system is stable if and only if the ROC of its system
function H(s) includes the entire imaginary axis of complex frequency plane.
(5) Memory
A memoryless system is a system of which its output for each value of the
independent variable at a given time is dependent only on the input at the same time. To the