^{1}

^{*}

^{1}

^{2}

In this paper an attempt has been made to find the aperture field distribution in a rectangular waveguide for non-sinusoidal, periodic excitations using Multiple Cavity Modeling Technique. The excitation functions, considered, are square, trapezoidal and clipped sine wave in nature. In the present analysis these time domain excitation functions have been represented in terms of a truncated Fourier series consisting of the fundamental frequency and its higher harmonics. Within the waveguide the fundamental frequency will give rise to a dominant mode excitation whereas the higher order modes will excite dominant as higher order modes. If the higher harmonics are assumed suppressed then the waveguide is subjected only to a dominant mode excitation. Results for dominant mode reflection coefficient (magnitude), VSWR and complex transmission coefficient have been computed and compared with theoretical data. The excellent agreement between them validates the analysis.

Waveguide and waveguide based components are used since World War II and they are still continued to be in use. These waveguide based elements have got wide applications in ground based, air borne and ship borne radars as well as in onboard satellite application in various frequency bands ranging from 1 GHz to 1000 GHz. The main reason for this is their high power handling capability, ability to sustain high environmental variations, ruggedness and the possibility of achieving high accuracy in fabrication. Modern radar system employs a number of waveguide components and circuit elements. Some of them are the irises, septum and windows, filters, waveguide Tee-junctions, waveguide power dividers, ortho-modal couplers and multiplexers etc. In addition to these waveguide components apertures cut in a ground plane and slots cut along the broad wall or narrow wall are extensively used in array antennas due to their large power handling capability, generation of ultra low side lobes and excellent polarization characteristics. Raw waveguides are also used for transmitting signals from one node to another in a high power microwave networks.

During post World War II, Lewin [

(MoM), Finite Difference Time Domain (FDTD) method etc. The total amount of work on these is so huge that even a brief literature survey of it will be a report of its own. However, comparatively, very few amounts of works are available on transient analysis and non-sinusoidal periodic excitations of waveguide.

In 1954 Karbowiak [

However, most of these analyses assume a pulse or impulse excitation. No attention was paid on the analysis of waveguide circuits under non-sinusoidal, periodic excitations like square wave, trapezoidal wave, clipped sine wave, triangular wave, saw tooth wave etc. Out of these the first three have huge technical significance in high power applications like radar. This is because in Continuous Wave Doppler radar if the gain and/or the input time harmonic signal of the amplifier is/are very high then the output signal may be clipped off resulting in a clipped sinusoidal wave. Depending on the clipping level it may be approximated as a trapezoidal or even a square wave.

In this paper, a methodology for the analysis of a waveguide under non-sinusoidal periodic excitation has been carried out and the aperture field distributions in the waveguide have been plotted for different excitation.

The proposed waveguide structure has been analyzed using Multiple Cavity Modeling Technique (MCMT) [13-15]. The cavity modeling of the structure with details of different regions and fictitious magnetic currents (equivalent source to electric field) at the apertures are shown in

distributed on apertures (and hence the fictitious magnetic currents) are unknown and to be determined. The component of electric field at the i^{th} aperture can be approximated in terms of unknown complex basis coefficients and known two dimensional piecewise traingular basis function as [13-15]^{}

and

The value of “M” in Equation (1) depends on the convergence and desired accuracy of the result. For the present analysis we have taken M = 25. Higher the value of “M” better is the accuracy and larger is the simulation time.

Since the basis function is known we need to find the value of the basis coefficients in Equation (1) to find the aperture electric field distribution/fictitious magnetic current distribution existing at the aperture. The fictitious magnetic current is basically the equivalent source of the aperture electric field and is related with it by the equivalent principle.

The boundary conditions (continuity of tangential component of magnetic field across the dielectric-dielectric boundary) at the apertures can be obtained using superposition principle and can be given by

The magnetic field scattered inside the cavity region due to the fictitious magnetic currents at the apertures can be determined by using cavity Green’s function of the electric vector potential. The cavity Green’s function has been derived by solving the Helmholtz equation for the electric vector potential for unit magnetic current source [

The tangential component of magnetic fields in the cavity region due to magnetic current in aperture “i” can be expressed as

where

whereand.

In Equations (7)-(10), 2a is the guide width, 2b is the guide height, and are the Neumann’s function, 2d is the cavity thickness/length of the waveguide, is the propagation constant for the (m, n) mode. The rest have their usual meanings.

The tangential component of magnetic fields in the waveguide region due to magnetic current in aperture “i” can be expressed as

where

The incident fields will be composed of a number TE and TM modes that are generated at the feed point discontinuity and are supported by the waveguide and can be written as

provided the incident field is assumed to be X-directed. If we further assume that the geometric distribution of the field source is such that the excitation is a linear combination of the supported modes then the amplitude and of Equations (19) and (20) can be obtained from the Fourier coefficients of the input signal. If be the rise and fall time and T be the time period then the Fourier coefficients for square, trapezoidal and clipped sine wave respectively can be given by (for all the cases and is zero)

Solving the boundary conditions, provided in Equations (3)-(6) by Galerkin’s specialization of Method of moment, where weighting function is same as expressed in equation Equations (1) and (2), the aperture electric field distribution can be obtained.

From the aperture electric field distribution, obtained using the present theory, we can easily calculate the complex reflection and transmission (T) coefficients using the following equations:

and

The electric field distribution at apertures of waveguide for square, trapezoidal and clipped sine excitations have been normalized with respect to the maximum aperture field strength and are shown in Figures 2-4 respectively. The figures show that the aperture electric field distributions are not purely sinusoidal. This is due to the presence of the higher order modes in the guide. However as the excitation changes from square to clipped sine wave it become more sinusoidal. This is because of the rapid convergence of the Fourier series and hence presence of lesser number of modes.

If we set the rise/fall time of the broken sine wave equal to T/4 then the excitation function changes to a sinusoid excitation. The dominant mode reflection coefficient (magnitude), VSWR and transmission coefficient for such case have been computed using Equations (24) and (25) and compared with the available theoretical data in _{11}|, VSWR and |S_{21}| are not shown in

The paper presents a methodology to find the aperture electric field distribution in the waveguide for square, trapezoidal and clipped sine wave excitation. The methodology is general and can be extended to other waveguide circuits excited by any deterministic and periodic waveform that can be extended in Fourier series.

It should be that noted our main objective of the present work is to propose a methodology to find the aperture field distribution in a waveguide circuit for non sinusoidal periodic excitation. This is because if we can find the aperture field distribution then from it we can calculate the complete characteristic of the circuit under the given excitation.

The proposed theory can be extended for any arbitrary not periodic waveform also. However for such cases we must know the strengths of the individual mode existing within the circuit.

The proposed theory is also applicable to other rectangular waveguide based circuits under periodic/nonperiodic non-sinusoidal excitations.

The work is supported by Department of Science and Technology, Government of India and the authors wish to express their gratitude for this support.