A transmission line is a physical structure that will guide an electromagnetic wave between two ports. Basically, the transmission systems are used to transfer energy from one point to another. It is very important in the RF and Microwave application. The theory covered all about the field analysis and basic circuit theory which are very important in microwave network analysis. To see the phenomenon of wave propagation on a transmission line, an approach using Maxwell's equations was applied. There are several type of transmission line such as two wire line, coaxial cable, waveguide and planar transmission lines. The so-called waveguide was used as transmission line in this project as a filter. This chapter is to give brief overview of the transmission line theory mainly on the waveguide to facilitate future discussion on other chapter.
2.1 Transmission Line Theory
Transmission line theory was analysed using circuit theory but there is difference between them which is electrical size. Assume that the physical dimensions of a network are much smaller than the electrical length, while transmission line may be considerable fraction of wavelength, or more. Thus, transmission line is a distributed-parameter network, where voltages and currents can vary in magnitude and phase over length. Thus any transmission line can be represented by a distributed electrical network as shown in figure 2.1. It comprises series inductors and resistor and shunt capacitors and resistors.
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Figure 2.1 Lumped-element equivalent circuit of transmission line
Figure 2.1 represented a two-wire line as a transmission line where R, L, G, C defined as follows:
R=Series resistance per unit length, in â„¦/m.
L=Series inductance per unit length, in H/m.
G=Shunt conductance per unit length, in S/m.
C=Shunt capacitance per unit length, in F/m.
From figure 2.1, simplified using Kirchhoff's voltage law and Kirchhoff's current law, obtained the following differential equation 2.1 and 2.2. These equations are in time domain form of transmission line equations. However, both equations being simplified into sinusoidal steady-state condition, with cosine-based phasors as in equation 2.3 and equation 2.4.(Pozar, 2005)
2.1.1 Wave propagation on a Transmission Line
From equation 2.3 and 2.4, both equations can be solved simultaneously to derive for wave equation in term of V (z) and I (z). Equation 2.5 and 2.6 is being simplified where Î³ is complex propagation constant, Î± is attenuation constant and Î² is phase constant.
2.1.2 Lossy and Lossless Transmission Line
Equation 2.7 above was for a general transmission line, including loss effect. For lossy cases, the attenuation constant is being considered in the propagation constant equation. However for the loss of the transmission line is very small and can be neglected will cause the attenuation to be approximately zero. Hence the equation will be simplified accordingly as shown in equation 2.8 into equation 2.9.
2.1.3 Classification of Wave Solutions
There are three type of wave propagation in cylindrical transmission line or waveguides, TEM, TE, and TM. The geometry is characterized by conductor boundaries parallel to z-axis. Hence, electric and magnetic field can be assumed that time-harmonic fields with an ejwt dependence as in equation 2.10 and 2.11.
and represent the transverse electric and magnetic field component, while ez and hz are the longitudinal electric and magnetic field components. The equation above is for the wave propagating in +z direction and for -z direction is can be obtained by replacing the Î² with -Î². In the cases of loss present, the propagation constant will be complex by replacing jÎ² with equation 2.7. By using the Maxwell's equation, for transverse field component in term of EZ and HX can be derived as shown in equation 2.12 to equation 2.15.
is defined as the cutoff wavenumber and k is wavenumber of the material filling the transmission line or waveguide region.
Transverse electromagnetic (TEM) wave are characterized by Ez=Hz=0. However by observing equation 2.12 to equation 2.15, the transverse field will also be zero, unless. Transverse electric (TE) waves, also referred to as H-waves are characterized by Ez=0 and Hz0 while Transverse magnetic (TM) waves also reffered as E-waves are characterized by Ez0 and Hz=0. Thus all the equation derived by the characteristics of each transverse wave is simplified in table 2.1.(Pozar, 2005)
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( TEM )
( TE )
( TM )
Table 2.1 Simplified Transverse Wave equation.
2.2 Microwave Filters
Filter had played an important role in many applications on modern world. The application are diverse, from traditional fixed telecommunication system to mobile, navigation, wireless, satellite communication system and remote-sensing application. Basically, the RF and microwave filter are widely used to discriminate between wanted and unwanted frequencies. The advancement of technologies affected the RF and microwave filter became more extensive in term of implementation in practical system. All characteristics need to be considered to create a filter that will give a satisfying and convenient result for user.
2.2.1 Type of Microwave Filters
Generally there are 4 types of filter function, lowpass, highpass, bandpass and bandstop filter. This filter is basically to allow the signal according to frequency that are wanted. For lowpass and highpass filter, the barrier of frequency depends on the cutoff frequency. Lowpass filter will allow the frequency below the cutoff frequency while the highpass filter allows the frequency beyond the cutoff frequency. However for the bandpass and bandstop filter depend on the fundamental frequency. Bandpass filter will allow the signal in the band of frequency but bandstop filter is otherwise.
Figure 2.1 Type of filter
2.2.2 Filters Response
There are other characteristics of filter called filter response. First type of filter response is maximally flat, also called binomial or Butterworth response. For example, lowpass filter is specified by equation 2.16. The representation of the equation is shown in graph form in figure 2.2. Secondly, for the response of Chebyshev or equal ripple, a Chebyshev polynomial is used to specify the insertion loss of a N-order low pass filter as in equation 2.17.
where N is the order of the filter, and wc is the cutoff frequency. The passband is extends from w = 0 to w = wc.
The equation is differing from Butterworth or binomial response because it provides the flattest possible passband response for a given order. This is shown is equation 2.17. Although the passband response will have ripple of amplitude 1+k2, as shown in figure 2.2, since TN(x) will cause it oscillates between for. Thus, k2 determines the passband ripple level. For a large x, , so far the insertion loss becomes
PLRwhich also increases at the rate of 20dB/decade. But the insertion loss for the Chebyshev response is (22N/4) larger than the Butterworth response. (Pozar, 2005)
0 0.5 1 1.5
1 + k2
Figure 2.2 Maximally flat and equal-ripple low-pass filter responses. (Pozar, 2005)
Figure 2.3 Low-pass prototype Chebyshev response and corresponding band-pass filter Chebyshev response. (G. Matthaei, 2000)
The characteristic of the Chebyshev lowpass filter model is shown in figure 2.3 where the LAR represents the attenuation tolerance or ripple for the lowpass filter response, w1' is the cutoff frequency, w1 and w2 are the pass band frequency. The sharper rate of cutoff will depend on the ripple and number of order. There are also common filter response, elliptic function and linear function which are not used in this project.
2.2.3 Method of Filter Design
The ideal filter network is a network that provides perfect transmission for all frequencies range in appropriate passband region. Filters designed using the lumped element circuit consist of filter synthesis techniques, image parameter and insertion loss method. For image parameter method consist of a cascade of simpler two-port filter section to provide desired cutoff frequencies and attenuation characteristics, but do not allow the specification of a frequency response over the complete operating range. Thus, although the procedure is relatively simple, the design filter by the image parameter method often must be iterated many times to achieve the desired results.
The other method is called the insertion loss method, used network synthesis techniques to design filters with a completely specified frequency response. The design is simplified by beginning with low-pass filter prototypes that are normalized in term of impedance and frequency. Transformations are applied to convert the prototype design to the desired frequency range and impedance level.(Pozar, 2005)
2.3 Waveguide as Transmission Lines
2.3.1 Types of Waveguide
Several types of waveguides that had been used are rectangular waveguide, circular waveguide, coaxial waveguide, Elliptical waveguide, Radical waveguide and spherical waveguide. All the waveguide have different type of cross-sectional area according to certain shape. Each of this waveguide has their own mathematical representation of electromagnetic field within a uniform or uniform region. In waveguide, only one mode is capable of propagation depend on the dimension and field excitation. However, the waveguide is completely characterized by the behaviour of the dominant mode of the waveguide, commonly the lowest mode.
2.3.2 Rectangular Waveguides
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Rectangular waveguides were one type of transmission lines and are used in many applications nowadays. It has played a large variety of components such as filters, couplers, detectors, isolators and attenuators. Rectangular waveguide are commercially available for various standard waveguide bands from 1GHz to over 220GHz as shown in appendix E. In term of development in technology, a lot of application move towards miniaturization and integration. The hollow rectangular waveguide can propagate TM and TE modes, but not TEM waves, since only one conductor is present.
Î¼ , Îµ
Figure 2.4 Geometry of a rectangular waveguide
Figure 2.4 shows geometry of rectangular waveguide where it is assumed that the waveguide is filled with a material of permittivity Îµ and permeability Î¼. The width, a is usually longest than the high, b of the rectangular waveguide. The value of a and b will cause different value of cutoff frequency and each mode (combination m and n) according to equation 2.19.
The mode with the lowest cutoff frequency is called the dominant mode. Hence, the lowest fc occurs for the TE10 (m = 1, n = 0) mode according to equation 2.20 is the dominant mode of the rectangular waveguide. At a given operating frequency f, only those mode having fc < f will propagate and mode with fc > f will lead to an imaginary Î² (or real Î±), meaning that all field components will decay exponentially away from the source of excitation as shown in equation 2.7 where complex propagation constant Î³, attenuation constant, Î± and phase constant Î². Such mode is referred to as cutoff, or evanescent, modes. Of more than one mode propagating, the waveguide is said to be overmoded.(Pozar, 2005)
Another important parameter is guide wavelength, defined as the distance between two equal phase planes along the waveguide. The value of guide wavelength can be found using equation 2.21 and 2.22 below. The guide wavelength will depend on the value of frequency f.
2.4 Microwave Resonator
Microwave resonators are used in variety of application, including filters, oscillators, frequency meters, and tuned amplifier. There are various implementation of resonators at microwave frequency using distributed elements such as transmission lines, rectangular waveguide, and dielectric cavities (Pozar, 2005). There are two types of resonant circuits, series and parallel. The filter structure as shown in figure 2.4 consists of series resonators alternating with shunt resonators, an arrangement which is difficult to achieve in a practical microwave structure. In a microwave filter, it is much more practical to use a structure which is approximates the circuit such as shown in figure 2.6 below. In this structure all of the resonators are of the same type, and an effect like alternating series and shunt resonator is achieved by the impedance inverters.
Figure 2.5 The odd number of order band-pass filter. (G. Matthaei, 2000)
By using equation from 2.23 to 2.26, value of capacitance and inductance in figure 2.5 can be calculated from value of g parameter. All calculation depend on the type of resonator being used either shunt or parallel resonators or series resonators.
For shunt resonators For series resonators
Lumped circuit elements are difficult to construct at microwave frequencies, hence it is usually desirable to realize the resonator in distributed-element forms rather than the lumped element forms. Basically, to establish the resonance properties of resonators it is convenient to specify their resonant frequency, w0 and their slope parameter. Equation 2.27 shows the reactance slope parameter for series-type resonator and equation 2.28 shows the susceptance slope parameter for shunt or parallel-type resonator.
Another important parameter for resonant circuit is its Q, or quality factor. For any resonator having series type of resonance with a reactance slope parameter,âˆ and series resistance, R has a Q shown in equation 2.29 but for resonator having shunt or parallel type of resonance with susceptance slope parameter, Ï and a shunt conductance, G has a Q shown in equation 2.30.
2.5 Impedance and Admittance Inverters
The concept of operation for impedance and admittance inverters is essentially form the inverse of the load impedance or admittance. Basically, they can be used to transform series-connected elements to shunt-connected element or vice versa. The Kuroda identities can also be used for the conversion but it's more useful for bandpass if using impedance (K) and admittance (J) inverters. A simplest form of inverters is a quarter wavelength of transmission line. If the resonators all exhibit series type of resonance and connected without impedance inverter, they will only operate like a single series of resonator. Beside a quarter wavelength line, there are also other circuit that can operate as inverter as shown in figure 2.6.
Impedance inverters Admittance inverters
J = wC
Figure 2.6 Impedance and admittance inverters. (Pozar, 2005)
Type of admittance and impedance inverters was shown in figure 2.6 where
Operation of impedance and admittance inverters
Implementation as quarter-wave transformers
Implementation using transmission lines and reactive elements
Implementation using capacitor networks.
For implementation using transmission lines and reactive elements in figure 2.6(c), the parameter for impedance inverter can be finding using equation 2.31 to equation 2.33 and for admittance inverter using equation 2.34 to equation 2.36.
Figure 2.7 A generalized band-pass filter circuit using impedance inverters
Figure 2.8 Reactance of jth resonator
Figure 2.7 shows a generalized circuit for a bandpass filter that have impedance inverter and series-type of resonator. The impedance inverter parameter K01, K12, to Kn, n+1 will correspond to desired shape of response according to the specification and can be find using equation 2.37 to 2.40. Equation 2.37 is reactance slope parameter from figure 2.8 response for the bandpass filter and selected randomly to be of any size corresponding to convenient resonator design. Normally, the value of termination RA, RB, and the fractional bandwidth, âˆ† may be specified as desired. The desired shape of response is then insured by specifying the impedance-inverter parameter. If the resonator of the filter consist of a lumped element, inductor and capacitor, and if the impedance inverter were not frequency sensitive, the equation below would be exact regardless the fractional bandwidth of the filter.(G. Matthaei, 2000)
Figure 2.9 A generalized band-pass filter circuit using admittance inverters
Figure 2.10 Susceptance of jth resonator
Figure 2.9 shows a generalized circuit for a bandpass filter that have admittance inverter and shunt-type of resonator. The impedance inverter parameter J01, J12, to Jn, n+1 will correspond to desired shape of response according to the specification and can be find using equation 2.41 to 2.44. Equation 2.41 is reactance slope parameter from figure 2.10 response for the bandpass filter and selected randomly to be of any size corresponding to convenient resonator design. Normally, the value of termination GA, GB, and the fractional bandwidth, âˆ† may be specified as desired. The desired shape of response is then insured by specifying the admittance-inverter parameter. If the resonator of the filter consist of a lumped element, inductor and capacitor, and if the impedance inverter were not frequency sensitive, the equation below would be exact despite the fractional bandwidth of the filter.(G. Matthaei, 2000)
Figure 2.11 The band-pass filter in figure 2.5 converted to use only series resonators and impedance inverters. (G. Matthaei, 2000)
2.6 Iris and H-plane Offset Rectangular waveguide filter
Iris or also known as cavity rectangular waveguide bandpass filter is example of one type of waveguide bandpass filter as shown in figure 2.13. The iris type of rectangular waveguide bandpass filter has two different design, inductive iris and capacitive iris. Material used depends on the application of the rectangular waveguide so that the waveguide fulfilled the requirement in many term and also the specification. The material must have the ability to hold the energy of the wave inside the waveguide and function. H-plane offset different with the iris in term of design. Both can be called as an iris, but to differentiate between the type of iris and also the coordination between windows of the waveguide filter.
Figure 2.12 Example of rectangular waveguide bandpass filter (Wu, 2009)
2.6.1 Filter Model of series of parallel inductive and capacitive
The Iris and H-plane rectangular waveguide are design from equivalent circuit transform from the prototype using the resonator and impedance or admittance inverter. This circuit representing a series of parallel or shunt inductive or capacitive and between it is electrical length. The value of inductor or capacitor can be calculated using the g parameter depending on the specification of the filter. All the values are normalized based on the characteristics impedance. Inductive iris was design where the iris metal plane is modelled as parallel inductive shunts between transmission lines of electrical length as shown in figure 2.13 where the value of inductor is based on the resonator and inverter of the response desired. Figure 2.14 shows the equivalent circuit of a capacitive iris and it is different with equivalent circuit in figure 2.13 in term of component used and also the equivalent circuit parameter that will be used to design the iris filter afterword.
Figure 2.13 Equivalent circuit of an Iris filter where the iris metal plane are modelled as parallel inductive shunts between transmission line of electrical length,Î¸
Figure 2.14 Equivalent circuit of an Iris filter where the iris metal plane are modelled as parallel capacitive shunts between transmission line of electrical length,Î¸
Figure 2.13 and 2.14, show that the equivalent circuit that can be used as a modelled of an iris and H-plane offset filter. The number of inductive or capacitive will depended on the number of order. The number of order determines the specification of the design. However, both inductive and capacitive will have different type of design in term of iris shape. To design the iris, symmetrical window needed to be used to find the value of the iris width. For the H-plane offset, asymmetrical window needed to be used to find the H-plane offset width.
2.6.2 Iris and H-plane Offset Configuration
The circuit shown in figure 2.13 and figure 2.14 are converting to iris design by using equivalent circuit parameter as shown in figure 2.17 to figure 2.20. Iris structure contains a geometrical discontinuity and design as a four terminal or two terminal pair. The description of the propagating modes is effected by representation of the input and output waveguides as transmission lines and by representation of the discontinuity as a four terminal-constant circuit as in figure 2.19 to figure 2.23. Quantitatively, the transmission line requires the indication of their characteristic impedance and propagation wavelength, the four terminal circuits and the location of the input output terminal.
Figure 2.17 Top view of cavity filter structure for shunt-inductance coupled waveguide filter
Figure 2.17 shows a top view of cavity filter for shunt inductance coupled waveguide filter. Basically, the cavity filter is being converted from the shunt or parallel inductance as shown in figure 2.13. Based on the figure, the important parameter needed to be calculated is the electrical length which is the length between iris and also the width of the iris, d. For this case, the iris thickness is being assumed to be approximately zero. Hence, the equivalent circuit that can be used to find the important parameter is shown in figure 2.19 to figure 2.20 using the equation 2.34 to equation 2.40. The iris that can be design is either capacitive iris using shunt-capacitance or inductive iris using shunt-inductance.
The other design of rectangular waveguide is H-plane offset rectangular waveguide bandpass filter. Basically, the design have the same procedure with the iris rectangular bandpass filter design but different in term of equivalent circuit parameter. From the definition, the design is only offsetting the H-plane and cause the rectangular waveguide has different size of obstacles. The obstacle will filter out the unwanted frequency according to the specification.
Figure 2.18 Top view of H-plane offset filter structure for shunt-inductance coupled waveguide filter
Based on figure 2.18, the H-plane offset filter structure are converted from figure 2.13 which is the shunt-inductance structure by using the equivalent circuit parameter of asymmetrical structure as shown in figure 2.21 and 2.22. The structure has the same width, b at all structure from input to output despite the size of the window cause by the offsetting of H-plane. The parameter of electrical length, Î¸ can be calculated using step same with the iris rectangular waveguide.
2.6.3 Symmetrical Window
Figure 2.19 Equivalent circuit parameter for capacitive two obstacles and window in rectangular waveguide (N.Marcuvitz, 1993)
For the symmetrical case d'=b - d:
Figure 2.19 shows the equivalent circuit parameter for the unsymmetrical case where. The window or the iris is formed by two obstacles which are up and down of the rectangular waveguide. However for this equivalent, there is restriction that is important to applying this formula. The equivalent circuit on figure 2.19 and equation 2.45 to equation 2.48 is valid in the range b/Î»g<1/2 for unsymmetrical case and b/Î»g<1 for the symmetrical case. For equation 2.45, its can apply only in the range 2b/ Î»g<1 with an estimated that occur less that 5% at the lowest wavelength range. Equation 2.46 is applicable in the range b/ Î»g<1 with an error of less than about 5% and in the range 2b/ Î»g<1 to within 1%. Equation 2.47 is a small part approximation that agrees with equation 2.46 to within 5% in the range d/b<0.5 and b/ Î»g<0.5. The small obstacle approximation which is the equation 2.48 agrees with equation 2.46 to within 5% in the range d/b>0.5 and b/ Î»g<0.4.
The quantities B Î»g/Y0b and Y0b/B Î»g from equation 2.46 are represent in the form of graph as shown in appendix B. The graph is plotted as a function of d/b with b/ Î»g as a parameter.
Figure 2.20 Equivalent circuit parameter for inductive two obstacles and symmetrical window in rectangular waveguide (N.Marcuvitz, 1993)
The design of iris rectangular bandpass filter using equivalent circuit parameter in figure 2.20 and equation 3.49 to equation 2.51 also have certain restriction needed to be follow. The equivalent is applicable in the wavelength range between 2/3a< Î»<2a. Equation 2.49 is estimated to be in error by less that 1% in the range a< Î»<2a but for range 2/3a< Î»<a the error is larger. The approximation form which is equation 2.39 valid in small-aperture range, agrees with equation 2.49 to within 4% for d<0.5a and a<0.92Î». For equation 2.51 is valid in the small-obstacle range of d'<0.2a and a<0.9Î» agrees with equation 2.49 to within 5%.
The equation 2.49 is represented in the form of graph as in appendix C where XÎ»g/Z0a is plotted as a function of d/a for the range 0 to 0.5 and for various values of a/Î».
2.6.4 Asymmetrical Window
Figure 2.21 Equivalent circuit parameter for capacitive one obstacles and window in rectangular waveguide (N.Marcuvitz, 1993)
Based on figure 2.22, the equivalent circuit parameter has the same restriction as to figure 2.19 except that the value of Î»g is replaced by Î»g/2 and also for the plotted graph, if the Î»g is replaced by Î»g/2, a plot of B Î»g/2Y0b as a function of d/b with 2d/Î»g as a parameter where B/Y0 is now the relative susceptance of a window formed by one obstacle.
Figure 2.22 Equivalent circuit parameter for inductive one obstacles and asymmetrical window in rectangular waveguide (N.Marcuvitz, 1993)
In this project, figure 2.23 is being used as the equivalent circuit parameter in the H-plane offset rectangular waveguide filter since the shunt-inductive is used. For figure 2.23, the equivalent circuit is applicable in the wavelength range a<Î»<2a. Equation 2.55 is valid in the wavelength range a<Î»<2a with estimated error of about 1%. Equation 2.56 valid for the small aperture range agrees to within 5% if d/a<0.3 and a/Î»<0.8. Furthermore, equation 2.57 is valid in the small obstacle range for d'/a<0.2 and a/Î»<0.8 to within 10%.
From equation 2.55, a graph can be obtained and plotted as shown in appendix D where the graph Z0a/XÎ»g is plotted as a function of d/a in the range of 0.1 to 0.7.
2.7 Fabrication Process
There are several ways to fabricate the waveguide for instance, using milling machine. However, type of fabrication process is selected based on the size, cost and time. Milling is the process of machining flat, curved, or irregular surfaces by feeding the work piece against rotating cutter. Milling machine is a tool that can be used to machine work piece or solid materials, ranging from aluminium to stainless steel. Basically, it can be classified as horizontal and vertical. This is referring to the machine orientation of the spindle. The milling machine can perform several numbers of operations such as cutting, planning and drilling. The precision of the design depend on the milling cutter and also the technology system of the machine. It can be operating manually or automatically using computer numerical control (CNC).
Milling machine has many types and each type is characterized by class of form, either vertical or horizontal and also the movable of the spindle. Types of milling machine such as knee-type, universal horizontal, ram-type, universal ram-type, and swivel cutter head ram-type milling machine. Milling machine tools and equipment such as milling cutters are usually made of high-speed steel and are available in a great variety of shapes and sizes for various purposes. Even the milling cutter may be made with various type such as saw teeth, helical milling cutters, metal slitting saw milling cutter, side milling cutters, end milling cutter, T-slot milling cutter, woodruff keyslot milling cutters, angle milling cutters, gear hob, concave and convex milling cutter, corner rounding milling cutter, and special shaped-formed milling cutter. Figure 2.23 shows the milling machine with its important part.
B: Vertical Head
F: Crossfeed Handle
G: Vertical Feed Crank
I: Vertical Positioning Screw
L: Table Handwheel
M: Table Transmission
N: Ram Type Overarm
O: Arbor Support