Resonators and Filters |
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Figure 7.5 Solving the quality factors of a resonator from the input admittance versus frequency on the Smith chart. Input admittance y in of an overcoupled resonator is presented.
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(7.12)
where Pa = I 2/4 is the available power from the current source and P L is the power coupled to the load.
7.1.4 Transmission Line Section as a Resonator
Let us consider a section of a short-circuited transmission line with a length l , shown in Figure 7.6. Its series resistance, series inductance, and parallel capacitance per unit length are R ′, L ′, and C ′, respectively. For an air-filled line the parallel conductance may be neglected. The input impedance of the line is
Z in = Z 0 tanh ( jbl + al ) = Z 0 |
tanh al + j tan bl |
(7.13) |
1 + j tan bl tanh a l |
We assume that the total loss is small so that tanh al ≈ al . Close to the
frequency f r , at which l = lg /2, tan bl = tan (p + p Df /f r ) = tan (pDf /f r ) ≈ p D f /f r . Now (7.13) simplifies to
148 Radio Engineering for Wireless Communication and Sensor Applications
Figure 7.6 Short-circuited l g /2-long transmission line and its equivalent circuit.
Z in ≈ Z 0 (al + jp D f /f r ) |
(7.14) |
The resistance is constant and the reactance is directly |
proportional |
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to the frequency deviation; thus a short-circuited lg /2-long |
transmission |
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line resembles a series resonant circuit. Because |
Z 0 = √L ′/C ′, |
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a = (R ′/2) √C ′/L ′, and bl = vr l √L ′C ′ = p, then |
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On the other hand, close to the resonance frequency, the input impedance of a series resonant circuit made of lumped elements is
Z in ≈ R + j 2L D f |
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By comparing (7.15) and (7.16), we see the relationship between the
distributed quantities and the lumped elements of the equivalent circuit: R = R ′l /2, L = L ′l /2, and C = 1/(v2r L ) = 1/(v2r Z 02 C ′l /2).
The reactance of a short-circuited 50-V line and that of a corresponding LC series resonant circuit are compared in Figure 7.7. We see that close to the resonance frequency these two circuits have nearly similar properties. The quality factor of the transmission-line resonator is
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As before, we can show that an open-circuited lg /4-long line corresponds to a series resonant circuit, whereas a short-circuited l g /4-long line and an open-circuited lg /2-long line correspond to a parallel resonant circuit.
Resonators and Filters |
149 |
Figure 7.7 The reactance X versus frequency for a short-circuited transmission line and an LC series resonant circuit. The length of the line is l g /2 at f r .
7.1.5 Cavity Resonators
Metal cavities can be used as resonators at microwave frequencies. A cavity resonator has a closed structure, except for the couplings to the external circuit, and thus has no radiative loss. The quality factor of a cavity resonator may by high—several thousands or even more.
Often a cavity resonator is made of a section of a waveguide or a coaxial line short-circuited at both ends. At resonance a standing wave is formed in the cavity as the wave bounces back and forth between the ends. Thus the length of the cavity is half of a wavelength or a multiple of that at the resonance frequency. A given cavity has an infinite number of resonance frequencies, unlike a resonator made of lumped elements.
Figure 7.8 shows three ways to couple a field into a cavity or from it: a loop, a probe, and a hole. A prerequisite for an efficient coupling is that
Figure 7.8 Couplings to a cavity: (a) loop; (b) probe; and (c) hole.
150 Radio Engineering for Wireless Communication and Sensor Applications
the fields of the resonance mode have some common components with the fields of the coupling element. Therefore, a loop at the maximum of the magnetic field perpendicular to the field or a probe at the maximum of the electric field along the field works as a good coupling element. In order for the hole coupling to be successful, the fields of the waveguide and cavity should have some common components at the coupling hole. Different resonance modes that may be excited at a given frequency can often be discriminated by choosing the proper position for the coupling element. The coupling coefficient b ci = Q 0 /Q ei at port i is used to describe the strength of the coupling.
Close to the resonance, a cavity resonator may be modeled with a parallel resonant or series resonant RLC circuit. A parallel resonant circuit may be transformed into a series resonant circuit and vice versa by changing the position of the reference plane at which the resonator input is assumed to be. Often it is more practical to treat a cavity resonator with its quality factors.
Let us consider the air-filled rectangular cavity shown in Figure 7.9 [2]. We can regard it as a section of a rectangular waveguide having short circuits at planes z = 0 and z = d . The phase constant of the TEnm and TMnm wave modes is
bnm = √ |
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where k 0 = v√m 0 e0 = 2p/l 0 . Since at resonance the length of the cavity is llg /2 (l is integer),
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Figure 7.9 Rectangular cavity resonator.
Resonators and Filters |
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151 |
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From (7.18) and (7.19) we solve |
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k 0 = k nml = √ |
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Resonance frequencies corresponding to these discrete values of k nml are
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This equation is valid for both the TEnm and TMnm wave modes. Resonance modes having the same resonance frequency but a different field distribution are called degenerate modes.
Let us study in more detail the resonance mode TE101 , that is, a resonance that is excited in a cavity that is half-wave long at the fundamental wave mode TE10 . This is the mode having the lowest resonance frequency if b < a < d . We find the field distribution by summing up waves propagating into the +z and −z directions:
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Because Ey = 0 at z = 0, E − = −E +. Ey must also be zero at z = d , which leads to b = p/d . By denoting E0 = −2jE +, we get
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152 Radio Engineering for Wireless Communication and Sensor Applications
The energy stored is the maximum energy of the electric field because then the energy of the magnetic field is zero. This energy is
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The loss can be calculated if the surface current Js and the surface resistance R s are known on all walls of the cavity. The surface current, or current per unit width, is
Js = n × H |
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where n is a unit vector perpendicular to the surface. Equations (7.25) through (7.27) are valid in case of ideal, lossless conductors but they can be applied with good accuracy in case of low-loss conductors. The power loss is obtained by integrating over all the surfaces of the cavity:
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By combining (7.28) and (7.30) we get the quality factor of TE101 mode as
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(7.32)
If the cavity is filled with a lossy dielectric having a permittivity of e
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This equation is valid for all resonance modes.