1 ENE 428 Microwave Engineering Lecture 11 Excitation of Waveguides and Microwave Resonator

2 Excitation of WGs-Aperture coupling  WGs can be coupled through small apertures such as for directional couplers and power dividers

3 A small aperture can be represented as an infinitesimal electric and/or magnetic dipole.  Both fields can be represented by their respective polarization currents.  The term ‘small’ implies small relative to an electrical wavelength. Fig 4.30

4 Electric and magnetic polarization Aperture shape ee mm Round hole Rectangular slot (H across slot)  e is the electric polarizability of the aperture.  m is the magnetic polarizability of the aperture. (x 0, y 0, z 0 ) are the coordinates of the center of the aperture.

5 From Maxwell’s equations, we have Thus since and has the same role as and, we can define equivalent currents as and Electric and magnetic polarization can be related to electric and magnetic current sources, respectively

6 Coupling through an aperture in the broad wall of a wg (1)  Assume that the TE 10 mode is incident from z < 0 in the lower guide and the fields coupled to the upper guide will be computed.

7 Coupling through an aperture in the broad wall of a wg (2)  The incident fields can be written as  The excitation field a the center of the aperture at x = a/2, y = b, z = 0 can be calculated.

8 Coupling through an aperture in the broad wall of a wg (3)  The equivalent electric and magnetic dipoles for coupling to the fields in the upper guide are Note that we have excited both an electric and a magnetic dipole.

9 Coupling through an aperture in the broad wall of a wg (4)  Let the fields in the upper guide be expressed as where A +, A - are the unknown amplitudes of the forward and backward traveling waves in the upper guide, respectively.

10 Coupling through an aperture in the broad wall of a wg (5)  By superposition, the total fields in the upper guide due to the electric and magnetic currents can be found for the forward waves as and for the backward waves as where

11 Microwave Resonator  A resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally น oscillates at some frequencies, called its resonant frequency, with greater amplitude than at others.  Resonators are used to either generate waves of specific frequencies or to select specific frequencies from a signal.  The operation of microwave resonators is very similar to that of the lumped-element resonators of circuit theory.

12 Basic characteristics of series RLC resonant circuits (1)  The input impedance is  The complex power delivered to the resonator is

13 Basic characteristics of series RLC resonant circuits (2)  The power dissipated by the resistor, R, is  The average magnetic energy stored in the inductor, L, is  The average electric energy stored in the capacitor, C, is  Resonance occurs when the average stored magnetic and electric energies are equal, thus

14 The quality factor, Q, is a measure of the loss of a resonant circuit.  At resonance,  Lower loss implies a higher Q  the behavior of the input impedance near its resonant frequency can be shown as

15 A series resonator with loss can be modeled as a lossless resonator   0 is replaced with a complex effective resonant frequency. Then Z in can be shown as  This useful procedure is applied for low loss resonators by adding the loss effect to the lossless input impedance.

16 Basic characteristics of parallel RLC resonant circuits (1)  The input impedance is  The complex power delivered to the resonator is

17 Basic characteristics of parallel RLC resonant circuits (2)  The power dissipated by the resistor, R, is  The average magnetic energy stored in the inductor, L, is  The average electric energy stored in the capacitor, C, is  Resonance occurs when the average stored magnetic and electric energies are equal, thus

18 The quality factor, Q, of the parallel resonant circuit  At resonance,  Q increases as R increases  the behavior of the input impedance near its resonant frequency can be shown as

19 A parallel resonator with loss can be modeled as a lossless resonator.   0 is replaced with a complex effective resonant frequency. Then Z in can be shown as

20 Loaded and unloaded Q  An unloaded Q is a characteristic of the resonant circuit itself.  A loaded quality factor Q L is a characteristic of the resonant circuit coupled with other circuitry.  The effective resistance is the combination of R and the load resistor R L.

21 The external quality factor, Q e, is defined.  Then the loaded Q can be expressed as

22 Transmission line resonators: Short- circuited /2 line (1)  The input impedance is

23 Transmission line resonators: Short- circuited /2 line (2)  For a small loss TL, we can assume  l << 1 so tan  l   l. Now let  =  0 + , where  is small. Then, assume a TEM line,  For  =  0, we have or

24 Transmission line resonators: Short- circuited /2 line (3)  This resonator resonates for  = 0 (l = /2) and its input impedance is  Resonance occurs for l = n /2, n = 1, 2, 3, …  The Q of this resonator can be found as

25 Transmission line resonators: Short- circuited /4 line (1)  The input impedance is  Assume tanh  l   l for small loss, it gives  This result is of the same form as the impedance of a parallel RLC circuit

26 Transmission line resonators: Short- circuited /4 line (2)  This resonator resonates for  = 0 (l = /4) and its input impedance is  The Q of this resonator can be found as