Chapter 6: Microwave

1) Applications 2) Series and Parallel Resonant Circuits 3) Loaded and Unloaded Q 4) Transmission Line Resonators 5) Rectangular Wave guide Cavities

6) Q of the TE10l Mode 7) Coupling 8) Gap Coupled Resonator 9) Excitation of Resonators 10) Summary Microwave Resonators Applications

Microwave resonators are used in a variety of applications such as: • Filters • Oscillators • Frequency meters • Tuned Amplifier

The operation of microwave resonators are very similar to that of the lumped-element resonators of circuit theory, thus we will review the basic of series and parallel RLC resonant circuits first. We will derive some of the basic properties of such circuits. Series Resonant Circuit

Near the frequency, a microwave resonator can be modeled as a series or parallel RLC lumped-element equivalent circuit.

1 ZRjLj  in C  1 2 1 2  1    Pin  I Zin  I  R  j L  j  2 2   C  1 Average magnetic energy: W  I 2 L m 4 1 1 Average electric energy: W  I 2 e 4  2 C

Resonance occurs when the average A series RLC resonator and its response. (a) The series RLC circuit. stored magnetic (Wm) and electric (b) The magnitude energies (We) are equal and Zin is purely versus frequency. real. Series Resonators

• The frequency in which 1  o  LC is called the resonant frequency. • Another important factor is the Quality Factor Q. Average Energy Stored Q  Energy Loss / Second  L 1 BW 1/ Q Q o R o RC Near the resonance     o  1 22  o ZRjLin (1 22 )  RjL ( )  LC  A series RLC resonator and its 2RQ  response. (a) The series RLC circuit. Rj   (b) The input impedance magnitude o 2 2  versus frequency.  o    o   o  2 

ZRjLin  2  Series Resonators

The input impedance magnitude versus frequency.

 L 1 Q o R o RC   BW Fractional Bandwidth is defined as:  BWQ 1/  o 2 and happens when the average (real) power delivered to the circuit is one-half that delivered at the resonance. • Bandwidth increases as R increases. • Narrower bandwidth can be achieved at higher quality factor (Smaller R). Parallel Resonators

1 11 ZjCin   RjL

Resonance occurs when the average stored magnetic and electric energies

are equal and Zin is purely real. The input impedance at resonance is equal to R.

Average Energy Stored Q  Energy Loss / Second

1   R o QRC o A parallel RLC resonator and its LC o L response. (a) The parallel RLC circuit. (b) The input impedance magnitude Note: Resonance frequency is equal to the versus frequency. series resonator case. Q is inversed. Parallel Resonators

The input impedance magnitude versus frequency. R QRC o BW 1/ Q o L • Bandwidth reduces as R increases.  • Narrower bandwidth can be achieved at higher quality factor (Larger R).

 1  1 R Close to the resonance frequency: Zin   2 j  C     R  1 2 j Q  /   o o 1 R  Zin  jC2( o ) Loaded and Unloaded

The Q factors that we have calculated were based on the characteristic of the resonant circuit itself, in the absence of any loading effect (Unloaded Q).

A resonant circuit connected to an external load, RL. In practice a resonance circuit is always connected to another circuitry, which will always have the effect of lowering the overall Q (Loaded Q).

 o L  for series connection  RL Qe   R  L for parallel connection  L  o  Loaded Q Factor

If the resonator is a series RLC and coupled to an external load resistor

RL, the effective resistance is:

R e  R  R L

If the resonator is a parallel RLC and coupled to an external load

resistor RL, the effective resistance is:

R e  R  R L / R  R L

Then the loaded Q can be written as:

111  QQQL e Note: Loaded Q factor is always smaller than Unloaded Q.

Transmission Line Resonators (Short Circuited) • Ideal lumped element (R, L and C) are usually impossible to find at microwave frequencies. • We can design resonators with transmission line sections with different lengths and terminations (Open or Short). • Since we are interested in the Q of these resonators we will consider the Lossy Transmission Line. For the special case of :   / 2

Z in Z o tanh   j  

tanh   j tan   Z  Z  A short-circuited length of lossy in o 1 j tanh  tanh  transmission line.

 L  Q  o  R 2 Transmission Line Resonators (Short Circuited)

n • The resonance occurs for   n 1,2,3, 2

ZZin o[/] j o 

 L  Q  o  R 2

A short-circuited length of lossy transmission line and the  distributions for n=1,   2 Transmission Line resonators

Zo Z in    j  /2 o

Z R  o  1 C  L  2   4 ooZ  o C

 Q  2

An open-circuited length of lossy transmission line, and the voltage distributions for n = 1 resonators. Rectangular waveguide cavity resonator  We can look at them as short circuit section of transmission line

22 2 mn  mn  ab Boundry conditions enforced

EExy0 for z  0, d l  mn dl mn l  1,2,3... d 2 222 lmn 2        dab     2 2 2 2 f  l   m   n           c  d   a   b 

then the resonance frequency A rectangular resonant cavity, and the electric field 222 distributions for the TE and TE resonant cl  m  n 101 102 f    modes. 2 dab   if b a d the lowest and dominant resonant TE (resp TM) mode will be TE101 (resp. TM110) Rectangular waveguide cavity resonator Q factor for TE10l

1 11 Q  QQcd 1 Q  d tan 3 2 ad 1 Q  c 2 2 R 23 3 23 3 s 22lab bd lad  ad

Rso 2 

    Resonators - coupling R Critical coupling occurs when 2Q  Q  Q L e u Z Zo i n Q If we define coupling coefficient g  u then we have Qe series RLC parallel RLC • undercoupled resonator if g<1 Z0 R • critically coupled resonator if g=1 (resonator matched to the feed line) g  g  R Z • overcoupled resonator if g>1 0

g  1 g  1 g  1

Series RLC circuit

Gap coupled microstrip resonator resonators  2 Serial RLC coupling acts as impedance inverter

 Z  tan l  b   c  z    j  where bc  ZoC Zo  bc tan l 

Resonance occurs when tanlb c  0 The coupling of the feed line to the resonator lowers its resonant frequency Gap coupled microstrip resonator resonators  

()  o ZZ()o  j 2Qb22 b coc

 RZ  o 2 bc  2Qbc 2Q

Zo  the coupling coeficient g R 2 2Qbuc

 b for critical coupling Zo  R c  2Q  for undercoupled resonator bc  2Q

 for overcoupled resonator bc  2Q