Chapter 6: Microwave Resonators

1) Applications2) Series and Parallel Resonant Circuits3) Loaded and Unloaded Q4) Transmission Line Resonators5) Rectangular Wave guide Cavities6) Q of the TE10l Mode7) Resonator Coupling8) Gap Coupled Resonator9) Excitation of Resonators10) 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 resonance frequency, a microwave resonator can be modeled as a series or parallel RLC lumped-element equivalent circuit.

1inZ R j L j

C

Resonance occurs when the average stored magnetic (Wm) and electric energies (We) are equal and Zin is purely real.

A series RLC resonator and its response. (a) The series RLC circuit. (b) The input impedance magnitude versus frequency.

CjLjRIZIP inin

121

21 22

Average magnetic energy: LIWm2

41

Average electric energy: C

IWe 22 1

41

A series RLC resonator and its response. (a) The series RLC circuit. (b) The input impedance magnitude versus frequency.

1o LC

Near the resonance o 2 2

2 2

1(1 ) ( )

2

oin

o

Z R j L R j LLC

RQR j

1o

o

LQ

R RC

2inZ R j L

1/BW Q

Series Resonators

• The frequency in which

is called the resonant frequency. • Another important factor is the Quality Factor Q.

222ooo

SecondLossEnergyStoredEnergyAverageQ

/

The input impedance magnitude versus frequency.

1/BW Q

Series Resonators

Fractional Bandwidth is defined as:

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).

1o

o

LQ

R RC

2BW

o

A parallel RLC resonator and its response. (a) The parallel RLC circuit. (b) The input impedance magnitude versus frequency.

1o LC

11 1

inZ j CR j L

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.

oo

RQ RCL

Parallel Resonators

SecondLossEnergyStoredEnergyAverageQ

/

Note: Resonance frequency is equal to the series resonator case. Q is inversed.

The input impedance magnitude versus frequency.

o

oo

RQ RCL

12 ( )in

o

Zj C

R

1/BW Q

Parallel Resonators

• Bandwidth reduces as R increases.• Narrower bandwidth can be achieved at higher quality factor (Larger R).

oin Qj

RCjR

Z

/21

21 1

Close to the resonance frequency:

A resonant circuit connected to an external load, RL.

for series connection

for parallel connection

o

Le

L

o

LR

QR

L

Loaded and Unloaded Q Factor

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).

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

Loaded Q Factor

If the resonator is a series RLC and coupled to an external load resistor RL, the effective resistance is:

Le RRR

If the resonator is a parallel RLC and coupled to an external load resistor RL, the effective resistance is:

LLe RRRRR /

Then the loaded Q can be written as:

1 1 1

L eQ Q Q

Note: Loaded Q factor is always smaller than Unloaded Q.

A short-circuited length of lossy transmission line.

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.

jZZ oin tanh

tanhtanh1tantanh

jjZZ oin

For the special case of : 2/

2

RLQ o

A short-circuited length of lossytransmission line and the voltage

distributions for n=1,

Transmission Line Resonators (Short Circuited)

• The resonance occurs for

2

RLQ o

,3,2,12

nn

2

[ / ]in o oZ Z j

An open-circuited length of lossy transmission line, and the voltage distributions for n = 1 resonators.

2Q

Transmission Line resonators

/ 2o

ino

ZZj

oZR

4 o oC

Z

CL

o2

1

A rectangular resonant cavity, and the electric field distributions for the TE101 and TE102 resonant modes.

Rectangular waveguide cavity resonator

We can look at them as short circuit section of transmission line

2 22

mnm na b

2

mnld

2 2 22l m n

d a b

2222

bn

am

dl

cf

2 2 2

then the resonance frequency

2c l m nf

d a b

Boundry conditions enforced 0 for 0,x yE E z d

mnd l ...3,2,1l

the lowest and dominant resonant TE (resp TM) modewill be TE101 (resp. TM110)if b a d

Rectangular waveguide cavity resonatorQ factor for TE10l

1

3

2 2 3 3 2 3 3

1 1

1tan2 1

2 2 2

2

c d

d

cs

s o

QQ Q

Q

adQ

R l a b bd l a d ad

R

Resonators - coupling

Critical coupling occurs when ueL QQQ 2

If we define coupling coefficiente

u

QQg then we have

• undercoupled resonator if g<1• critically coupled resonator if g=1 (resonator matched to the feed line)• overcoupled resonator if g>1

Series RLC circuit

series RLC parallel RLC

RZg 0

0ZRg

R

Zo Zi

n

1g 1g1g

2

CZblbblj

ZZz oc

c

c

o

where

tantan

Serial RLCcoupling capacitor actsas impedance inverter

Resonance occurs when tan 0cl b

Gap coupled microstrip resonator resonators

The coupling of the feed line to the resonator lowers its resonant frequency

Gap coupled microstrip resonator resonators

2the coupling coeficient 2

oZR

u c

gQ b

2 2

( )( )

2o

oc o c

Z Z jQb b

22oc

R ZQb

for critical coupling oZ R

2cbQ

for overcoupled resonator

for undercoupled resonator 2cbQ

2cbQ

2cbQ