EC-333: Microwave Techniques

Module#3: Introduction to Microwave Resonator

Karun Rawat
Karun.rawat.in@ieee.org

RF Group, Department of Electronics and Communications Engineering

Indian Institute of Technology Roorkee, India

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 Important: Copyright Disclaimer
➢ These slides have been prepared from several text-books and
copyright materials which is meant for the fair-use in teaching only.

➢ Various sources from where the material have been taken are
mentioned in the respective slides.

➢ The slides should only be used by the students and shall not be
distributed (offline as well as online) other than students registered
and TAs of the subject-ECN333 (Department of Electronics &
Communication, I.I.T Roorkee).

➢ The instructor prohibits the use of this material for any purpose
other than teaching (fair use only).

➢ The instructor is not responsible for further redistribution of such

material by students and TAs.
2/15
 Important
➢ To all members of Radio Amplifier and Power Transceiver Lab, RF Group,
I.I.T Roorkee.

➢ These slides have been prepared from the following text books and are
being used for class lecture/demonstration only:

➢ D.M. Pozar, Microwave Engineering.

➢ Online Resources

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 Fundamental Resonant Circuits
Series Resonance Circuit Parallel Resonance Circuit

BW: half-power fractional bandwidth

Courtesy: David. M. Pozar, Microwave Engineering, Wiley 4/15

 Series Resonant Circuits
Series Resonance Circuit

Zin

Power dissipated in Resistor R:

Av. Electric Energy stored in C:

Av. Magnetic Energy stored in C:

Courtesy: David. M. Pozar, Microwave Engineering, Wiley 5/15

 Series Resonant Circuits
Series Resonance Circuit

Zin

At Resonance: Wm=We and resonance occurs at

The Quality factor is defined as:

At Resonance, the Q-factor is:

Courtesy: David. M. Pozar, Microwave Engineering, Wiley Note: Q0 increases as R decreases. 6/15
 Series Resonant Circuits: Behaviour of
Resonator near Resonant Frequency
Analyzing behavior of resonator at: where is small

Since,

Therefore,
Since is small,
(  = 0 +   +  = 2 = 0 +  +  )
 0 L 
Therefore,  Q0 = 
 R 
If a Resonator with loss is treated as lossless resonator, then the resonant
frequency of the lossless resonator will be
Courtesy: David. M. Pozar, Microwave Engineering, Wiley 7/15
 Parallel Resonant Circuits
Parallel Resonance Circuit

At Resonance: Wm=We and resonance

occurs at

At Resonance, the Q-factor is: Analyzing behavior of resonator at:

where is small

For lossy resonator treated as lossless the

resonant frequency of the lossless resonator: 8/15
 Loaded & Unloaded Q
➢ The Resonator quality factor at resonance i.e. Q0 as discussed in
previous slides is UNLOADED Q.
➢This unloaded Q is a characteristic of resonator itself which is obtained
in absence of any loading effect caused by external circuitry.
➢If resonator is loaded, the overall Q is called loaded Q (or QL), which
is less than the unloaded Q.
➢If the resonator is a series RLC circuit, the load
resistor RL adds in series with R, the effective
resistance is R + RL .
➢If the resonator is a parallel RLC circuit, the
effective resistance is RRL/(R + RL ).
➢If external Q is defined as Qe, then

Courtesy: David. M. Pozar, Microwave Engineering, Wiley 9/15

 Summary of Series and Parallel Resonators

Courtesy: David. M. Pozar, Microwave Engineering, Wiley 10/15

 Transmission Line Resonators: Short-
circuited λ/2 Line (1/2)
Short-circuited λ/2 Line
Z0: characteristic impedance, β: propagation
constant and α: attenuation constant.
Using Identity of Hyperbolic Tangent

Assuming αl <<1, tanh αl ≈ αl.

For l= λ/2 at ω = ω0, and ω = ω0 + Δω:

 0 2 2 0    0   0  
 l= l = =   l= l= l= 
0 0 2  0 v p 0 v p 0 
 vp   vp

Courtesy: David. M. Pozar, Microwave Engineering, Wiley 11/15

 Transmission Line Resonators: Short-
circuited λ/2 Line (2/2)
Short-circuited λ/2 Line

  l 
  1
  0 
This is of the form of Series Resonator
with Equivalent input impedance:

The Equivalence is :

Courtesy: David. M. Pozar, Microwave Engineering, Wiley 12/15

 Transmission Line Resonators: Open-
circuited λ/2 Line
Open-circuited λ/2 Line
Input impedance of open circuit lossy line:

For l= λ/2 at ω = ω0, and let ω = ω0 + Δω.

Then,

Since,
This is equivalent to Parallel RLC Resonant Circuit with the following
Equivalence:

Courtesy: David. M. Pozar, Microwave Engineering, Wiley 13/15

 Transmission Line Resonators: Short-
circuited λ/4 Line
Input impedance of short circuit lossy line:

This was obtained by expanding tanh function and multiplying

numerator and denominator by –j cotβl:

For l= λ/4 at ω = ω0, and let ω = ω0 + Δω, and considering TEM line:

For :αl <<,

This is equivalent to Parallel RLC Resonant Circuit with the following

Equivalence:
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Short Circuited Half Wave Transmission
Z = jX at f  f
Line
in 0

Z in = jZ 0 tan  l
Z in = 0
λ/2 2 0
at f 0 Z in = jZ 0 tan = jZ 0 tan 
0 2

Z in = − jX at f  f 0

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Courtesy: electronics-tutorial.ws/accircuits/series resonances
Short Circuited Quarter Wave
Transmission Line
Z in = jX at f  f 0
Z in = ; Yin = 0 Z in = jZ 0 tan  l
at f 0 λ/4
2 0 
Z in = jZ 0 tan = jZ 0 tan
0 4 2

Z in = − jX at f  f 0

Courtesy: electronics-tutorial.ws/accircuits/series resonances 16/15