Magical Lambda

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Ms. Sumadhura Gunupati

Editor's note: this page was submitted by Ms. Sumadhura Gunupati, a student of Dr. Raghavan, at National Institute of Technology, Trichy.  It is a comprehensive look at many microwave engineering concepts that can be described in terms of wavelength (lambda). Many thanks for this content, and all of your kind words!

Lambda, when thought about, strikes a simple and rudimental idea of wavelength. Nothing new?! Well, there is so much distinct about Lambda which we will see shortly. If we can delve deeper into the concepts, we can perceive its aptitude as a design parameter. Starting from transmission lines to all the modern microwave technology, the design can be narrowed down and talked in terms of ‘λ’.

While designing a component we come across many equations and terms and it’s the hardest thing to remember all of them. What if we don’t have to remember!! Let’s note that length of any component can be designed in terms of lambda while width from impedance. We will put aside all the other terms and keep our focus only on λ for a while, so that we can notice its wonders.

The well known relation between frequency and wavelength is: λ = c/f. Certain types of λ, we are already familiar with, are HALF WAVELENGTH(λ/2), QUARTER WAVELENGTH (λ/4) AND λ/8. What applications can we derive from these?

Well, when it comes to connecting two devices whose impedances are quite conflicting, Quarter wavelength solves our problem. A Quarter wave transmission line can be used to match or TRANSFORM these impedances and thus called IMPEDANCE TRANSFORMER. Devices like DIRECTIONAL COUPLER, RAT-RACE COUPLER, exhibit such behaviour as their internal design is done in terms of λ/4.  Waveguide bends can be radiative. To reduce this effect, we can design the bend’s length or curvature as a multiple of λ/4.


Have we ever wondered if there is any frequency at which transmission line ceases to work?? If yes, the answer is BRAGG Frequency. The low pass nature of transmission line is a product of this.

BRAGG Frequency = 1/ (pi* sqrt (LC)),

And it occurs when the unit cell is 1/3(λ).

(To know more about Bragg frequency, refer to

The above mentioned are just a handful. Like so, one can find many such λ’s hiding in the design, doing their work. The exceedingly popular SMITH CHART is all based on λ/2 as Normal impedance or admittance repeats for every λ/2 distance. Antenna design is done using λ/2 (STRONG RADIATION PATTERNS ARE GENERATED).  Let’s have a much closer look at antennas.




L = λ/2

Radiation resistance is obtained around 73ohm.

L < nλ/2

Radiation resistance is around 50 ohm which is highly useful in matching components.

λ/10 < L < λ/50

L < = λ/50.



Number of lobes ≃  +1

When h increases beyond λ, minor lobes increase

For a vertical dipole above an infinite perfect electric conductor, if h = 0.4585λ

The Directivity is 6.566 which is four times greater than that of an isolated element.


Radius = (5/32) λ

ISOTROPIC antenna can be almost mimicked.

The circumference of Loop antenna = 1.48λ.


The maximum directivity of 4.63 dB can be achieved.

 Radius of loop =λ/25(0.04 λ)

The electrical size is almost 24 times larger than the physical size.

Radius <

Radius >  

The directivity is 1.761Db.

The directivity is 0.677(C/λ).

Length L of the patch is usually λ0/3 < L < λ0 βˆ•2 


h β‰ͺ λ0, usually 0.003  λ0h ≤ 0.05 λ0 .

Rectangular microstrip patch antenna.

vertical beam width in degrees is 51 λ / b

horizontal beam width in degrees is 70 λ / a

a and b are wide and narrow dimensions.

Gain of the Horn radiator is G = 10 A / (λ^2).

  • The ground plane for the helical antenna is in general designed with a diameter of 3 λ/4.
  • The directivity of a horn antenna is given by:

D =

  • The aperture of the corner reflector (Da) is usually made between one and two wavelengths (λ < Da < 2λ) and spacing (g) between wires is usually g ≤ λβˆ•10.

In antennas, almost everything is talked about in terms of lambda. Radiation patterns, number of lobes, Directivity, Beam widths and many more parameters can be related to λ.



The size of the DRA is proportional to πœ†0/√πœ–π‘Ÿ

where πœ†0 = 𝑐/𝑓 0 being the free-space wavelength at the resonant frequency 𝑓0

                πœ–π‘Ÿ denotes the relative permittivity of the material forming the radiating structure.



  • The NRD waveguide is similar to H waveguide, except that the distance chosen between the metal plates is less than a wavelength in NRD waveguide.
  • The polarisation of electric filed orients parallel to the metal plates due to which the conduction losses are reduced.
  •  Due to the dielectric slab, the electromagnetic field is confined in the vicinity of the dielectric region, whereas in the outside region, for suitable frequencies, the electromagnetic field decays exponentially.


  • The vias diameter d shall be:

                      d < (λg/5)

  • The distance between center of the vias (pitch b) shall be:

                       b <= 2.5d

  • If the distance (pitch) between PTH vias is less than 1/10th of the wavelength, then they will behave as approximate sidewalls.
  • Guided wavelength is given by:


f is the resonant frequency.

c is the speed of light.

  • w and l are the width and length of a single SIW cavity.
  • weff and leff are the effective width and length of the SIW structure.


Metamaterials is another area with extensive scope for research. We all have surely heard of THE INVISIBILITY CLOAK (Hi there HARRY POTTER fans). Have we wondered once, whether it’s possible to build such device in real life (without magic!!)?? Metamaterial can deliver a solution to this question.

Metamaterials are artificial periodic structures which exhibit UNCONVENTIONAL properties like negative refractive index, negative permeability and negative permittivity. So negative this structure is!!! These unusual properties can be advantageous in many ways. Recently, many microwave devices are being designed using meta materials.

The non-conventional properties that are seen in metamaterials are achieved by designing each unit cell such that its size is less than λg. Refraction is the dominating effect and for this to be possible, Effective homogeneity condition (size of unit cell = λg/4) is to be satisfied.

By electing the dimensions of unit cell in terms of ‘λ’, we were able to build an amazingly potential structure. Wonderful!! Don’t you think??


Contrast to the metamaterials is the PBG structure which exhibit Bragg’s diffraction.

2PSin(θ) = mλg

Thus, Braggs angles are also a function of λg.

Photonic band gap structures are also periodic structures with the lattice period chosen to be around λg/2 or a multiple of λg/2.


Conventional microstrip antennas had some limitations like single operating frequency, low impedance bandwidth, low gain, larger size, and polarization problems. DGS (Defected Ground Structures) are one of the prominent solutions to solve these limitations.

A single defect (unit cell) or a number of periodic and aperiodic defects configurations may be comprised in DGS. Thus, periodic and/or aperiodic defects etched on the ground plane of planar microwave circuits are referred to as DGS. DGS slots are modelled as LC circuit where the inductance and capacitance values depend on the cut off and resonant wavelengths.


A powerful yet plain structure. Powerful as FSS is used in defence strategies and plain because its operation is similar to a filter. Periodicity is employed to attain the required operation.

Frequency selective surfaces, also called Spatial filters, have their inter element spacing in the order of 0.4λ. Resonance plays a key role in the operation. One can obtain resonance at desired frequency when the slot size in unit cell is in order of half of wavelength.

Bandwidth of almost all types of FSSs can be varied by altering the inter-element spacing, which must be small in terms of the wavelength. If the inter-element spacing is greater than half of the wavelength, it will result in the early onset of the grating lobes, pushing the main resonance towards downside with varying incidence angles. Layered design is also opted often.

FSS based antennas, absorbers, filters are the future of microwave technology.








The description given so far gives us detail about lambdas other than the regular quarter wave and half wave. The applications we land at by picking design parameters in terms of LAMBDA. To conclude, there are still many more devices and structures whose operation can be associated with Lambda, a simple yet high-potential term.

Hoping I didn’t bore the readers out…Thank you.


I express my sincere thanks to Prof. Dr. S Ragavan for introducing me to the Microwave world. I also like to extend my vote of thanks to the editor, Dr. Steve, for giving every enthusiast an opportunity to express their views on a platform like MICROWAVES 101. Microwaves 101 has been so helpful and informative to the students of ECE in NIT Tiruchirappalli. Not just for students, to all the people who love microwaves this is a place one can learn and teach as well. I am so glad that I got a chance to contribute back.


[1] Paper written by Prof .Dr.S Raghavan titled  ‘WONDERFUL  LAMBDA’.

[2] R. Mongia, A. Ittibipoon, and M. Cuhaci, “Low profile dielectric resonator antennas using a very high permittivity material,” Electronics Letters, vol. 30, no. 17, pp. 1362–1363, 1994.

[3] Bozzi, Maurizio & Perregrini, Luca & Wu, Ke & P, Arcioni. Current    and Future Research Trends in Substrate Integrated Waveguide Technology. Radioengineering. 18.

[4] Keyrouz, Shady & Caratelli, Diego. (2016). Dielectric Resonator Antennas: Basic Concepts, Design Guidelines, and Recent Developments at Millimeter-Wave Frequencies. International Journal of Antennas and Propagation. 2016. 1-20. 10.1155/2016/6075680.

[5] M. Pasian, S. Monni, A. Neto, M. Ettorre and G. Gerini, "Frequency    Selective Surfaces for Extended Bandwidth Backing Reflector Functions," in IEEE Transactions on Antennas and Propagation, vol. 58, no. 1, pp. 43-50, Jan. 2010, doi: 10.1109/TAP.2009.2036185.

[6] KHANDELWAL, MUKESH & Kanaujia, Binod & Kumar, Sachin. (2017). Defected Ground Structure: Fundamentals, Analysis, and Applications in Modern Wireless Trends. International Journal of Antennas and Propagation. 2017. 1-22. 10.1155/2017/2018527.

[7] Pathak, N.P., Basu, A. & Koul, S.K. Full wave analysis of non-radiative dielectric waveguide modulator for   the determination of   electrical equivalent circuit. Pramana - J Phys 7165–75 (2008).

[8] Bates, Bevan D and Kym Jordan, Nonradiative Dielectric Waveguide Millimetre-Wave Integrated Circuitsβ€―: a Review (DSTO Australia, Electronics Research Laboratory, 1990).

[9] Si-Gyun Jeong, Do-Kyeong Hwang, Yong-Chae Jeong and Chul- Dong Kim, "Amplifier design using /spl lambda//4 high impedance bias line with defect ground structure (DGS)," 2002 IEEE MTT-S International Microwave Symposium Digest (Cat. No.02CH37278), Seattle, WA, USA, 2002, pp. 1161-1164 vol.2, doi: 10.1109/MWSYM.2002.1011854.

[10] Latrach, Mohamed & Rmili, Hatem & Sabatier, C. & Seguenot,   E. & Toutain, S.. (2010). Design of a new type of metamaterial radome for low frequencies. Microwave and Optical Technology Letters. 52. 1119 - 1123. 10.1002/mop.25127.

[11] Archana Sharma1, Kavita Khare2 S.C.Shrivastava3. International  Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering Vol. 2, Issue 6, June 2013 Dielectric Resonator Antenna for X band Microwave Application.

[12] International Journal of Advanced Research in Biology  Engineering Science and Technology (IJARBEST) Vol. 2, Special Issue 14, May 2016 119.Design and Analysis of Substrate Integrated Waveguide Wideband Filter for X-Band Frequency Applications.

[13] Breed, Gary. (2008). An Introduction to Defected Ground Structures in Microstrip Circuits. High Freq Electron. 7.

[14] Ke Wu, D. Deslandes and Y. Cassivi, "The substrate integrated circuits - a new concept for high-frequency electronics and optoelectronics," 6th International Conference on Telecommunications in Modern Satellite, Cable and Broadcasting Service, 2003. TELSIKS 2003., Nis, Yugoslavia, 2003, pp. P-III, doi: 10.1109/TELSKS.2003.1246173.

[15] Mahajan, Karan and Sourabh S. Mahajan. “Design and Analysis of Rectangular Conducting Slots Based Substrate Integrated Waveguide.” (2015).

[16] R. Anwar, L. Mao, and H. Ning, “Frequency Selective Surfaces: A Review,” Applied Sciences, vol. 8, no. 9, p. 1689, Sep. 2018.

[17] L. H. Weng, Y.-C. Guo, X.-W. Shi, and X.-Q. Chen, "An Overview on Defected Ground Structure," Progress In Electromagnetics Research B, Vol. 7, 173-189, 2008.

[18] Fei-Ran Yang , Yongxi Qian , Roberto Coccioli & Tatsuo Itoh (1999) Analysis and Application of Photonic Band-Gap (PBG) Structures for Microwave Circuits, Electromagnetics, 19:3, 241-254, DOI: 10.1080/02726349908908642.


Author : Sumadhura Gunupati