Waveguides and its Types Field view and structures
- 1. Branch - Electronics Engineering & Related Branches Microwave Engineering UNIT – 2 Waveguide: Rectangular Waveguide and their Applications, Modes, Propagation constant and Cut-off Frequency Presented By Avishisht Kumar
- 2. Learning outcomes • What are Waveguides and its types? • Rectangular Waveguides: Fields Inside • Modes of Propagation • Transverse Magnetic Mode (TM-Mode). E = (Ex; Ey; Ez) and H = (Hx; Hy; 0). • Transverse Electric Mode (TE-Mode). E = (Ex; Ey; 0) and H = (Hx; Hy; Hz). • Cut-off frequency and Propagation Constant • Field Views inside Rectangular Waveguide
- 3. • A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. • A hollow metallic tube of the uniform cross section for transmitting electromagnetic waves by successive reflections from the inner walls of the tube is called as a Waveguide • There are five types of waveguides are: What are Waveguides and its types? 1. Rectangular waveguide 2. Circular waveguide 3. Elliptical waveguide 4. Single ridged waveguide 5. Double ridged waveguide
- 4. Rectangular Waveguides: Fields inside Using phasors & assuming waveguide filled with • lossless dielectric material and • walls of perfect conductor, the wave inside should obey… ∇2 𝐸 + 𝑘2 𝐸 = 0 ∇2 𝐻 + 𝑘2 𝐻 = 0 where 𝑘2 = 𝜔2 𝜇𝜀 𝑐 Then applying on the z-component… 𝜕2 𝐸𝑧 𝜕𝑥2 + 𝜕2 𝐸𝑧 𝜕𝑦2 + 𝜕2 𝐸𝑧 𝜕𝑧2 + 𝑘2 𝐸𝑧 = 0 Solving by method of Separation of Variables: 𝐸𝑧(𝑥, 𝑦, 𝑧) = 𝑋(𝑥)𝑌(𝑦)𝑍(𝑧) from where we obtain: 𝑋′′ 𝑋 + 𝑌′′ 𝑌 + 𝑍′′ 𝑍 = −𝑘2 𝛻2 𝐸𝑧 + 𝑘2 𝐸𝑧 = 0
- 5. 𝑋(𝑥) = 𝑐1 cos 𝑘 𝑥 𝑥 + 𝑐2 sin 𝑘 𝑥 𝑥 𝑌(𝑦) = 𝑐3 cos 𝑘 𝑦 𝑦 + 𝑐4 sin 𝑘 𝑦 𝑦 𝑍(𝑧) = 𝑐5 𝑒 𝛾𝑧 + 𝑐6 𝑒−𝛾𝑧 ℎ2 = 𝛾2 + 𝑘2 = 𝑘 𝑥 2 + 𝑘 𝑦 2 Rectangular Waveguides: Fields inside 𝑋′′ 𝑋 + 𝑌′′ 𝑌 + 𝑍′′ 𝑍 = −𝑘2 −𝑘 𝑥 2 − 𝑘 𝑦 2 + 𝛾2 = −𝑘2 which results in the expressions: 𝑋′′ + 𝑘 𝑥 2 𝑋 = 0 𝑌′′ + 𝑘 𝑦 2 𝑌 = 0 𝑍′′ − 𝛾2 𝑍 = 0 z yyxxz z yyxxz zz yyxxz eykBykBxkBxkBH eykAykAxkAxkAE z ececykcykcxkcxkcE sincossincos ,fieldmagneticfor theSimilarly sincossincos :direction-intravelingwaveat thelookingonlyIf sincossincos 4321 4321 654321
- 6. From the previous slide equations we can conclude: TEM (Ez=Hz=0) can’t propagate. TE (Ez=0) transverse electric In TE mode, the electric lines of flux are perpendicular to the axis of the waveguide TM (Hz=0) transverse magnetic, Ez exists In TM mode, the magnetic lines of flux are perpendicular to the axis of the waveguide. HE hybrid modes in which all components exists Modes of Propagation
- 7. z yyxxz eykAykAxkAxkAE sincossincos 4321 ,axE ,byE z z 0at0 0at0 TM Mode Boundary Conditions z oy z ox z oy z ox e b yn a xm E a m h j H e b yn a xm E b n h j H e b yn a xm E b n h E e b yn a xm E a m h E sincos cossin cossin sincos 2 2 2 2 0 sinsin z zj oz H ey b n x a m EE 𝐸 𝑥 = − 𝛾 ℎ2 𝜕𝐸 𝑧 𝜕𝑥 𝐻 𝑥 = 𝑗𝜔𝜀 ℎ2 𝜕𝐸 𝑧 𝜕𝑦 𝐸 𝑦 = − 𝛾 ℎ2 𝜕𝐸 𝑧 𝜕𝑦 𝐻 𝑦 = − 𝑗𝜔𝜀 ℎ2 𝜕𝐸 𝑧 𝜕𝑥 TM Mode
- 8. ,axE ,byE y x 0at0 0at0 z yyxxz eykBykBxkBxkBH sincossincos 4321 TE Mode Boundary Conditions z oy z ox z oy z ox e b yn a xm H b n h j H e b yn a xm H a m h j H e b yn a xm H a m h j E e b yn a xm H b n h j E sincos cossin cossin sincos 2 2 2 2 0 coscos z zj oz E ey b n x a m HH 𝐸 𝑥 = − 𝑗𝜔𝜇 ℎ2 𝜕𝐻 𝑧 𝜕𝑦 𝐻 𝑥 = − 𝛾 ℎ2 𝜕𝐻 𝑧 𝜕𝑥 𝐸 𝑦 = − 𝑗𝜔𝜇 ℎ2 𝜕𝐻 𝑧 𝜕𝑥 𝐻 𝑦 = − 𝛾 ℎ2 𝜕𝐻 𝑧 𝜕𝑦 TE Mode
- 9. Cut-off Frequency and propagation Constant • The cutoff frequency occurs when • Evanescent: • Means no propagation, everything is attenuated • Propagation: • This is the case we are interested since is when the wave is allowed to travel through the guide. 2 22 222 b n a m kkk yx When 𝜔𝑐 2 𝜇𝜀 = 𝑚𝜋 𝑎 2 + 𝑛𝜋 𝑏 2 then 𝛾 = 𝛼 + 𝑗𝛽 = 0 or 𝑓𝑐 = 1 2𝜋 1 𝜇𝜀 𝑚𝜋 𝑎 2 + 𝑛𝜋 𝑏 2 When 𝜔2 𝜇𝜀 < 𝑚𝜋 𝑎 2 + 𝑛𝜋 𝑏 2 𝛾 = 𝛼 and 𝛽 = 0 When 𝜔2 𝜇𝜀 > 𝑚𝜋 𝑎 2 + 𝑛𝜋 𝑏 2 𝛾 = 𝑗𝛽 and 𝛼 = 0
- 10. TE mode E and H Field particle View TE mode E and H Field Wave Vectors Figure showing the particle view of the fields Left side- Electric Field Right Side- Magnetic Field Figure showing the Vector view of the fields Left side- Electric Field Right Side- Magnetic Field Field Views inside Rectangular Waveguide
- 11. E Field 3D View Field Views inside Rectangular Waveguide TE12 Mode E Field Vectors
- 12. TE Mode E and H Field combined View TE32 Mode E and H Field Radiation View Field Views inside Rectangular Waveguide
- 13. References 1- commons.wikimedia.org 2- https://www.wikipedia.org 3- documents.pub_rectangular-wave-guides 4- physics stackexchange com 5- manson.gmu.edu 6- http://www.falstad.com/embox/guide.html