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ECE 6130 Rectangular Waveguides
Text Sections: 3.
Chapter 3, Problem 3 (See Appendix I) and Derive the TM modes of a rectangular waveguide following the methods described here for TE modes.
Rectangular Waveguides
Recall: Method of solution:
- Solve Helmholtz equation for either Hz (TE) or Ez (TM). This can be done analytically or numerically. In the analytical case, you guess the form of the solution, which will have several unknown constants (like magnitude, phase, number of cycles)
- Use 3.19 to 3.23 to find transverse components from Ez or Hz.
- Solve for the constants from the boundary conditions. In metal boundaries, these are that tangential E and normal H = 0 on the boundary. Now you have Ez or Hz.
- Use Maxwell’s equation to find the other E or H components.
T E Solution
- Solve Helmholtz wave equation:
a) Use Method of Separation of Variables: h (^) x (x,y) = X(x) Y(y) b) Substitute into wave equation (Helmholtz equation):
where kc^2 = k (^) x^2 + k (^) y^2 c) Separate the Variables:
d) “Guess” the form of the solution:
2 2 ( , )^0
2 2
2 ⎟⎟^ = ⎠
k h x y x y c z
2 2
2
d Y X dx
d X Y (^) c
2 2
2
2 2
2
⎥^ =
k Y dy
d Y X
k X dx
d X Y
y
x
hz(x,y) = (A cosk (^) x x + B sin kx x) (C cos k (^) yy + D sin k (^) yy)
The unknown constants are ABCD. Also kx and k (^) y.
- Solve for the unknown constants from boundary conditions. a) Define the boundary conditions Tangential E fields =0 on the metal surfaces (walls of the waveguide) e (^) x = 0 at y=0,b e (^) y = 0 at x=0,a b) Obtain appropriate expressions for the boundary condition fields From equations 3.19: Ex =(-jωμ / k (^) c^2 ) ∂Hz / ∂y Ey =(jωμ / k (^) c^2 ) ∂Hz / ∂x
So: e (^) x (x,y) =(-jωμ / k (^) c^2 ) ky (A coskx x + B sin k (^) x x) (- Csin k (^) yy + D cos k (^) yy) e (^) y(x,y) = (jωμ / k (^) c^2 ) kx (-A sink (^) x x + B cos k (^) x x) (C cos kyy + D sin k (^) yy)
c) Use boundary conditions to solve for ABCD: Substitute ex = 0 at y=0,b into equations above. e (^) x (x,0) =(-jωμ / k (^) c^2 ) ky (A coskx x + B sin k (^) x x) (- Csin k (^) y0 + D cos k (^) y0)= So, D = 0 Note that a “trivial solution” also exists if ky = e (^) x (x,b) =(-jωμ / k (^) c^2 ) ky (A coskx x + B sin k (^) x x) (- Csin k (^) yb + 0 cos kyb)= when ky = nπ/b and k (^) x = mπ/a
Substitute ey =0 at x=0,a into equations above: e (^) y(0,y) = (jωμ / k (^) c^2 ) kx (-A sink (^) x 0 + B cos k (^) x 0) (C cos kyy + D sin k (^) yy) = So, B= e (^) y(x,a) = (jωμ / k (^) c^2 ) kx (-A sink (^) x a + B cos kx a) (C cos kyy + D sin k (^) yy)= So, kx = mπ/a
Now we can simplify the form: e (^) x (x,b) =(-jωμ / k (^) c^2 ) (nπ/b) (-AC) cos( mπx/a ) sin (nπy/b ) e (^) x (x,b) =A (^) mn cos( mπx/a ) sin (nπy/b )
d) Apply constants to H: hz(x,y) = (A cosk (^) x x + 0 sin kx x) (C cos k (^) yy + 0 sin k (^) yy) Hz(x,y) = AC cos (mπx/a) cos (nπy/b) = A (^) mn cos (mπx/a) cos (nπy/b) e -jβz
e) Ex, Ey, Hx,Hy components are found from equations 3.19 again.
TE (^) mn modes:
Modes are numbered m-n, indicating how many cosine wave cycles are in the waveguide.
