ECE 546 Lecture 03 Waveguides Spring 018 Jose E. Schutt-Aine Electrical & Computer Engineering Universit o Illinois jesa@illinois.edu ECE 546 Jose Schutt Aine 1
Parallel-Plate Waveguide Maxwell s Equations E E 0 E x E + x E + x = - E x x E E E + + = - x E E E E + + = - x E ECE 546 Jose Schutt Aine
TE Modes For a parallel-plate waveguide, the plates are ininite in the - extent; we need to stud the propagation in the -direction. The ollowing assumptions are made in the wave equation 0, but 0 and 0 x Assume E onl These two conditions deine the TE modes and the wave equation is simpliied to read E E + = - x E ( ) ECE 546 Jose Schutt Aine 3
Phasor Solution General solution (orward traveling wave) j jxx jxx E ( x, ) e Ae Be At x = 0, E = 0 which leads to A + B = 0. Thereore, A = -B = E o /j, where E o is an arbitrar constant j E ( x, ) Ee sin x o x a is the distance separating the two PEC plates ECE 546 Jose Schutt Aine 4
Dispersion Relation At x = a, E (x, ) = 0 Ee o j sin a 0 x This leads to: x a= m, where m = 1,, 3,... x m a Moreover, rom the dierential equation ( ), we get the dispersion relation x which leads to m a ECE 546 Jose Schutt Aine 5
Guidance Condition m a where m = 1,, 3... Since propagation is to take place in the direction, or the wave to propagate, we must have > 0, or m a This leads to the ollowing guidance condition which will insure wave propagation m a ECE 546 Jose Schutt Aine 6
Cuto Frequenc The cuto requenc c is deined to be at the onset o propagation c m a c v c = a m Each mode is reerred to as the TE m mode. It is obvious that there is no TE 0 mode and the irst TE mode is the TE 1 mode. The cuto requenc is the requenc below which the mode associated with the index m will not propagate in the waveguide. Dierent modes will have dierent cuto requencies. ECE 546 Jose Schutt Aine 7
Magnetic Field or TE Modes From E = - j we have 1 j xˆ ˆ ˆ x 0 0 E 0 which leads to j x Ee o sin xx jx j Ee o cos xx The magnetic ield or TE modes has components ECE 546 Jose Schutt Aine 8
E & Fields or TE Modes As can be seen, there is no component, thereore, the TE solution has E, x and onl. From the dispersion relation, it can be shown that the propagation vector components satis the relations = sin, x = cos where is the angle o incidence o the propagation vector with the normal to the conductor plates. ECE 546 Jose Schutt Aine 9
Phase and Group Velocities The phase and group velocities are given b v p c 1 c and v g c 1 c The eective guide impedance is given b: E TE x 1 o c ECE 546 Jose Schutt Aine 10
Transverse Magnetic (TM) Modes The magnetic ield also satisies the wave equation: Maxwell s Equations + = 0 x x x + + = - x + + = - x + + = - x x ECE 546 Jose Schutt Aine 11
For TM modes, we assume 0, but 0 and 0 x Assume onl These two conditions deine the TM modes and the equations are simpliied to read + = - x TM Modes General solution (orward traveling wave) j jxx jxx ( x, ) e Ae Be ECE 546 Jose Schutt Aine 1
Electric Field or TM Modes From = -j E we get E 1 j xˆ ˆ ˆ x 0 0 0 This leads to j jxx jxx Ex( x, ) e Ae Be x j jxx jxx E ( x, ) e Ae Be ECE 546 Jose Schutt Aine 13
TM Modes Fields At x=0, E = 0 which leads to A = B = o / where o is an arbitrar constant. This leads to j (,) x e cos x o x j Ex(,) x e o cos xx jx j E( x, ) oe sinxx At x=a,e =0which leads to x a = m, where m = 0, 1,, 3,... ECE 546 Jose Schutt Aine 14
E & Fields or TM Modes x m a This deines the TM modes which have onl, E x and E components. The eective guide impedance is given b: TM Ex o 1 The electric ield or TM modes has components c ECE 546 Jose Schutt Aine 15
E & Fields or TM Modes TE DISPERSION RELATION, GUIDANCE CONDITION AND CUTOFF EQUATIONS FOR A PARALLEL-PLATE WAVEGUIDE ARE TE SAME FOR TE AND TM MODES. This deines the TM modes; each mode is reerred to as the TM m mode. It can be seen rom that m=0 is a valid choice; it is called the TM 0, or transverse electromagnetic or TEM mode. For this mode and, ECE 546 Jose Schutt Aine 16
TEM Mode x =0 and =. There are no x variations o the ields within the waveguide. The TEM mode has a cuto requenc at DC and is alwas present in the waveguide. j oe E e e j j x o o E 0 The propagation characteristics o the TEM mode do not var with requenc The TEM mode is the undamental mode on a parallel-plate waveguide ECE 546 Jose Schutt Aine 17
Power or TE Modes Time-Average Ponting Vector TE modes 1 P ReˆE ˆ ˆ x * * x 1 Re P E * 1 Eo E o P Reˆ sin ˆ xxxj xcos xxsin xx Eo P ˆ sin xx ECE 546 Jose Schutt Aine 18
Power or TM Modes TM modes 1 P RexˆE ˆ ˆ * x E 1 o o P Re ˆ cos ˆ xxxj xsin xxcos xx o P ˆ cos xx The total time-average power is ound b integrating <P> over the area o interest. ECE 546 Jose Schutt Aine 19