Parallel Plate Transmission Line y c a Assume e jz er b x Partially loaded parallel Plate waveguide variation , no variation with x 2 2 k in the air region 0 2 2 2 TM modes : t ez k c ez 0 , k c 2 2 k in the dielectric region for air region p Let k c for the dielectric region 2 p 2 (e r 1)k 02 je r k 0Y0 e z 2 y hx ( y ) jk Y ez 02 p y e z ( y ) C1 sin y e z ( y ) C 2 sin p (b y ) For dielectric region for air region 0 ya a yb C1 sin a C 2 sin pc er 1 C1 cos a C 2 cos pc p tan a e r p tan pc Transceden tal equation must be solved simultaneo usly with 2 p 2 (e r 1) k 02 , p k 02 p 2 k 2 l 2 Most of the modes will be nonpropaga ting if is imaginary The variation is e z and the field decays exponentia lly. The value of between k 0 and k can occur if p jp0 Let 0 to be the correspond ing value of then : 0 tan 0 a e r p0 tan p0 c 20 p02 (e r 1)k 02 Low Frequency Solution When the frequency is low, k 02 is very small number , 0 and p0 are very small 0 a e r p02 c 2 (e r 1) k 02 p02 e r p02 c a 2 ( e 1 ) k 2 r 0a or p0 The solution for is a e rc k p0 2 0 2 e rb k0 e e k0 a e rc e e is the effective dielectric constant y c b er a x -W e e LC W L, C are the static inductance and capacitanc e per meter. The time average stored magnetic energy is Wm 0 0 1 2 Wm H dxdy WbJ LI z 0 w 4 2 4 b I z 2WJ z L 0 2W b w 2 x 2 z C2 C1 sin 0 a / j sinh p0 c jC1 0 a / p0 c 0 ya ez C1 0 y j b ey C1 jC1 0 (e r 1)c je r k 0Y0 (e r c a )e r hx C1 jY0C1 0 (e r 1)c Ca Cd e 0 2W e 0e r 2W The capacitanc e C , Ca , Cd Ca Cd c a e 0e r 2W C e rc a e 0e r 0 b LC e rc a LC The capacitanc e C e 0e r 2W C e rc a e 0e r 0 b LC e rc a Ca Cd Ca Cd , Ca e 0 2W c LC Field expression s : C2 C1 sin 0 a / j sinh p0 c jC1 0 a / p0 c 0 ya ez C1 0 y ey j b C1 jC1 0 (e r 1)c , Cd e 0e r 2W a je r k 0Y0 (e r c a )e r hx C1 jY0C1 0 (e r 1)c In the air region : (b y ) e z C1 0 a c j 0 a b e y 2 C1 jC1e r (e r 1)c p0 c hx j k 0Y0 0 C1 jY0C1 b V e y dy jC1 0 I z 2WJ z 2WH x (e r c a )e r (e r 1)c b (a e r c) (e r 1)c The characteri stic impedance is : Z0 V Zc I 2W ( a e r c )b er L C In the low frequency limit, the dominant mode of propagatio n becomed a TEM mode (quasi - TEM mode) At hight frequency the mode of propagatio n is an E mod High Frequency Solution: At high frequency k 0 and 0 , p0 are large. tanh p0 c 1 0 tanh 0 a e r p0 e r (e r 1)k 02 20 20 p02 (e r 1)k 02 The solution is independen t of b e z ( y ) C1 sin 0 y for 0 y a sinh p0 (b y ) e z ( y ) C 2 j sinh p0 (b y ) C1 sin 0 a sinh p0 (b a ) e p0 ( b y ) C1 sin 0 a p0 ( ba ) C1 sin 0 ae p0 ( y a ) for a y b e The field decays exponentia lly away from the air - dielectric surface and does not depend on b as long as p0 c p0 (b a ) is large. The field is guided by the dielectric sheet. This type of mode is called surface wave mode. Microstrip Transmission Line y w H x r ( x, y , z ) r s ( x, z ) ( y H ) J ( x, y , z ) J s ( x, z ) ( y H ) B A, E jB j A E jA H jD J For anisotropi c dielectric : D e 0e r ( E x aˆ x E z aˆ z ) e 0e y E y aˆ y A A 2 A j 0 D 0 J D e 0e r [ j ( Ax aˆ x Az aˆ z ) aˆ x aˆ z ] x z e 0e y ( jAy aˆ y aˆ y ) y let A je 0e r 0 (Lorentz condition) - 2 A j 0 [ je 0e r A e 0 e r e 0 (e y e r ) jaˆ y Ay aˆ y ] 0 J y J does not have a y component 2 Ax e r ( y )k 02 Ax 0 J x 2 Az e r ( y )k 02 Az 0 J z e r Ay e y ( y )k Ay j 0e 0 (e y e r ) r y 2 2 0 j 0e 0 (e y e r ) ( H )e r 1) ( y H ) y D r 2 2 e r 2 2 e y e r2 k 02 y y y x Ay r j (e y 1) Ay ( H ) ( y H ) j (e y e r ) e0 y Boundary conditions: lim 0 H H Ax Ax dy 2 y y 2 H H lim 0 H H 0 J sx ( x, z ) ( y H )dy 0 J sx Ax y H 0 J sx y y , H Ax 0 H z y Ay Az y , H 0 J sz H Az 0 H x y H j 0e 0 (e r 1) ( H ) H H ey y H rs j (e y 1) Ay ( H ) e0 In the substrate region away from the interface we have : ( 2 e r k 02 ) Ax 0 ( 2 e r k 02 ) Az 0 ( 2 e y k 02 ) Ay j 0e 0 (e y e r ) y Ay 2 2 e y 2 2 2 e r k 0 j (e y e r ) 2 2 x z e r y y In the air region e r e y 1 For an isotropic substrate e r e y d 2 ez 2 ez 0 2 dy for 0 y a d 2 ez 2 p ez 0 for a y b 2 dy ez ( y ) 0 at y 0, b , e z ( y ) is continuous at y a H x is continuous at y a ey ( y) j 2 j p2 ez y e z y e r ez 2 y y a 1 ez 2 p y For dielectric region for air region y a