Electromagnetics IIUnit 4 Resonant Modes, Bilinear
Transformation between Impedance and Reflection
CoefficientLecturer: rofessor !ean"#u $iangTe%t boo&: '( '(
Rao, )Elements of Engineering Electromagnetics,* si%t+ ed(, earson,
,--4(1Unless noted, t+e course materials are licensed under
Creati.e Commons /ttribution"'onCommercial"0+are/li&e 1(-
Taiwan 2CC B3"'C"0/ 1(-4 Transmission Line wit+ 0+ort"circuited
Load2 V z ( ) V+e jz+Vejz , jV+sinzI z ( ) V+Z- e jz VZ- ejz
,V+Z-coszV - ( ) - V V+( )( )( )( )( )-in - inininimpedance
:taninput impedance : 5 tan-,, 6, 6, 4VzZ z jZ zIzZ l jZ z jXZ l n
l nnZ l n l t z lz -Z-,Z z ( )Zi nResonant Modes or 'atural Modes
V2z4 V+e jz+Vejz ,V+j sinzI2z4 V+Z- e jzVZ- ejz,V+Z- j coszB( C( at
z -,V2z -4 - V V+ V2z, t4 Re V2z4ejt{ } ,V+sinzcos t + 7, ( ) I2z,
t4 Re I2z4ejt{ } ,V+Z-coszcos t + ( ) sin t + ( ) P2z, t4 V z,t ( )
I z, t ( ) V+,Z-sin,zsin , t + ( ) [ ]P2z, t4 6TP2z, t4-Tdt - no
net power l 14 4l1 c 1,l c f 1c4l 2894 oscillation fre:uenc;( )B(
C( at, 2 4 - : oscillation modes4cos - , 6 , 6, ,, 1,(((, , 6z lI z
lnll l n nn 3Input Reactance4X j LX j6C4,14lIm Zin{ }
Xincapaciti.einducti.eTwo Transmission lines wit+ 0+ort"circuited
Loads5in6 -6 6 6in, -, , ,tantanZ jZ lZ jZ l'6 6 6, , ,,,ff V6 V,I6
I,I6V6 + I,V, -Z-6tan6l6 + Z-,tan,l, --6 6 6 6 -, , , ,tan , tan ,
- Z f l Z f l + / discrete number of : resonant fre:uenc;, natural
fre:uenc; of oscillationfZi n 6Zi n , I,I6V6V,+ + l6l,Z- ,,,Z-
6,66,6( ),,,( )Transmission Line wit+ an /rbitrar; Load6V z ( ) V+e
jz+VejzVL V+ +VI z ( ) V+Z- e jz VZ- ejz IL V+Z- VZ-)ZLV z ( ) V+e
jz6+VV+e, jz
1]1 V+e jz6+ z ( ) 1]I z ( ) V+Z- e jz6 z ( ) 1]( )( )( )( )( )(
)( )( )-in -6666V z zZ z ZIz zlZ Z l Zl+ + ZL VLILLV7V+Z-6+ L6 L L
ZL Z-ZL + Z- z ( ) Le, jzz lz -Z-,Z z ( )Zi n z (
)ZLVL+ILLReflection Coefficient at /n; z7 z ( ) Z z ( ) 7 Z- 6Z z (
) 7 Z- +6 Z z ( ) 6Z z ( ) +6 Z z ( ): normali9ed impedanceL L ejV
z ( ) V+6+ z ( )I z ( ) V+Z-6 z ( )Lt o w a r d l o a d 2 z 4t o w
a r d g e n e r a t o r 2 z 4R e { }I m { } 6 , - ( ), z6 + L6 + z
( ) z ( ) L e, j z " p l a n eCalculate /dmittance in"plane 8( )(
)( )( ) ( )( )( )( )( )( ) ( ),4, 7 4 ,7 46 6 7 47 4 66 6 7 47 4j z
j zL Lz e e zz zZ z Z zz zY z Z z t t tt + + t t
