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Course: ECE 3030, Fall 2007
School: Cornell
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20 Lecture Transmission Lines: The Basics In this lecture you will learn: Transmission lines Different types of transmission line structures Transmission line equations Power flow in transmission lines Appendix ECE 303 Fall 2006 Farhan Rana Cornell University Guided Waves So far in the course you have been dealing with waves that propagated in infinite size media For many applications it is desirable...

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20 Lecture Transmission Lines: The Basics In this lecture you will learn: Transmission lines Different types of transmission line structures Transmission line equations Power flow in transmission lines Appendix ECE 303 Fall 2006 Farhan Rana Cornell University Guided Waves So far in the course you have been dealing with waves that propagated in infinite size media For many applications it is desirable to have electromagnetic energy be guided in much the same way as water flow is guided by having it flow in pipes Transmission lines are the simplest structures that guide electromagnetic waves Transmission line: In this course a transmission line would be any two arbitrary shaped metal conductors that are very long and uniform in at least one dimension y A transmission line made up of two metal conductors that are very long and uniform in the z-direction x ECE 303 Fall 2006 Farhan Rana Cornell University 1 Common Transmission Line Structures - I W b a d Co-axial Cable 2 C= b log a L= Parallel-Plate Transmission Line C= W d d W o b log 2 a L = o LC = o ECE 303 Fall 2006 Farhan Rana Cornell University LC = o Capacitance and Inductance per unit length for each structure are shown Common Transmission Line Structures - II radius a d d/2 o radius a Parallel Metal Wires For d >> a : Metal Wire Over a Ground Plane For d >> a : C= d log a o C= 2 o d log a L= o d log a L= 2 o d log a LC = o o LC = o o ECE 303 Fall 2006 Farhan Rana Cornell University Capacitance and Inductance per unit length for each structure are shown 2 y Transmission Line Voltages Suppose the potential difference between the two conductors of a transmission line at location z is V(z) then E-field line integral in a plane parallel to the x-y plane at the location z is independent of the contour taken, and is related to the potential difference V(z) as: x C1 C2 r r r r V (z ) = E . ds = E . ds c1 c2 The charge per unit length Q(z) on the transmission line at location z is: Q (z ) = C V (z ) Capacitance per unit length Two points with different z-values can have different potentials: y r r r r V (z1) = E . ds E . ds = V (z2 ) c1 c2 V(z1) C1 V(z2) C2 The conductors are no longer equipotentials z z1 z2 ECE 303 Fall 2006 Farhan Rana Cornell University y Transmission Line Currents Suppose the total current in the upper conductor in the +z-direction at location z is I(z) and in the lower conductor is I(z) H x The H-field flux per unit length (z) enclosed between the two conductors at the location z is related to the current I(z) as: (z ) = L I (z ) y Inductance per unit length I(z1) I(z2) z1 z2 z Two points with different z-values can have different currents I ( z 1 ) I (z 2 ) ECE 303 Fall 2006 Farhan Rana Cornell University 3 y C + V(z,t) I(z,t) H Faraday's Law + V(z+z,t) - z z + z z Use Faraday's law for the contour C: r r r r E . ds = - o H . da t c V (z + z , t ) - V (z , t ) = - ( z , t ) z t V (z + z , t ) - V (z , t ) L I (z , t ) =- z t V (z , t ) I (z , t ) = -L z t (z , t ) = L I (z , t ) (1) ECE 303 Fall 2006 Farhan Rana Cornell University Current-Charge Continuity Equation y I(z,t) Q(z,t) + ++ V(z,t) - -I(z+z,t) z z + z Use the principle of conservation of charge (current-charge continuity equation): z If current is varying in space, there must be charge either piling up or piling down somewhere I ( z , t ) - I ( z + z , t ) = Q (z , t ) z t I ( z + z , t ) - I ( z , t ) C V (z , t ) =- z t I (z , t ) V (z , t ) = -C t z Q (z , t ) = C V (z , t ) (2) ECE 303 Fall 2006 Farhan Rana Cornell University 4 Transmission Line Equations The following two equation describe the propagation of guided electromagnetic waves on transmission lines (also called the Telegrapher's Equations): V (z , t ) I (z , t ) = -L z t I (z , t ) V (z , t ) = -C z t (1) (2) Wave equations: (1) (2) 2 V (z , t ) z 2 = -L 2 I (z , t ) z t (3) 2 I (z , t ) 2 V (z , t ) = -C t z t 2 2 V (z , t ) z 2 (4) Equation of a wave traveling with a velocity = v = 1 LC (3) and (4) = LC 2 V (z , t ) t 2 2 I (z , t ) t 2 A similar equation can be derived for the current: 2 I (z , t ) z 2 = LC ECE 303 Fall 2006 Farhan Rana Cornell University Nature of Guided Waves in Transmission Lines - I 2 V (z , t ) z 2 = LC 2 V (z , t ) t 2 2 I (z , t ) z 2 = LC 2 I (z , t ) t 2 The guided wave consists of E-fields and H-fields together with charges and currents on the conductors that all move together in sync with a velocity given by: v = 1 LC + ++ y The charges satisfy the boundary conditions for the E-fields ECE 303 Fall 2006 Farhan Rana Cornell University ++ + ++ + - -- - -- -- - -- - + ++ + ++ ++ + -- - - -- z 5 Nature of Guided Waves in Transmission Lines - II 2 V (z , t ) z 2 = LC 2 V (z , t ) t 2 2 I (z , t ) z 2 = LC 2 I (z , t ) t 2 The guided wave consists of E-fields and H-fields together with charges and currents on the conductors that all move together in sync with a velocity given by: v = 1 LC y The charges satisfy the boundary conditions for the E-fields The currents satisfy the boundary conditions for the H-fields The wave is called a TEM wave since both the E-field and H-field point in a direction transverse to the direction of propagation ECE 303 Fall 2006 Farhan Rana Cornell University E-Fields and H-fields for Common Transmission Lines H - E ++ + + + + ++ - E + + + + + + + - H - y x Co-axial Cable y x Parallel-Plate Transmission Line Notice that at each point E (r , t ) H (r , t ) points in the +z-direction indicating energy flow in the +z-direction ECE 303 Fall 2006 Farhan Rana Cornell University r r r r ++ + ++ + - -- - -- -- - -- - + ++ + ++ + ++ ++ + -- - - -- z 6 E-Fields and H-fields for Common Transmission Lines H y x x +++ +++ +++ +++ y H E E ----- Metal Wire Over a Ground Plane Parallel Wires Notice Metal that at each point E (r , t ) H (r , t ) points in the +z-direction indicating energy flow in the +z-direction ECE 303 Fall 2006 Farhan Rana Cornell University r r r r Current and Voltage Phasors Convert to phasors: V (z , t ) = Re V (z ) e j t I (z , t ) = Re I (z ) e j t [ [ ] ] Transmission line equations in phasor notation: V (z , t ) I (z , t ) = -L z t I (z , t ) V (z , t ) = -C z t Wave equations in phasor notation: V (z ) = - j L I (z ) z I (z ) = - j C V (z ) z 2 V (z , t ) z 2 = LC 2 V (z , t ) t 2 2 V (z ) z 2 2 I (z ) z 2 = - 2 LC V (z ) 2 I (z , t ) z 2 = LC 2 I (z , t ) t 2 = - 2 LC I (z ) ECE 303 Fall 2006 Farhan Rana Cornell University 7 Solutions of Transmission Line Equations Start with the complex wave equation: y 2 V (z ) z 2 = - 2 LC V (z ) V+ z Assume a solution of the form of a traveling wave: V (z ) = V+ e - j k z A wave traveling in the +z-direction Substitute in the complex wave equation: 2 V (z ) z 2 = - 2 LC V (z ) - k 2 V+ e - j k z = - 2 LC V+ e - j k z k 2 = 2 LC k = LC Dispersion relation of a wave traveling with a velocity = v = 1 LC ECE 303 Fall 2006 Farhan Rana Cornell University Impedance of a Transmission Line Voltage is: V (z ) = V+ e - j k z Find the current from the transmission line equation: V (z ) = - j L I (z ) z Where Zo, given by: - j k V+ e - j k z = - j L I (z ) I (z ) = k L V+ e - j k z V I (z ) = + e - j k z Zo Zo = L k = L C is called the characteristic impedance of the transmission line So a voltage-current wave propagating in the +z-direction on a transmission line is specified completely by: V (z ) = V+ e - j k z V I (z ) = I + e - j k z = + e - j k z Zo ECE 303 Fall 2006 Farhan Rana Cornell University 8 Backward Waves on a Transmission Line A voltage-current wave propagating in the +z-direction on a transmission line is specified completely by: V (z ) = V+ e - j k z V I (z ) = I + e - j k z = + e - j k z Zo A voltage-current wave propagating in the -z-direction on a transmission line is specified completely by: V (z ) = V- e + j k z y V I (z ) = I - e + j k z = - - e + j k z Zo Notice the ve sign ECE 303 Fall 2006 Farhan Rana Cornell University Forward and Backward Waves on a Transmission Line y V+ I+ V- I- z In general, voltage on a transmission line is a superposition of forward and backward going waves: V (z ) = V+ e - j k z + V- e + j k z The corresponding current is also a superposition of forward and backward going waves: V V I (z ) = I + e - j k z + I - e + j k z = + e - j k z - - e + j k z Zo Zo ECE 303 Fall 2006 Farhan Rana Cornell University ++ + ++ + - -- - -- -- - -- - + ++ + ++ + ++ ++ + -- - - -z 9 Parallel-Plate Transmission Lines: Fields, Voltages, and Currents H E + + + W + + + + d - If the voltage and the current waves are: V (z ) = V+ e - j k z V I (z ) = I + e - j k z = + e - j k z Zo then the E-field and the H-field are (ignoring the fringing fields): y x Parallel-Plate Transmission Line r V ^ E (z ) = - y + e - j k z d r I+ - j k z ^ e H (z ) = x W Given the amplitude(s) of the voltage and/or current waves, the E-field and the H-field associated with the wave can be found ECE 303 Fall 2006 Farhan Rana Cornell University Energy Flow and Power on a Transmission Line Consider a transmission line with a voltage-current wave going in the +z-direction: y V+ I+ z How much is the total time-average power (not power per unit area) carried by the wave in the +z-direction? r r The area integral is over the entire x-y plane ^ Pz (t ) = S (r , t ) . z dxdy (or any plane parallel to the x-y plane) r r 1 ^ = Re S (r ) . z dxdy 2 [ ] It can be show that this integral equals: Pz (t ) = V 2 1 1 * Re V+ I + = Re +* 2 2 Zo [ ] And if there is a also backward wave then: Pz (t ) = V 2 V 2 1 1 * * Re V+ I + - V- I - = Re +* - -* 2 2 Zo Zo ECE 303 Fall 2006 Farhan Rana Cornell University [ ] 10 Waves in Free Space and Waves in Transmission Lines Free Space Transmission Lines r r r r E (r ) = - j o H (r ) r r r r H (r ) = j o E (r ) r r r r 2 E (r ) = - 2 o o E (r ) r r r r 2 H (r ) = - 2 o o H (r ) r r ^ E (r ) = x Eo e - j k z V (z ) = - j L I (z ) z I (z ) = - j C V (z ) z 2 V (z ) z 2 2 I (z ) z 2 = - 2 LC V (z ) = - 2 LC I (z ) V (z ) = V+ e - j k z I (z ) = ECE 303 Fall 2006 Farhan Rana Cornell University r r E ^ H (r ) = y o e - j k z o V+ - j k z e Zo Appendix: Energy Flow and Power on a Transmission Line y This appendix offers a proof of the power flow formula for arbitrary transmission lines r r ^ Pz (t ) = S (r , t ) . z dxdy r r 1 ^ = Re S (r ) . z dxdy 2 r r r r 1 ^ = Re E (r ) H * (r ) . z dxdy 2 x [ ] [ ] By assumption both E- and H-fields have only transverse components (i.e. no component in the z-direction) ^ =x ^ ^ ^ +y +z = T + z z x y z From Faraday's Law: r r E = - j o H r r ^ T + z E = - j oH z ECE 303 Fall 2006 Farhan Rana Cornell University 11 Appendix: Energy Flow and Power on a Transmission Line r r ^ T + z E = - j oH z r T E = 0 y x Therefore one may write the E-field as the transverse gradient of a scalar potential: r E = -T Where by assumption the potential satisfies: 1st conductor - 2nd conductor = V+ e - jkz From Amperes's Law: r r ^ H = z J z + j E r ^ ^ T + z H = z J z + j E z r ^ T H = z J z ECE 303 Fall 2006 Farhan Rana Cornell University Appendix: Energy Flow and Power on a Transmission Line - jkz 1st conductor J z dxdy = I + e y 2nd conductor J z dxdy = - I + e - jkz r r 1 ^ Pz (t ) = Re E H * . z dxdy 2 r 1 ^ = Re - T H * .z dxdy 2 0 r 1 ^ = Re T - H * .z dxdy 2 r 1 ^ + Re T H * .z dxdy 2 1 * = Re J z dxdy 2 1 * = Re 1st conductor - 2nd conductor I + e jkz 2 1 * = Re V+ I + 2 [ [ ] x [ ] ] [ ] [ ] [( [ ) ] ] ECE 303 Fall 2006 Farhan Rana Cornell University 12
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