Chapter 7 TEM Transmission Lines MIT OpenCourseWare

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conductors The voltage between two plates v z t for this sinusoidal wave can be found by. integrating E z t over the distance d from the lower plate which we associate here with the. voltage v to the upper plate,v t z x i E z t d E o d cos t kz V 7 1 3. 0 y contour C 1 3,Figure 7 1 1 Parallel plate TEM transmission line. Although this computed voltage v t z does not depend on the path of integration connecting. the two plates provided it is at constant z it does depend on z itself Thus there can be two. different voltages between the same pair of plates at different positions z Kirchoff s voltage law. says that the sum of voltage drops around a loop is zero this law is violated here because such a. loop in the x z plane encircles time varying magnetic fields H z t as illustrated In contrast. the sum of voltage drops around a loop confined to constant z is zero because it circles no H t. therefore the voltage v z t computed by integrating E z between the two plates does not. depend on the path of integration at constant z For example the integrals of E ds along. contours A and B in Figure 7 1 1 b must be equal because the integral around the loop 1 2 4 3. 1 is zero and the path integrals within the perfect conductors both yield zero. If the electric and magnetic fields are zero outside the two plates and uniform between them. then equal and opposite currents i t z flow in the two plates in the z direction The surface. current is determined by the boundary condition 2 6 17 Js n H A m 1 If the two. conducting plates are spaced close together compared to their widths W so that d W then the. fringing fields at the plate edges can be neglected and the total current flowing in the plates can. be found from the given magnetic field H z t y Eo o cos t kz and the integral form. of Ampere s law,C Hids A J D t in da 7 1 4, If the integration contour C encircles the lower plate and surface A at constant z in a clockwise. right hand sense with respect to the z axis as illustrated in Figure 7 1 1 then Din 0 and the. current flowing in the z direction in the lower plate is simply. i z t W Jsz z t W H y z t WE o o cos t kz A 7 1 5, An equal and opposite current flows in the upper plate. Note that the computed current does not depend on the integration contour C chosen so long. as C circles the plate at constant z Also the current flowing into a section of conducting plate at. z1 does not generally equal the current flowing out at z2 seemingly violating Kirchoff s current. law the sum of currents flowing into a node is zero This inequality exists because any section. of parallel plates exhibits capacitance that conveys a displacement current D t between the. two plates the right hand side of Equation 2 1 6 suggests the equivalent nature of the. conduction current density J and the displacement current density D t. Such a two conductor structure conveying waves that are purely transverse to the direction. of propagation i e Ez Hz 0 is called a TEM transmission line because it is propagating. transverse electromagnetic waves TEM waves Such lines generally have a physical cross. section that is independent of z This particular TEM transmission line is called a parallel plate. Because there are no restrictions on the time structure of a plane wave any v t can. propagate between parallel conducting plates The ratio between v z t and i z t for this or any. other sinusoidal or non sinusoidal forward traveling wave is the characteristic impedance Zo of. the TEM structure, v z t i z t o d W Zo ohms characteristic impedance 7 1 6.
In the special case d W Zo equals the characteristic impedance o of free space 377 ohms. Usually W d in order to minimize fringing fields yielding Zo 377. Since the two parallel plates can be perfectly conducting and lossless the physical. significance of Zo ohms may be unclear Zo is defined as the ratio of line voltage to line current. for a forward wave only and is non zero because the plates have inductance L per meter. associated with the magnetic fields within the line The value of Zo also depends on the. capacitance C per meter of this structure Section 7 1 3 shows 7 1 59 that Zo L C 0 5 for any. lossless TEM line and 7 1 19 shows it for a parallel plate line The product of voltage and. current v z t i z t represents power P z t flowing past any point z toward infinity this power is. not being converted to heat by resistive losses it is simply propagating away without reflections. It is easy to demonstrate that the power P z t carried by this forward traveling wave is the. same whether it is computed by multiplying v and i or by integrating the Poynting vector. S E H W m 2 over the cross sectional area Wd of the TEM line. P z t v z t i z t E z t d H z t W E z t H z t Wd,S Wd 7 1 7. The differential equations governing v and i on TEM lines are easily derived from Faraday s. and Ampere s laws for the fields between the plates of this line. E t H y z E x z t 7 1 8,H t E x z H y z t 7 1 9, Because all but one term in the curl expressions are zero these two equations are quite simple. By substituting v Exd 7 1 3 and i HyW 7 1 5 7 1 8 and 7 1 9 become. dv dz d W di dt Ldi dt 7 1 10,di dz W d dv dt Cdv dt 7 1 11. where we have used the expressions for inductance per meter L Hy m 1 and capacitance per. meter C F m 1 of a parallel plate TEM line see 3 2 11 32 and 3 1 10 This form of the. differential equations in terms of L and C applies to any lossless TEM line as shown in Section. These two differential equations can be solved for v by eliminating i The current i can be. eliminated by differentiating 7 1 10 with respect to z and 7 1 11 with respect to t thus. introducing d2i dt dz into both expressions permitting its substitution That is. d 2 v dz 2 Ld 2i dt dz 7 1 12,d 2i dz dt C d 2 v dt 2 7 1 13. Combining these two equations by eliminating d2i dt dz yields the wave equation. d 2 v dz 2 LC d 2 v dt 2 d 2 v dt 2 wave equation 7 1 14. Wave equations relate the second spatial derivative to the second time derivative of the. same variable and the solution therefore can be any arbitrary function of an argument that has. the same dependence on space as on time except for a constant multiplier That is one solution. to 7 1 14 is,v z t v z ct 7 1 15, where v is an arbitrary function of the argument z ct and is associated with waves.
propagating in the z direction at velocity c This is directly analogous to the propagating waves. Note 3 2 11 gives the total inductance L for a length D of line where area A Dd The inductance per unit. length L d W in both cases, characterized in Figure 2 2 1 and in Equation 2 2 9 Demonstration that 7 1 15 satisfies. 7 1 14 for c 0 5 follows the same proof provided for 2 2 9 in 2 2 10 12. The general solution to 7 1 14 is any arbitrary waveform of the form 7 1 15 plus an. independent arbitrary waveform propagating in the z direction. v z t v z ct v z ct 7 1 16, The general expression for current i z t on a TEM line can be found for example by. substituting 7 1 16 into the differential equation 7 1 11 and integrating over z Thus using. the notation that v q dv q dq,di dz Cdv dt cC v z ct v z ct 7 1 17. i z t cC v z ct v z ct Zo 1 v z ct v z ct 7 1 18, Equation 7 1 18 defines the characteristic impedance Zo cC 1 L C for the TEM line. Both the forward and backward waves alone have the ratio Zo between v and i although the sign. of i is reversed for the negative propagating wave because a positive voltage then corresponds to. a negative current These same TEM results are derived differently in Sections 7 1 3 and 8 1 1. The characteristic impedance Zo of a parallel plate line can be usefully related using 7 1 18. to the capacitance C and inductance L per meter where C W d and L d W for parallel. plate structures 7 1 10 11,Zo L ohms d characteristic impedance 7 1 19.
All lossless TEM lines have this simple relationship as seen in 8 3 9 for R G 0 It is also. consistent with 7 1 6 where o 1 c o o 0 5, The electric and magnetic energies per meter on a parallel plate TEM line of plate. separation d and plate width W are 33,We t z 1 E t z 1 Wd J m 1 7 1 20. Wm t z 1 H t z 1 Wd J m 1, Italicized symbols for We and Wm J m 1 distinguish them from We and Wm J m 3. Substituting C cW d and L d W into 7 1 20 and 7 1 21 yields. We t z 1 Cv 2 J m 1 TEM electric energy density 7 1 22. Wm t z 1 Li 2 J m 1 TEM magnetic energy density 7 1 23. If there is only a forward moving wave then v t z Zoi t z and so. We t z 1 Cv2 1 CZo 2i 2 1 Li 2 Wm t z 7 1 24, These relations 7 1 22 to 7 1 24 are true for any TEM line. The same derivations can be performed using complex notation Thus 7 1 10 and 7 1 11. can be written,j I z j L I z 7 1 25,W j V z j CV z 7 1 26.
Eliminating I z from this pair of equations yields the wave equation. 2 LC V z 0 wave equation 7 1 27, The solution to the wave equation 7 1 27 is the sum of forward and backward propagating. waves with complex magnitudes that indicate phase,V z V e jkz V e jkz 7 1 28. I z Yo V e jkz V e jkz 7 1 29, where the wavenumber k follows from k2 2LC which is obtained by substituting 7 1 28 into. k LC 2 7 1 30, The characteristic impedance of the line as seen in 7 1 19 is. Zo L 1 ohms 7 1 31, and the time average stored electric and magnetic energy densities are.
We 1 C V 2 J m Wm 1 L I 2 J m 7 1 32, The behavior of these arbitrary waveforms at TEM junctions is discussed in the next section. and the practical application of these general solutions for arbitrary waveforms is discussed. further in Section 8 1 Their practical application to sinusoidal waveforms is discussed in. Sections 7 2 4,Example 7 1A, A certain TEM line consists of two parallel metal plates that are 10 cm wide separated in air by. d 1 cm and extremely long A voltage v t 10 cos t volts is applied to the plates at one end. z 0 What currents i t z flow What power P t is being fed to the line If the plate. resistance is zero where is the power going What is the inductance L per unit length for this. Solution In a TEM line the ratio v i Zo for a single wave where Zo od W see 7 1 6. and o 0 5 377 ohms in air Therefore i t z Zo 1v t z W d o 10 cos t. kz 0 1 0 01 377 10 cos t kz 0 27 cos t z c A P vi v2 Zo. 2 65 cos2 t z c W The power is simply propagating losslessly along the line. toward infinity Since c LC 0 5 3 108 and Zo L C 0 5 37 7 therefore L. Zo c 1 3 10 7 Henries m 1,7 1 3 TEM waves in non planar transmission lines. TEM waves can propagate in any perfectly conducting structure having at least two non. contacting conductors with an arbitrary cross section independent of z as illustrated in Figure. 7 1 2 if they are separated by a uniform medium characterized by and The parallel plate. TEM transmission line analyzed in Section 7 1 2 is a special case of this configuration and we. shall see that the behavior of non planar TEM lines is characterized by the same differential. equations for v z t and i z t 7 1 10 and 7 1 11 when expressed in terms of L and C This. result follows from the derivation below, We first divide the del operator into its transverse and longitudinal z axis components. T z z 7 1 33,parallel plate TEM line E,cable arbitrary.
Figure 7 1 2 TEM lines with arbitrary cross sections. where T x x y y Faraday s and Ampere s laws then become. E T ET z z ET HT t 7 1 34,H T HT z z HT ET ET t 7 1 35. The right hand sides of these two equations have no z components and therefore the transverse. curl components on the left hand side are zero because they lie only along the z axis. T E T T H T 0 7 1 36, Moreover the divergences of ET and HT are also zero since z HT z ET 0 and. H 0 T HT z z HT 7 1 37,E 0 T ET z z ET 7 1 38, Since the curl and divergence of ET and HT are zero both these fields must independently. satisfy Laplace s equation 4 5 7 which governs electrostatics and magnetostatics these field. solutions will differ because their boundary conditions differ Thus we can find the transverse. electric and magnetic fields for TEM lines with arbitrary cross sections using the equation. solving and field mapping methods described in Sections 4 5 and 4 6. The behavior of E and H for an arbitrary TEM line can be expressed more simply if we. first define the line s capacitance per meter C and the inductance per meter L C is the charge. Q per unit length divided by the voltage v between the two conductors of interest and L is the. flux linkage per unit length divided by the current i Capacitance inductance and flux. linkage are discussed more fully in Sections 3 1 and 3 2. Cross section of,a TEM line c,a contours C,conductor a i. conductor b x, Figure 7 1 3 Integration paths for computing TEM line voltages and currents.
To compute Q and we consider a differential element of length along the z axis of the TEM. line illustrated in Figure 7 1 3 and then compute for Q and respectively surface and line. integrals encircling the central positively charged conducting element a in a right hand sense. relative to z To compute the voltage v we integrate ET from element a to element b and to. compute the current i we integrate HT in a right hand sense along the contour C circling. conductor a,C Q v 1 E T n da,A ab ET ds capacitance m 7 1 39. L i z HT ds HT ds,inductance m 7 1 40,HT z ds HT ds Hm 1. It is also useful to define G the line conductance per meter in terms of the leakage current. density J A m 1 conveyed between the two conductors by the conductivity of the medium. Chapter 7 TEM Transmission Lines 7 1 TEM waves on structures 7 1 1 Introduction Transmission lines typically convey electrical signals and power from point to point along arbitrary paths with high efficiency and can also serve as circuit elements In most transmission lines the electric and magnetic fields point purely transverse to the direction of propagation such waves are called

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