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Transmission Lines 9/27/2017 Course Instructor Dr. Raymond C. Rumpf Office: A‐337 Phone: (915) 747‐6958 E‐Mail: [email protected] EE 4347 Applied Electromagnetics Topic 4a Transmission Lines Transmission Lines These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Slide 1 Lecture Outline • Introduction • Transmission Line Equations • Transmission Line Wave Equations • Transmission Line Parameters – and – Characteristic Impedance, Z0 • Special Cases of Transmission Lines – General transmission lines – Lossless lines – Weakly absorbing lines – Distortionless lines • Examples – RG‐59 coaxial cable – Microstrip design Transmission Lines Slide 2 1 9/27/2017 Introduction Transmission Lines Slide 3 Map of Waveguides (LI Media) • Confines and transports waves. Waveguides • Supports higher‐order modes. • Has one or less conductors. • Usually what is implied by Transmission Lines “Pipes” the label “waveguide.” • Contains two or more conductors. • No low‐frequency cutoff. • Thought of more as a circuit clement Metal Shell Pipes Dielectric Pipes Homogeneous Inhomogeneous • Enclosed by metal. • Composed of a core and a cladding. • Has TEM mode. • Supports only • Does not support TEM mode. • Symmetric waveguides have no • Has TE and TM quasi‐(TEM, TE, • Has a low frequency cutoff. low‐frequency cutoff. modes. & TM) modes. Homogeneous Channel Waveguides • Supports TE and TM modes • Confinement along two axes. coaxial microstrip • TE & TM modes only supported in circularly symmetric guides. rectangular circular dual‐ridge Single‐Ended stripline coplanar optical Fiberphotonic crystal rib Inhomogeneous • Supports TE and TM modes Slab Waveguides buried parallel coplanar strips only if one axis is uniform. plate • Otherwise supports quasi‐TM • Confinement only along one axis. and quasi‐TE modes. • Supports TE and TM modes. • Interfaces can support surface waves. Differential shielded pair slotline no uniform axis uniform axis dielectric Slab large‐area interface (no TE or TM) (has TE and TM) parallel plate Transmission Lines Slide 4 2 9/27/2017 Transmission Line Parameters RLGC We can think transmission lines as being composed of millions of tiny little circuit elements that are distributed along the length of the line. In fact, these circuit element are not discrete, but continuous along the length of the transmission line. Transmission Lines Slide 5 RLGC Circuit Model It is not technically correct to represent a transmission line with discrete circuit elements like this. However, if the size of the circuit z is very small compared to the wavelength of the signal on the transmission line, it becomes an accurate and effective way to model the transmission line. z Transmission Lines Slide 6 3 9/27/2017 L‐Type Equivalent Circuit Model Distributed Circuit Parameters There are many possible circuit models for R (/m) Resistance per unit length. transmission lines, but most produce the same Arises due to resistivity in the equations after analysis. conductors. L (H/m) Inductance per unit length. Arises due to stored magnetic R z L z energy around the line. G (1/m) Conductance per unit length. Arises due to conductivity in the dielectric separating the conductors. Gz Cz 1 G R C (F/m) Capacitance per unit length. Arises due to stored electric energy between the conductors. z zz Transmission Lines Slide 7 Relation to Electromagnetic Parameters Every transmission line with a homogeneous fill has: , , LC G C Transmission Lines Slide 8 4 9/27/2017 Fundamental Vs. Intuitive Parameters Fundamental Parameters Intuitive Parameters Electromagnetics Electromagnetics , , n, , , , tan Transmission Lines Transmission Lines RLGC, , , Z0 , , , VSWR The fundamental parameters are the The intuitive parameters provide intuitive most basic parameters needed to solve a insight about how signals behave on a transmission line problem. transmission line. However, it is difficult to be intuitive They isolate specific information to a about how they affect signals on the line. single parameter. An electromagnetic analysis is needed to determine R, L, G, and C from the The intuitive parameters are calculated geometry of the transmission line. from R, L, G, and C . Transmission Lines Slide 9 Example RLGC Parameters RG‐59 Coax CAT5 Twisted Pair Microstrip R 36 mΩ m R 176 mΩ m R 150 mΩ m L 430 nH m L 490 nH m L 364 nH m G 10 m G 2 m G 3 m C 69 pF m C 49 pF m C 107 pF m Z0 75 Z0 100 Z0 50 Surprisingly, almost all transmission lines have parameters very close to these same values. Transmission Lines Slide 10 5 9/27/2017 Transmission Line Equations Transmission Lines Slide 11 E & H V and I Fundamentally, all circuit problems are electromagnetic problems and can be solved as such. All two‐conductor transmission lines either support a TEM wave or a wave very closely approximated as TEM. An important property of TEM waves is that E is uniquely related to V and H and uniquely related to E. VEd I Hd L L This let’s us analyze transmission lines in terms of just V and I. This makes analysis much simpler because these are scalar quantities! Transmission Lines Slide 12 6 9/27/2017 Transmission Line Equations The transmission line equations do for transmission lines the same thing as Maxwell’s curl equations do for unguided waves. Maxwell’s Equations Transmission Line Equations H VI E RI L t zt E I V H GV C t zt Like Maxwell’s equations, the transmission line equations are rarely directly useful. Instead, we will derive all of the useful equations from them. Transmission Lines Slide 13 Derivation of First TL Equation (1 of 2) R z L z + 2 3 + I zt, Vzt , 1 Gz Cz 4 Vz zt, ‐ ‐ z zz Apply Kirchoff’s voltage law (KVL) to the outer loop of the equivalent circuit: Izt, Vzt , IztRz , Lz Vz zt, 0 t 1 2 4 3 Transmission Lines Slide 14 7 9/27/2017 Derivation of First TL Equation (2 of 2) We rearrange the equation by bringing all of the voltage terms to the left‐hand side of the equation, bringing all of the current terms to the right‐hand side of the equation, and then dividing both sides by z. Izt, Vzt,, IztRz Lz Vz zt ,0 t Vz zt,, Vzt Izt , RI z, t L zt In the limit as z 0, the expression on the left‐hand side becomes a derivative with respect to z. Vzt,, Izt RI z, t L zt Transmission Lines Slide 15 Derivation of Second TL Equation (1 of 2) R z L z 1 2 + I zt, I zzt , + 3 4 Vzt , Gz Cz Vz zt, ‐ ‐ z zz Apply Kirchoff’s current law (KCL) to the main node the equivalent circuit: Vz zt, Izt , Iz zt, GzVz zt, Cz 0 t 1 2 3 4 Transmission Lines Slide 16 8 9/27/2017 Derivation of Second TL Equation (2 of 2) We rearrange the equation by bringing all of the current terms to the left‐hand side of the equation, bringing all of the voltage terms to the right‐hand side of the equation, and then dividing both sides by z. Vz zt, IztIzztGzVzztCz,, , 0 t Iz zt,, Izt Vz zt , GV z z, t C zt In the limit as z 0, the expression on the left‐hand side becomes a derivative with respect to z. Izt,, Vzt GV z, t C zt Transmission Lines Slide 17 Transmission Line Wave Equations Transmission Lines Slide 18 9 9/27/2017 Starting Point – Telegrapher Equations We start with the transmission line equations derived in the previous section. Vzt ,, Izt Izt ,, Vzt RI z, t L GV z, t C time‐domain ztzt For time‐harmonic (i.e. frequency‐domain) analysis, we Fourier transform the equations above. dV z dI z R jLIz GjCVz frequency‐domain dz dz Note: Our derivative d/dz became an ordinary derivative because z is the only independent variable left. These last equations are commonly referred to as the telegrapher equations. Transmission Lines Slide 19 Wave Equation in Terms of V(z) dV z dI z R jLIz Eq. (1)GjCVz Eq. (2) dz dz To derive a wave equation in terms of V(z), we first differentiate Eq. (1) with respect to z. dV2 z dIz RjL Eq. (3) dz2 dz Second, we substitute Eq. (2) into the right‐hand side of Eq. (3) to eliminate I(z) from the equation. dV2 z R jL G jCVz dz 2 Last, we rearrange the terms to arrive at the final form of the wave equation. dV2 z RjLGjCVz 0 dz 2 Transmission Lines Slide 20 10 9/27/2017 Wave Equation in Terms of I(z) dV z dI z R jLIz Eq. (1)GjCVz Eq. (2) dz dz To derive a wave equation in terms of just I(z), we first differentiate Eq. (2) with respect to z. dI2 z dVz GjC Eq. (3) dz2 dz Second, we substitute Eq. (1) into the right‐hand side of Eq. (3) to eliminate V(z) from the equation. dI2 z GjCRjLIz dz 2 Last, we rearrange the terms to arrive at the final form of the wave equation. dI2 z GjCRjLIz 0 dz 2 Transmission Lines Slide 21 Propagation Constant, In our wave equations, we have a common term GjCRjL . Define the propagation constant to be j GjCRjL Given this definition, the transmission line equations are written as dV2 z 2Vz 0 dz2 dI2 z 2 Iz 0 dz2 Transmission Lines Slide 22 11 9/27/2017 Solution to the Wave Equations If we hand the wave equations off to a mathematician, they will return with the following solutions. 2 dV z zz 2Vz 0 Vz Ve Ve dz2 00 2 dI z zz 2 Iz 0 I zIeIe dz2 00 Forward wave Backward wave Both V(z) and I(z) have the same differential equation so it makes sense they have the same solution.
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