Lecture -- Transmission Line Equations

1 9/21/20191 Electromagnetics:Electromagnetic Field TheoryTransmission line EquationsLecture Outline Introduction Transmission line Equations Transmission line Wave EquationsSlide 2129/21/20192 Slide 3 IntroductionMap of Waveguides (LI Media)Slide 4 Transmission Lines Contains two or more conductors. No low frequency cutoff. Thought of more as a circuit clement Confines and transports waves. Supports higher order modes. Has TEM mode. Has TE and TM Pipes Has one or less conductors. Usually what is implied by the label waveguide. Metal Shell PipesDielectric PipesInhomogeneousHomogeneous Enclosed by metal. Does not support TEM mode. Has a low frequency cutoff. Supports TE and TM modes Supports TE and TM modes only if one axis is uniform. Otherwise supports quasi TM and quasi TE WaveguidesSlab Waveguides Composed of a core and a cladding. Symmetric waveguides have no low frequency cutoff.

2 Confinement only along one axis. Supports TE and TM modes. Interfaces can support surface waves. Confinement along two axes. TE & TM modes only supported in circularly symmetric Slabinterfaceoptical Fiberribdual ridgeno uniform axis(no TE or TM)WaveguidesHomogeneousInhomogeneous Supports only quasi (TEM, TE, & TM) EndedDifferentialburied parallel platecoplanar stripsphotonic crystalshielded pairlarge area parallel plateuniform axis(has TE and TM)349/21/20193 Signals in Transmission Lines: CoaxSlide 5 Signals in Transmission Lines: MicrostripSlide 6569/21/20194 Signals in Transmission Lines: Twisted PairSlide 7 Transmission line Parameters RLGCS lide 8It can be useful to think of 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 ModelSlide 9It is not technically correct to represent a Transmission line with discrete circuit elements like , if the size of the circuit zis very small compared to the wavelength of the signal on the Transmission line , it becomes an accurate and effective way to model the Transmission L Type Equivalent Circuit ModelSlide 10 Distributed Circuit ParametersR( /m)Resistance per unit length.

3 Arises due to resistivity in the (H/m)Inductance per unit length. Arises due to stored magnetic energy around the (1/ m)Conductance per unit length. Arises due to conductivity in the dielectric separating the (F/m)Capacitance per unit length. Arises due to stored electric energy between the zRz Lz Gz Cz There are many possible circuit models for Transmission lines, but most produce the same Equations after 9109/21/20196L Type Equivalent Circuit ModelSlide 11 Distributed Circuit ParametersR( /m)Resistance per unit length. Arises due to resistivity in the (H/m)Inductance per unit length. Arises due to stored magnetic energy around the (1/ m)Conductance per unit length. Arises due to conductivity in the dielectric separating the (F/m)Capacitance per unit length. Arises due to stored electric energy between the zRz Lz Gz Cz There are many possible circuit models for Transmission lines, but most produce the same Equations after L Type Equivalent Circuit ModelSlide 12 Distributed Circuit ParametersR( /m)Resistance per unit length.

4 Arises due to resistivity in the (H/m)Inductance per unit length. Arises due to stored magnetic energy around the (1/ m)Conductance per unit length. Arises due to conductivity in the dielectric separating the (F/m)Capacitance per unit length. Arises due to stored electric energy between the zRz Lz Gz Cz There are many possible circuit models for Transmission lines, but most produce the same Equations after 11129/21/20197L Type Equivalent Circuit ModelSlide 13 Distributed Circuit ParametersR( /m)Resistance per unit length. Arises due to resistivity in the (H/m)Inductance per unit length. Arises due to stored magnetic energy around the (1/ m)Conductance per unit length. Arises due to conductivity in the dielectric separating the (F/m)Capacitance per unit length. Arises due to stored electric energy between the zRz Lz Gz Cz There are many possible circuit models for Transmission lines, but most produce the same Equations after Relation to Electromagnetic ParametersSlide 14LC , , GC Every Transmission line with a homogeneous fill has.

5 13149/21/20198 Example RLGCP arametersSlide 15036 m m430 nH m10 m69 pF m75 RLGCZ 0176 m m490 nH m2 m49 pF m100 RLGCZ Surprisingly, almost all Transmission lines have parameters very close to these same m m364 nH m3 m107 pF m50 RLGCZ RG 59 CoaxCAT5 Twisted PairMicrostripSlide 16 Transmission line Equations15169/21/20199E& H Vand ISlide 17 Fundamentally, all circuit problems are electromagnetic problems and can be solved as two conductor Transmission lines either support a TEM wave or a wave very closely approximated as important property of TEM waves is that Eis uniquely related to Vand Hand uniquely related to LIHd This reduces analysis of Transmission lines to just Vand I. This makes analysis much simpler because these are scalar quantities! Transmission line EquationsSlide 18 The Transmission line Equations do for Transmission lines the same thing as Maxwell s curl Equations do for s EquationsTransmission line EquationsHEt EHt VIRI Lzt IVGV Czt Like Maxwell s Equations , the Transmission line Equations are rarely directly useful.

6 Instead, we will derive all of the useful Equations from ,Vzt ,IztLzt ,IztRz ,Vz zt Derivation of First TL Equation (1 of 2)Slide 19zz zRz Lz Gz Cz + ,Vzt+ ,Vz zt Apply Kirchoff svoltage law (KVL) to the outer loop of the equivalent circuit:1234 ,Izt12340 Derivation of First TL Equation (2 of 2)Slide 20We 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. ,,,,0,,,,IztVzt IztRz LzVz zttVz zt VztIztRI z tLzt In the limit as z 0, the expression on the left hand side becomes a derivative with respect to z. ,,,VztIztRI z tLzt 19209/21/201911 ,0Vz ztCzt ,GzVz zt ,Iz zt Derivation of Second TL Equation (1 of 2)Slide 21zz zRz Lz Gz Cz + ,Vzt+ ,Vz zt 1234 Apply Kirchoff scurrent law (KCL) to the main node the equivalent circuit: ,Izt ,Izt1234 ,Izzt Derivation of Second TL Equation (2 of 2)Slide 22We 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.

7 ,,, ,0,,,,Vz ztIztIz ztGzVz ztCztIz zt IztVz ztGV zz t Czt In the limit as z 0, the expression on the left hand side becomes a derivative with respect to z. ,,,IztVztGV z t Czt 21229/21/201912 Slide 23 Transmission line Wave EquationsStarting Point Telegrapher EquationsSlide 24 Start with the Transmission line Equations derived in the previous section. ,,,VztIztRI z tLzt ,,,IztVztGV z t Czt time domainFor time harmonic ( frequency domain) analysis, Fourier transform the Equations above. dV zRjLIzdz dI zGjCVzdz frequency domainNote: The derivative d/dzbecame an ordinary derivative because zis the only independent variable last Equations are commonly referred to as the telegrapher Equation in Terms of V(z)Slide 25To derive a wave equation in terms of V(z), first differentiate Eq. (1) with respect to z. dV zRjLIzdz dI zGjCVzdz Eq.

8 (1)Eq. (2) 22dV zdI zRjLdzdz Eq. (3)Second, substitute Eq. (2) into the right hand side of Eq. (3) to eliminate I(z)from the equation. 22dV zRjL G jCVzdz Last, rearrange the terms to arrive at the final form of the wave equation. 220dV zRjLGjCVzdz Wave Equation in Terms of I(z)Slide 26To derive a wave equation in terms of I(z), first differentiate Eq. (2) with respect to z. dV zRjLIzdz dI zGjCVzdz Eq. (1)Eq. (2) 22dI zdV zGjCdzdz Eq. (3)Second, substitute Eq. (1) into the right hand side of Eq. (3) to eliminate V(z). 22dI zGjCRjLIzdz Last, rearrange the terms to arrive at the final form of the wave equation. 220dI zGjCRjLIzdz 25269/21/201914 Propagation Constant, Slide 27 Define the propagation constant to be jGjCRjL Given this definition, the Transmission line Equations are written as 2220dV zVzdz 2220dI zIzdz In the wave Equations , there is the common term.

9 Solution to the Wave EquationsSlide 28If the wave Equations are handed off to a mathematician, they will return with the following solutions. 2220dV zVzdz 2220dI zIzdz 00 zzVz Ve Ve 00 zzIzIe Ie Both V(z)and I(z) have the same differential equation so it makes sense they have the same waveBackward wave2728