Transmission Line MenagerieWaveguides and Transmission LinesCascade of T-Lines (I)Cascade of T-Lines (II)Transmission CoefficientConservation of EnergyBounce DiagramJunction of Parallel T-LinesReactive Terminations (I)Reactive Terminations (II)Reactive Terminations (III)Solution for Reactive TermLaplace Domain Solution ILaplace Domain Solution (II)Time Harmonic Steady-State Why Sinusoidal Steady-State?Generalized Distributed Circuit ModelTime Harmonic Telegrapher's EquationsSin. Steady-State (SSS)Voltage/Current Lossless Line for SSSBack to Time-Domain Passive T-Line/Wave SpeedEECS 117Lecture 3: Transmission Line Junctions / TimeHarmonic ExcitationProf. NiknejadUniversity of California, BerkeleyUniversity of California, Berkeley EECS 117 Lecture 3 – p. 1/23Transmission Line Menageriestriplinemicrostriplinecoplanarrectangularwaveguidecoaxialtwo wiresT-Lines come in many shapes and sizesCoaxial usually 75Ω or 50Ω (cable TV, Internet)Microstrip lines are common on printed circuit boards(PCB) and integrated circuit (ICs)Coplanar also common on PCB and ICsTwisted pairs is almost a T-line, ubiquitous forphones/EthernetUniversity of California, Berkeley EECS 117 Lecture 3 – p. 2/23Waveguides and Transmission LinesThe transmission lines we’ve been considering havebeen propagating the “TEM” mode or TransverseElectro-Magnetic. Later we’ll see that they can alsopropagation other modesWaveguides cannot propagate TEM but propagationTM (Transverse Magnetic) and TE (Transverse Electric)In general, any set of more than one losslessconductors with uniform cross-section can transmitTEM waves. Low loss conductors are commonlyapproximated as lossless.University of California, Berkeley EECS 117 Lecture 3 – p. 3/23Cascade of T-Lines (I)Z01Z02z =0i1i2v1v2Consider the junction between two transmission linesZ01and Z02At the interface z = 0, the boundary conditions are thatthe voltage/current has to be continuousv+1+ v−1= v+2(v+1− v−1)/Z01= v+2/Z02University of California, Berkeley EECS 117 Lecture 3 – p. 4/23Cascade of T-Lines (II)Solve these equations in terms of v+1The reflection coefficient has the same form (easy toremember)Γ =v−1v+1=Z02− Z01Z01+ Z02The second line looks like a load impedance of valueZ02Z01Z02z =0i1+v1−University of California, Berkeley EECS 117 Lecture 3 – p. 5/23Transmission CoefficientThe wave launched on the new transmission line at theinterface is given byv+2= v+1+ v−1= v+1(1 + Γ) = τv+1This “transmitted” wave has a coefficientτ = 1 + Γ =2Z02Z01+ Z02Note the incoming wave carries a powerPin=|v+1|22Z01University of California, Berkeley EECS 117 Lecture 3 – p. 6/23Conservation of EnergyThe reflected and transmitted waves likewise carry apower ofPref=|v−1|22Z01= |Γ|2|v+1|22Z01Ptran=|v+2|22Z02= |τ|2|v+1|22Z02By conservation of energy, it follows thatPin= Pref+ Ptran1Z02τ2+1Z01Γ2=1Z01You can verify that this relation holds!University of California, Berkeley EECS 117 Lecture 3 – p. 7/23Bounce DiagramConsider the bouncediagram for the follow-ing arrangementZ01Z02RsRL`1`2TimeSpacetd2td3td4td5td6tdv+1τ1v+1ΓLτ1v+1ΓLτ1τ2v+1ΓsΓLτ1τ2v+1Γjv+1ΓjΓsv+1ΓsΓ2jv+1`1`1+ `2td1University of California, Berkeley EECS 117 Lecture 3 – p. 8/23Junction of Parallel T-LinesZ01Z02z =0Z03Again invoke voltage/current continuity at the interfacev+1+ v−1= v+2= v+3v+1− v−1Z01=v+2Z02+v+3Z02But v+2= v+3, so the interface just looks like the case oftwo transmission lines Z01and a new line with char.impedance Z01||Z02.University of California, Berkeley EECS 117 Lecture 3 – p. 9/23Reactive Terminations (I)RsVs`Z0, tdLLet’s analyze the problem intuitively firstWhen a pulse first “sees” the inductance at the load, itlooks like an open so Γ0= +1As time progresses, the inductor looks more and morelike a short! So Γ∞= −1University of California, Berkeley EECS 117 Lecture 3 – p. 10/23Reactive Terminations (II)So intuitively we might expect the reflection coefficientto look like this:1 2 3 45-1-0.50.51t/τThe graph starts at +1 and ends at −1. In between we’llsee that it goes through exponential decay (1st orderODE)University of California, Berkeley EECS 117 Lecture 3 – p. 11/23Reactive Terminations (III)Do equations confirm our intuition?vL= Ldidt= Lddtv+Z0−v−Z0And the voltage at the load is given by v++ v−v−+LZ0dv−dt=LZ0dv+dt− v+The right hand side is known, it’s the incomingwaveformUniversity of California, Berkeley EECS 117 Lecture 3 – p. 12/23Solution for Reactive TermFor the step response, the derivative term on the RHSis zero at the loadv+=Z0Z0+ RsVsSo we have a simpler casedv+dt= 0We must solve the following equationv−+LZ0dv−dt= −v+For simplicity, assume at t = 0 the wave v+arrives atloadUniversity of California, Berkeley EECS 117 Lecture 3 – p. 13/23Laplace Domain Solution IIn the Laplace domainV−(s) +sLZ0V−(s) −LZ0v−(0) = −v+/sSolve for reflection V−(s)V−(s) =v−(0)L/Z01 + sL/Z0−v+s(1 + sL/Z0)Break this into basic terms using partial fractionexpansion−1s(1 + sL/Z0)=−11 + sL/Z0+L/Z01 + sL/Z0University of California, Berkeley EECS 117 Lecture 3 – p. 14/23Laplace Domain Solution (II)Invert the equations to get back to time domain t > 0v−(t) = (v−(0) + v+)e−t/τ− v+Note that v−(0) = v+since initially the inductor is anopenSo the reflection coefficient isΓ(t) = 2e−t/τ− 1The reflection coefficient decays with time constantL/Z0University of California, Berkeley EECS 117 Lecture 3 – p. 15/23Time Harmonic Steady-StateCompared with general transient case, sinusoidal caseis very easy∂∂t→ jωSinusoidal steady state has many importantapplications for RF/microwave circuitsAt high frequency, T-lines are like interconnect fordistances on the order of λShorted or open T-lines are good resonatorsT-lines are useful for impedance matchingUniversity of California, Berkeley EECS 117 Lecture 3 – p. 16/23Why Sinusoidal Steady-State?Typical RF system modulates a sinusoidal carrier(either frequency or phase)If the modulation bandwidth is much smaller than thecarrier, the system looks like it’s excited by a puresinusoidCell phones are a good example. The carrier frequencyis about 1 GHz and the voice digital modulation is about200 kHz(GSM) or 1.25 MHz(CDMA), less than a 0.1% ofthe bandwidth/carrierUniversity of California, Berkeley EECS 117 Lecture 3 – p. 17/23Generalized Distributed Circuit ModelZ0Z0Z0Z0Z0Y0Y0Y0Y0Y0Z0: impedance per unit length (e.g. Z0= jωL0+