EE340–Electromagnetic TheoryTransmission lines

Pradeep Kumar K

Department of Electrical EngineeringIndian Institute of Technology Kanpur

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Outline

1 Distributed vs Lumped Circuits

2 When is wire a wire?

3 Transmission ala Kirchoff

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Distributed vs Lumped Circuits

Three regimes

Lumped circuit regimephysical dimensions� wavelength of signalMaxwell’s equations simplified considerably: wave nature ignored;Kirchoff’s laws, Inductance, Capacitance

Optics regimephysical dimensions� wavelength of signalMaxwell’s equations simplified considerably: Waves become rays;Snell’s law, mirrors, lenses, polarizers

Transmission line/distributed circuit regimephysical dimensions ≈ wavelength of signalMaxwell’s equations cannot be simplified: waves are waves; T-lines,microwave circuits, optical fiber

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Distributed vs Lumped Circuits

Maxwell and Kirchoff

Let’s unleash MaxwellSet µ0 and ε0 both to∇× E = 0 and ∇× H = JV =

∫E · d l = 0 (KVL)

∇ · J = 0 (KCL)

Speed of light in air= 1√µ0ε0

=∞ if µ0 and ε0 are zero Instantaneoustransmission across circuit elementLight=EM wave (Duck test)

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Distributed vs Lumped Circuits

In pictures: Lumped and distributed circuits

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When is wire a wire?

Waveforms on IC interconnects

Interconnects in ICConnecting leads between driver and receiver in ICGood interconnect minimizes distortion and adds little noiseAll interconnects are transmission lines

Signal on interconnection varies along its lengthForget this fact at your own peril in high-speed IC designs

E. Bogatin, IEEE Microwave Mag, Aug 2011

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When is wire a wire?

Time and frequency domain pictures of signals

Signals on interconnects are digital waveformsRise-time tr decides spectral bandwidth and is more importantthan clock frequency

90% energy contained within ≈ 0.24/tr ; Conservative estimate0.35/tr is also used

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When is wire a wire?

Highest frequency and shortest wavelength

fmax = 0.35/tr commonly taken as highest frequency of interestCorresponding to fmax , shortest wavelength λshort = v/fmax

Distributed regime: physical length l � λshort

“Much less" is arbitrary, say 0.1For fmax=30 GHz in air, λshort = 1cm; l should be no more than0.1cm; People sized objects can’t be treated as lumpedSpeed of EM waves in PCB less than speed in air; 30→14 GHz

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Transmission ala Kirchoff

Transmission line model

Stray currents through imperfect dielectrics cause current to bedifferent from source and load ends

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Transmission ala Kirchoff

Transmission line model

Stray currents through imperfect conductors cause current to bedifferent from source and load ends

Inductance + resistance (=impedance) and capacitance +conductance (=admittance) along the line cannot be ignored

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Transmission ala Kirchoff

Transmission line model

T-line modeled as circuit containing distributed R,L,C, and Gparameters organized into an infinite number of unit cellsWithin unit cell of length ∆z � λshort , KCL and KVL hold (Why?)

Distributed parameters L and C are calculated using EM theorylater in course

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Transmission ala Kirchoff

Transmission line equations

Apply KVL to unit cell

Apply KCL to node N

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Transmission ala Kirchoff

Solving T-line equations: Lossless case (R = G = 0)

First-order T-line equations combined to get second-order PDE

∂2V (z, t)∂z2 = LC

∂2V (z, t)∂t2 =

1u2

p

∂2V (z, t)∂t2 ; up =

1√LC

Solution: V (z, t) = V+(t − z/up)︸ ︷︷ ︸Forward

+ V−(t + z/up)︸ ︷︷ ︸Backward

(Check)

Current I(z, t) = I+(t − z/up) + I−(t + z/up)

From T-line equations, Z0 = V+

I+ is characteristic impedance

In terms of line parameters, Z0 =√

L/C

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Transmission ala Kirchoff

Sinusoidal excitation of T-lines: General case

Phasors:Signal of frequency ω rad/s at any z, voltage V (z, t) = R{V(z)ejωt}For fixed z, V(z) is complex number, called phasor

∂V (z, t)/∂t →= −ωR{V(z)ejωt} = R{jωV(z)ejωt}

∂/∂t → jω

T-line equation in terms of phasors:

∂2V(z)

∂z2 = (R + jωL) (G + jωC) V(z) = γ2V(z)

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