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LIMITS TO COMPUTATION SPEED Devices: Emitter Gate Carrier transit and diffusion times (f = ma, v < c) Field-effect RC ≅ ε/σ; RL, LC time constants transistors Beyond scope of 6.013 (read Section 8.2) Interconnect, short lines <<λ: 2r Wire resistance R ∝ D/r2 Capacitance C = εA/d ∝ D2/d τ = RC ∝ D3/r2d ≅ const. if D:r:d = const. D R is high for polysilicon, C is high for thin gaps L/R and τ = LC scale well with size and do not limit speed Drain Charge Carriers D d D Interconnect, long lines >~λ/8: Propagation delay: c = 1/ με < 3 ×108 [m s-1] (ε might be ~2εo) Reflections at wire and device junctions, unless carefully designed Resistive loss Radiation and cross-talk (3-GHz clocks imply 30-GHz harmonics) L11-1 WIRED INTERCONNECTIONS Transverse EM Transmission Lines: TEM: Ez = Hz = 0 Parallel wires Coaxial cable z Parallel plates Stripline Arbitrary cross-section ≠ f(z) L11-2 PARALLEL-PLATE TRANSMISSION LINE Boundary Conditions: H × E = S E // = H⊥ = 0 at perfect conductors Uniform Plane Wave Solution: x-polarized wave propagating in +z direction in free space y E = xˆ E+ t - z c H = yˆ ( 1 ) E+ t - z ηo c ( ) ( ) z H z I(z) = H(z)W, independent of path C Js (z) = n̂ × H(z) [A m-1] E x Currents in Plates: v∫C H • ds = ∫∫A J • da = I(z) Surface Currents Js(A m-1): σ=∞ W C H n̂ z I(z) I(z) W L11-3 TRANSMISSION LINE VOLTAGES Voltages between plates: Since Hz = 0 ⇒ v∫c E • ds = 0 at fixed z, Φ1 2 ∫1 E • ds = Φ1 − Φ 2 = V(z) y V(z) is uniquely defined σ=∞ + E c z V(z) Φ2 x ds d Surface charge density ρs(z) [C m-2]: n̂ • εE(z) = ρs (z) (Boundary condition; from ∇ • D = ρ Integrate E,H to find v(t,z),i(t,z) ) 2 ( ) ˆ + (t - z c) v(t,z) = ∫1 E • ds = d × E+ t - z/c here, where E = xE ˆ + ( t- z c ) i(t,z) = v∫c H • ds = (W/ηo )E+ ( t - z/c ) , where H = yE v(t,z) = Zo i(t,z) [if there is no backward propagating wave] Zo = ηod/W [ohms] "Characteristic impedance" Note: v(z) violates KVL, and i(z) violates KCL L11-4 TELEGRAPHER’S EQUATIONS W Equivalent Circuit: LΔz i(z) LΔz i(z+Δz) LΔz z y CΔz v(t,z) CΔz v(t,z+Δz) CΔz Δz L [Henries m-1], C [Farads m-1] + v(t,z) x i(t,z) E • zˆ = H • zˆ = 0 (TEM) Difference Equations: di(z) dt dv(z) i(z+Δz) - i(z) = −CΔz dt v(z+Δz) - v(z) = −LΔz d Limit as Δz → 0: Wave Equation: dv(z) di(z) = −L dz dt di(z) dv(z) = −C dz dt d2 v = LC d2 v dz2 dt 2 L11-5 SOLUTION: TELEGRAPHER’S EQUATIONS Wave Equation: d2v = LC d2 v 2 2 dz Solution: dt v(z,t) = f+(t – z/c) + f-(t + z/c) f+ and f- are arbitrary functions Substituting into Wave Equation: (1/c2) [f+″(t – z/c) + f-″(t + z/c)] = LC [f+″(t – z/c) + f-″(t + z/c)] Therefore: Current I(z,t): Recall: Therefore: Therefore: di(z) dv(z) = −C = -C[f+′(t – z/c) + f-′(t + z/c)] dz dt i(z,t) = cC [f+(t – z/c) – f-(t + z/c)] cC = C LC = C L= Yo "Characteristic admittance" Zo = 1/Yo = ohms “Characteristic impedance” i(z,t) = Yo[f+(t – z/c) – f-(t + z/c)] C / L Where: c = 1 LC = 1 με L11-6 ARBITRARY TEM LINES Can we estimate L and C? s ≠ f(z) s Φi E d 1 L LC A Zo = = where LC = με C C d εA C = capacitance/m = d ε = C = nC for n parallel square incremental capacitors (C = ε [Fm-1]) C = C /m for m capacitors C in series m Therefore: C = nC /m = nε/m [Farads/m] N cells Zo = LC = με = ηo m ohms ηo = μ/ε]; Ζosingle cell = 377Ω [ C nε/m n L11-7 TRANSMISSION LINE VOLTAGES v(z,t) Velocity c + vs(t) Zo ohms, c [m/s] 0 z Matching boundary conditions: v(t) and I(t) are continuous at z = 0 v(z,t) = vs(t – z/c) 1 i(z,t) = v (t – z/c) Zo s L11-8 MIT OpenCourseWare http://ocw.mit.edu 6.013 Electromagnetics and Applications Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.