LIMITS TO COMPUTATION SPEED Devices: Emitter Gate Drain Carrier transit and diffusion times (f = ma, v < c) Field-effect ChargeRC ≅ε/σ; RL, LC time constants transistors Carriers Beyond scope of 6.013 (read Section 8.2) Interconnect, short lines <<λ: DD Wire resistance R ∝ D/r2 2r dCapacitance C = εA/d ∝ D2/d τ = RC ∝ D3/r2d ≅ const. if D:r:d = const. R is high for polysilicon, C is high for thin gaps D L/R and τ = LC scale well with size and do not limit speed 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) L11-1WIRED INTERCONNECTIONS Transverse EM Transmission Lines: TEM: Ez = Hz = 0 Parallel wires Coaxial cable z Parallel Stripline Arbitrary cross-section plates ≠ f(z) L11-2PARALLEL-PLATE TRANSMISSION LINE Boundary Conditions: E// = H⊥= 0 at perfect conductors W σ=∞ z E S=E×H x H z Uniform Plane Wave Solution: x-polarized wave propagating in +z direction in free space =ˆ t -z y Ex E + ( c )) 1 z=ˆ ) E + (Hy( ηo t -c H z I(z) I(z) C nˆ Currents in Plates: v∫C • ∫∫ A J I(z) Hds da = • = I(z) = H(z)W, independent of path C Surface Currents Js(A m-1):-1J (z) =×nˆ H(z) [A m ] s W L11-3L11-4 TRANSMISSION LINE VOLTAGES Voltages between plates: 2 1 21 Eds V(z) • =Φ −Φ =∫ ⇒ • =∫vz c Since H = 0 E ds 0 at fixed z, V(z) is uniquely defined Surface charge density ρs(z) [C m-2]: (Boundary condition; from )•ε =ρsˆn E(z) (z) ∇•=ρD ( ) ( )= • = × =∫2 + +1 ˆv(t,z) E ds d E t - z/c here, where E xE t - z c Integrate E,H to find v(t,z),i(t,z) Φ1 Φ2 d σ=∞ V(z)-+ E ds c x y z ( ) ( )= • = η ∫v o o o o + +c ˆi(t, z) H ds (W/η )E t - z/c , where H = yE t- z c v(t,z) = Z i(t,z) [if there is no backward propagating wave] Z = d/W [ohms] "Characteristic impedance" Note: v(z) violates KVL, and i(z) violates KCLL11-5 TELEGRAPHER’S EQUATIONS Equivalent Circuit: L [Henries m-1], C [Farads m-1] -+v(t,z) d y x z i(t,z) ˆ ˆEz H z 0• =• = W i(z) i(z+Δz)LΔz v(t,z) v(t,z+Δz) Δz LΔz LΔz CΔz CΔz CΔz Difference Equations: Limit as Δz → 0:=− Δ =− Δ di(z) v(z+Δz) - v(z) L z dt dv(z) i(z+Δz) - i(z) C z dt dv(z) di(z) Ldz dt di(z) dv(z) Cdz dt =− =− 2 2 2 2 dv dvLC dz dt =Wave Equation: (TEM)L11-6 SOLUTION: TELEGRAPHER’S EQUATIONS Wave Equation: Solution: Substituting into Wave Equation: v(z,t) = f+(t – z/c) + f-(t + z/c) f+ and f-are arbitrary functions (1/c2) [f+ ″(t – z/c) + f-″(t + z/c)] = LC [f+ ″(t – z/c) + f-″(t + z/c)] Therefore: c1 LC 1= = με Current I(z,t): Recall: = -C[f+ ′(t – z/c) + f-′(t + z/c)] Therefore: i(z,t) = cC [f+(t – z/c) – f-(t + z/c)] Where: = = ocC C LC C L= Y "Characteristic admittance" Therefore: i(z,t) = Yo[f+(t – z/c) – f-(t + z/c)] Zo = 1/Yo = ohms “Characteristic impedance”L/C 2 2 2 2 dv dvLC dz dt = di(z) dv(z) Cdz dt=−ARBITRARY TEM LINES Can we estimate L and C? s s Φi E≠ f(z) L LC 1d A d C = capacitance/m = εA = ε Zo = C = C where LC = με dC = nC for n parallel square incremental capacitors (C = ε [Fm-1]) C = C/m for m capacitors C in series mTherefore: C = nC /m = nε/m [Farads/m] N cells Zo = LC = με =η m ohms [ηo= μ/ε]; Ζosingle cell = 377Ω C n/m ε on L11-7TRANSMISSION LINE VOLTAGES Velocity c v(z,t) vs(t) Zo ohms, c [m/s] + -z0 Matching boundary conditions: v(t) and I(t) are continuous at z = 0 v(z,t) = vs(t – z/c) i(z,t) = Z1 vs(t – z/c) o L11-8MIT OpenCourseWare http://ocw.mit.edu 6.013 Electromagnetics and Applications Spring 2009 For information about citing these materials or our Terms of Use, visit:
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