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Lumped Vs. Distributed Circuits ECE 391 supplemental notes - #1 1 Oregon State University ECE391– Transmission Lines Spring Term 2014 Lumped vs. Distributed Circuits Lumped-Element Circuits: • Physical dimensions of circuit are such that voltage across and current through conductors connecting elements does not vary. • Current in two-terminal lumped circuit element does not vary (phase change or transit time are neglected) 2 Oregon State University ECE391– Transmission Lines Spring Term 2014 1 Lumped vs. Distributed Circuits Distributed Circuits: • Current varies along conductors and elements; • Voltage across points along conductor or within element varies è phase change or transit time cannot be neglected Example: 25 cm f = 300MHz vp=c è∞ c 3×108 m λ = = s = 1m f 300 ×106 1 s current wavelength λ = 1 period in space distance 3 Oregon State University ECE391– Transmission Lines Spring Term 2014 Example: Hybrid Microwave Integrated Circuit (MIC) distributed element transmission line interconnection lumped element èlow loss 4 Oregon State University ECE391– Transmission Lines Spring Term 2014 2 When Do We Need to Consider Transmission Lines Efects digital domain analog domain 100% λ 90% 10% 0% distance tr tf risetime falltime flight time (delay time) td=d/vp maximum dimension d tr, tf vs td d vs λ 5 Oregon State University ECE391– Transmission Lines Spring Term 2014 Lumped vs. Distributed td = length/velocity Z0, td VS RL vs (t) = v0 u(t) §! Rule-of-thumb (heuristic) Delay time td = length of line / velocity Rise time of signal tr (fall time tf) Signal path can be treated as lumped element" distributed element" if! tr td > 6 if! tr td < 2.5 6 Oregon State University ECE391– Transmission Lines Spring Term 2014 3 Example • CMOS Buffer with tr = tf = 0.5 ns • FR-4 PCB (velocity ≈ 0.45 c) • Note: VDD signal ε vp ≈ c εr r gnd Determine maximum distance d so that tr/td > 6 tr/td = tr/(d/vp) > 6 è d < tr vp/6 d < 0.5ns * (0.45*30cm/ns)/6 = 1.125 cm ≈ 11 mm 7 Oregon State University ECE391– Transmission Lines Spring Term 2014 TL Problem Classification “faster board” risetime 0 2.5td 6td clock reset signals simulate and check after no problems with critical timing (or layout of ckt go for it! edge sensitivity) simulate and non-critical signals consider quick no problems check after (e.g. data lines with hand analysis go for it! layout of ckt tsu ≈ 6 td) simulate and status signals; slow, check after no problems expected insensitive lines layout of ckt In this region, other consid. begin to surface (EMI, crosstalk) 8 Oregon State University ECE391– Transmission Lines Spring Term 2014 4 Sinusoidal Signals V0 cos(2π ft) = V0 cos(ωt) z phase constant V cos(ω(t − t ) ω 2π 0 d β = = = V0 cos(ωt −ωtd ) vp λ z = V0 cos(ωt −ω ) = V0 cos(ωt − βz) vp In practice: lumped if td < 0.1 T (d < 0.1 λ) safely lumped if d < 0.01λ 9 Oregon State University ECE391– Transmission Lines Spring Term 2014 Illustration CH1 CH2 time 10 Oregon State University ECE391– Transmission Lines Spring Term 2014 5 Example: Lumped vs. Distributed For example, the spatial dependence z of the current i(z,t) = I0 cos(ω t − βz) in a conductor can be neglected if z λ << 1 . Example: 1.25 1 f =100MHz 0.75 f =1GHz 0.5 0.25 f =10GHz 0 -0.25 -0.5 -0.75 Normalized current I(z,t=0) current Normalized -1 -1.25 -4 -3 -2 -1 0 1 2 3 4 Distance (cm) resistor 11 Oregon State University ECE391– Transmission Lines Spring Term 2014 Concept: Electrical Length • Electrical length (θ, E) is a measure of the physical length expressed in terms of wavelength λ z θ = E = 2 π (in radians) λ z θ = E = 3 6 0 0 (in degrees) λ z E = (as fraction of wavelength) λ 12 Oregon State University ECE391– Transmission Lines Spring Term 2014 6 Transmission Line Examples coaxial line two-wire line (also twisted-pair) microstrip stripline coplanar strip (CPS) coplanar waveguide (CPW) 13 Oregon State University ECE391– Transmission Lines Spring Term 2014 Transmission Lines/Interconnects § What is a transmission line? – lumped vs. distributed circuit? – how many conductors? § Types of transmission lines – on-chip (à transmission line?) – off chip – PCB, packages – cables (coax, twisted pair, …) – etc. § Applications of transmission lines – interconnections § signal transmission § power transmission – circuit elements/functional components (at high frequencies) § filters (e.g. coupled lines, stubs, …) § couplers § power dividers § matching networks – … 14 Oregon State University ECE391– Transmission Lines Spring Term 2014 7 Interconnect Technologies Inter- Line Line Max. connection Width Thickness Length On-chip Type (µm) (µm) (cm) On-Chip 0.5-2 0.5-2 0.3-1.5 Package Thin-Film 10-25 5-8 20-45 75-10 Ceramic 16-25 20-50 PCB 0 60-10 PCB 8-70 40-70 0 cables, etc. Source: IBM (plus changes) 15 Oregon State University ECE391– Transmission Lines Spring Term 2014 Transmission Line – Characteristic Dimensions Cross-sectional Dimensions << wavelength D << λ D L D L L < λ L >> λ Lengths vary from fractions of a wavelength to many wavelengths 16 Oregon State University ECE391– Transmission Lines Spring Term 2014 8 Model for Transmission Line Note: here R,L,G,C generic equivalent circuit model represent total values per section 17 Oregon State University ECE391– Transmission Lines Spring Term 2014 Transmission Line Parameters Cross-sectional view of typical uniform interconnects: § Capacitance between conductors, C (F/m) § Inductance of conductor loop, L (H/m) § Resistance of conductors (conductor loss), R (Ω/m) § Shunt conductance (dielectric loss), G (S/m) R,L,G,C are specified as per-unit-length parameters 18 Oregon State University ECE391– Transmission Lines Spring Term 2014 9 Derivation of Transmission Line Equations ∂i(z,t) −[v(z + Δz,t)− v(z,t)]= LΔz ∂t ∂v(z + Δz,t) −[i(z + Δz,t)−i(z,t)]= CΔz ∂t Note: here L,C are per-unit-length parameters 19 Oregon State University ECE391– Transmission Lines Spring Term 2014 Transmission Line Equations Lossless transmission line i ∂i(z,t) −[v(z + Δz,t)− v(z,t)]= LΔz ∂t ∂v(z + Δz,t) −[i(z + Δz,t)−i(z,t)]= CΔz ∂t after taking Δz → 0 Telegrapher’s Equations ∂v(z,t) ∂i(z,t) ∂i(z,t) ∂v(z,t) − = L − = C ∂z ∂t ∂z ∂t 20 Oregon State University ECE391– Transmission Lines Spring Term 2014 10 Wave Equations ∂v(z,t) ∂i(z,t) ∂i(z,t) ∂v(z,t) − = L − = C ∂z ∂t ∂z ∂t Wave Equations ∂2v(z,t) ∂2v(z,t) 1 ∂2v(z,t) 2 = LC 2 = 2 2 ∂z ∂t vp ∂t ∂2i(z,t) ∂2i(z,t) 1 ∂2i(z,t) 2 = LC 2 = 2 2 ∂z ∂t vp ∂t 21 Oregon State University ECE391– Transmission Lines Spring Term 2014 General Wave Solutions ∂ 2v ∂ 2v 1 ∂ 2v 2 = LC 2 = 2 2 ∂z ∂t v p ∂t General Solution for Voltage + − v(z,t) = v (z − vpt) + v (z + vpt) + − = v (t − z / vp ) + v (t + z / vp ) +z direction -z direction 1 Velocity of Propagation v = (m/s ) p LC 22 Oregon State University ECE391– Transmission Lines Spring Term 2014 11 Illustration of Space–Time Variation of Single Traveling Wave + − v(z,t) = v (z − vpt) + v (z + vpt) + − = v (t − z / vp ) + v (t + z / vp ) v p v p time distance 23 Oregon State University ECE391– Transmission Lines Spring Term 2014 Wave Solutions for Current + − i(z,t) = i (t − z / vp ) +i (t + z / vp ) ∂v(z,t) ∂i(z,t) ∂i(z,t) ∂v(z,t) − = L − = C ∂z ∂t ∂z ∂t 1 + − + − {v (t − z / vp ) − v (t + z / vp )}= L{i (t − z / vp ) + i (t + z / vp )} vp 1 − − + − {i (t − z / vp ) −i (t + z / vp )}= C{v (t − z / vp ) + v (t + z / vp )} vp v+ (t − z / v ) v− (t + z / v ) i(z,t) = p − p Z0 Z0 1 L Characteristic Impedance Z0 = vp L = = (Ω) vpC C 24 Oregon State University ECE391– Transmission Lines Spring Term 2014 12 Summary of Transmission Line Parameters • Capacitance per-unit-length C (F/m) • Inductance per-unit-length L (H/m) • Characteristic impedance Z 0 (Ω) • Velocity of propagation v p (m/s) • Per-unit-length delay time t p = 1 v p (s/m) • Delay time (TD) t d = l v p = l t p (sec) L 1 Z = v = t = LC t = l LC 0 C p LC p d L = Z0 vp = Z0 t p C =1 (Z0v p ) =t p Z0 25 Oregon State University ECE391– Transmission Lines Spring Term 2014 Properties of Ideal Transmission Lines § C from 2D electro-static solution § L from 2D magneto-static solution § Velocity of propagation 1 1 c0 vp = = = c0 ≈ 30cm/nsec LC µ0µrε 0ε r ε r (neglecting magnetic materials) 26 Oregon State University ECE391– Transmission Lines Spring Term 2014 13 Propagation Speeds for Typical Dielectrics Dielectric Rel. Dielectric Constant Propagation Delay time speed per unit length εr (cm/nsec) (ps/cm) Polyimide 2.5 – 3.5 16-19 53 - 62 Silicon dioxide 3.9 15 66 Epoxy glass (PCB) 5.0 13 75 Alumina (ceramic) 9.5 10 103 27 Oregon State University ECE391– Transmission Lines Spring Term 2014 Example T-Line Structures Parallel-Plate Line Coaxial Line w t ε r a ε r s b c 2πε 0ε r µ ⎛ b ⎞ w s C = L = 0 ln⎜ ⎟ C = ε ε L = µ ln(b a) 2π ⎝ a ⎠ 0 r s 0 w µ ln(b a) Z0 = s 2ρ ε ε 2π µ R = 0 r Z0 = DC wt ρ ρ ε0ε r w RDC = 2 + 2 2 π a π (c − b ) 28 Oregon State University ECE391– Transmission Lines Spring Term 2014 14 Examples (cont’d) r r s s 2ρ ρ R = R = DC π r 2 DC π r 2 πε 2πε C = C = cosh −1(s 2r) cosh −1(s r) µ µ L = 0 cosh −1(s 2r) L = 0 cosh −1(s r) π 2π 1 µ0 −1 1 µ0 −1 Z0 = cosh (s 2r) Z0 = cosh (s r) π ε 2π ε 29 Oregon State University ECE391– Transmission Lines Spring Term 2014 Examples (cont’d) stripline microstrip W ε0 t W εr εr t b h 1 µ ⎛ 1+ w b ⎞ 1 µ ⎛ 4h ⎞ Z ln Z0 = ln⎜ ⎟ 0 = ⎜ ⎟ 4 ε ⎝ w b + t b ⎠ 2π ε 0ε eff ⎝ d ⎠ 1 C = ε r d = 0.536w + 0.67t c0Z0 Z0 L = ε εeff ≈ 0.475ε r + 0.67 r c 0 (very approximate!) 30 Oregon State University ECE391– Transmission Lines Spring Term 2014 15 Alternative Formulas w ε0 t w A ε b r h εr t B 60Ω ⎛ 4b ⎞ Z0,sym = ln⎜ ⎟ 87Ω ⎛ 5.98h ⎞ ⎜ ⎟ Z = ln ε r ⎝ 0.67π (t + 0.8w) ⎠ 0 ⎜ ⎟ ε r +1.41 ⎝ 0.8w + t ⎠ for w/b < 0.35 and t/b < 0.25 for 0.1 < w/h < 2 and 1 < εr < 15 Z0,sym (2A, w,t,ε r )*Z0,sym (2B, w,t,ε r ) Z0,offset ≅ 2 Z0,sym (2A, w,t,ε r ) + Z0,sym (2B, w,t,ε r ) more accurate if 2A # 2A+t and 2B # 2B+t 31 Oregon State University ECE391– Transmission Lines Spring Term 2014 16 .
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