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EE 201C Homework 1 Solution

Calculate inductance and capacitance among three modelled wires for signal analysis and simulation purposes. Solve for self and mutual inductance, ground and coupling capacitance, and determine resistance for Spice net-list.

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EE 201C Homework 1 Solution

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  1. EE 201C Homework 1 Solution Fang Gong gongfang@ucla.edu

  2. Homework • Given three wires, each modeled by at least 2 filaments, find the 3x3 matrix for (frequency-independent) inductance between the 3 wires. We assume that the ground plane has infinite size and is 10 um away for the purpose of capacitance calculation. l l l T W W W S S H • wire width: W=6um, wire thickness: T=4um, wire length: l=6000um, • wire spacing: S = 10um, distance to ground: H=10um, • Copper electrical resistivity 0.0175 Ωmm2/m (room temperature), • µ =1.256×10−6H/m, • free space 0=8.85×10 -12F/m

  3. l l l T W W W S S (1) Inductance Calculation • 1.1 discretize the wires into filaments, and calculate the self-inductance for each filament using formula.

  4. (1) Inductance Calculation (cont.) • 1.2 Calculate the mutual inductance for each pair of filaments • 1.3 calculate the self-inductance and mutual-inductance for wires: L1_self=L2_self=L3_self=3.66e-8 H L12_mutual=L21_mutual=2.7e-8 H L23_mutual=L32_mutual=2.7e-8 H L13_mutual=L31_mutual =2.37e-8 H Problem? the inductance of wire will increase along with the number of filaments.

  5. l l T W W S Field solver (fasthenry): L matrix =[ 9.1040e-09 6.7526e-09 5.9179e-09 6.7527e-09 9.1037e-09 6.7523e-09 5.9182e-09 6.7525e-09 9.1034e-09] (1) Inductance Calculation (cont.) • L_self and L_mutual of wire should be calculated as follows: • m is the number of filaments in wire k • n is the number of filaments in wire l • Reason: current in source filament and magnetic flux though target filament are reduced. • For our problem, all inductance should divided by 2*2=4 as L11=L22=L33=9.15e-9 H; L12=L21=L23=L32=6.75e-9 H; L13=L31 =5.925e-9 H

  6. (2) Capacitance • 2.1 Using formula [Sakurai-Tamaru, ED’83] to calculate ground capacitance and coupling capacitance. For a single wire above ground, the ground capacitance per unit length is For three wires above ground, the total capacitance of middle wire (=C20+2*C21) per unit length is Ground cap.for single wire Coupling cap.

  7. C2 C4 C1 C3 C5 (2) Capacitance (cont.) • For our problem: (1) Ground capacitance of middle wire: C3=e*(1.15*(width/h)+2.8*(thickness/h)^0.222)*length =1.579e-13 F; (2) Coupling capacitance of middle wire C2=C4=e*(0.03*(width/h)+0.83*(thickness/h)-0.07*(thickness/h)^0.222)*(s/h)^-1.34*length = 1.5552e-014 F; (3) Ground capacitance of edge wires: C1=C5=(single wire + three wire)/2 = 1.7350e-013 F; CAPACITANCE MATRIX, (1e-15 F) 1 2 3 1%GROUP1 1 166.7 -19.76 -3.139 2%GROUP1 2 -19.76 157.7 -19.85 3%GROUP1 3 -3.139 -19.85 166.8 C3=1.577e-13 F C1=C5=1.667e-13 F C2=C4=1.976e-14 F

  8. (3) Resistance • Copper electrical resistivity 0.0175 Ωmm2/m (room temperature), R1 = R2 = R3 = 4.375 Ω

  9. (4) Simulation • Generate net-list for Spice. K12 = L12/sqrt(L11*L22) • Different rising time lead to different waveforms.

  10. (4) Simulation • RCL model

  11. Thanks.

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