⚡ Inductive Crosstalk — Aggressor → Victim (Completed: R₂, C₂)

Figure 0 — the hand-drawn source model (R₁L₁C₁ primary → M → R₂L₂C₂ victim): step drive, I₁ with slope V_agg/L₁ & turning point ≈2L₁/R₁, induced EMF, and the victim across C₂. This report formalizes and simulates it.

Table 1 (image) — ranked case table (Case A far / Case B near / live Case C) rendered from the hand-drawn model above; live interactive version in § 4.

1. Introduction — what changed

The victim is no longer an open probe. Adding R₂ and C₂ turns the secondary into its own series L₂–R₂–C₂ loop, driven by the coupled EMF M dI₁/dt. What we now measure is the voltage across C₂ — the induced EMF after an R₂C₂ low-pass filter. Two consequences: (1) the victim waveform stops looking like dI₁/dt and instead tracks the integral of the secondary current, i.e. it follows I₁(t); (2) the peak is heavily attenuated — here a 500 mV EMF spike becomes a 100 mV bump (≈5× smaller) — a real, measurable victim. Everything below is first-principles: coupled KVL, a closed-form estimate, and a full 4-state simulation that confirms it.

2. Completed schematic & loop equations

3. Transient: I₁, induced EMF, filtered victim (zoomed)

Because L₁=0.88 nH is tiny, all the action is in the first ~100 ps — so the x-axis is zoomed to ps. On I₁(t) the red line is the initial slope S=V_agg/L₁=1.13 mA/ps, the purple marker is 2L₁/R₁=35 ps, and the blue dot is the turning point (slope=0) at t≈101 ps where I₁≈V_agg/R₁=19.7 mA.

Figure 2 — Top: I₁(t) with initial slope V_agg/L₁, the 2L₁/R₁ time, and the turning point (slope=0); the τ=R₁C₁=5 ns decay continues off-plot. Middle: coupled EMF V_L2=M dI₁/dt (peak M V_agg/L₁=500 mV). Bottom: victim across C₂ after the light R₂C₂ filter — reaches 100 mV (Case B: d=0.088 mm, k=0.50, M=442 pH).

4. Pin list — grouped by function

Before any coupling check we sort the package/IC pins into functional groups. Each group has a typical signal-integrity role — noisy switching nets are natural aggressors; low-tolerance sense/reference nets are natural victims. (Representative multiphase-buck example — edit the two tables below to match your design.)

5. Aggressor info

Each row is a driver pad that injects noise. dV is the output swing; dV/dt and f describe the edge/rate; R_out is the driver output resistance (it sets the peak loop current I_1,pk≈dV/R_out); W·T·L set the trace self-inductance. Even an infinite dV/dt is checkable — the loop inductance limits dI₁/dt, so the checker uses a worst-case voltage step (the ∞-slew limit); a finite dV/dt only relaxes the result.

6. Victim info

Each row is a receiver pad we must protect. V_err is its noise tolerance (the pass/fail limit); R_in, C_in form the receiver low-pass that integrates & attenuates the coupled EMF; d is the center-to-center spacing to the aggressor (same layer, or the layer-to-layer distance); W·T·L set the victim trace self-inductance and the parallel-run coupling length.

7. Spec Checker — SI mechanisms (inductive crosstalk first)

The checker auto-parses the aggressor & victim tables above and evaluates every aggressor→victim pair with the simple closed-form limit V_victim,pk = M·V_agg / (R₁·R₂·C₂) (R₁=driver R_out, R₂=receiver R_in, C₂=receiver C_in). This first release scores the inductive-crosstalk mechanism; the other SI mechanisms are scaffolded and will light up in later revisions.

Per-victim verdict — worst aggressor for each victim (inductive only):

All aggressor→victim pairs (worst first):

8. First-principles derivation

Coupled KVL with the mutual inductance matrix (states \(I_1,V_{C_1},I_2,V_{C_2}\)):

Secondary is R-dominated for this fast pulse (\(\omega L_2\ll R_2\)), so the induced EMF drops mostly across \(R_2\): \(\;I_2\approx -\dfrac{M}{R_2}\dot I_1\). Feeding that into the cap and integrating:

That boxed formula is the slow-filter limit (τ₂ >> pulse). Two caveats: (1) the primary R₁ must appear — it sets I_1,pk=V_agg/R₁; your sketch's box "2M V_agg/R₂C₂" dropped it (and used a rectangular area, hence the factor 2). (2) With the light filter chosen here (τ₂=50 ps ≈ the ~100 ps pulse) we are not in the slow-filter limit, so that formula over-reads — only the full 4-state solve is exact.

9. Design spec & simulation results

Primary: V_agg=1 V, R₁=50 Ω, L₁=0.88 nH (1 mm wire), C₁=100 pF; nominal coupling = Case B M=441.7 pH (k=0.50); victim R₂=0 kΩ, C₂=1 pF.

10. Python model (coupled 4-state)

Self-contained — copy, paste, run with nothing but the Python standard library (no numpy / scipy / matplotlib). It builds the mutual inductance from the geometry, RK4-solves the 4-state ODE, and prints an ASCII waveform with a V_err spec line — the trace beyond spec is red, the part within spec green, with the peak marked (*) and labeled — reproducing the default-populated Spec Checker case (§5–7): aggressor PWM_HDRV into ISEN_CS1 (R_in=200 Ω → 3.18 mV, FAIL) and ISEN_CS2 (R_in=1000 Ω → 0.85 mV, PASS vs the 1 mV budget). Same file in the repo: inductive_crosstalk_model.py.

# Coupled 4-state inductive-crosstalk model — copy, paste, run (pure standard library).
#   python3 this_file.py          # no numpy / scipy / matplotlib needed; prints an ASCII waveform
#
# Reproduces the DEFAULT case populated in the Spec Checker (section 7):
#   aggressor PWM_HDRV : dV=5 V, Rout=100 ohm, 30x10 um trace, 4000 um long
#   victims  ISEN_CS1 (Rin=200 ohm) and ISEN_CS2 (Rin=1000 ohm), Cin=1 pF, d=100 um, 150 um long
# The waveform is drawn with a Verr spec line; the part of the trace BEYOND spec is red, the
# part within spec is green (ANSI colors — most terminals show them).
import math
mu0 = 4e-7*math.pi
RED, GREEN, RESET = "\033[91m", "\033[92m", "\033[0m"

def Lself(L,w,t): return mu0*L/(2*math.pi)*(math.log(2*L/(w+t))+0.5+0.2235*(w+t)/L)  # trace self-L
def Mmut(d,L):    return mu0*L/(2*math.pi)*(math.asinh(L/d)-math.sqrt(1+(d/L)**2)+d/L) # parallel mutual

# --- default-populated geometry / drive ---
Vagg, R1, C1 = 5.0, 100.0, 100e-12
La = Lself(4000e-6, 30e-6, 10e-6)      # aggressor self-inductance (4 mm trace)
Lv = Lself( 150e-6, 30e-6, 10e-6)      # victim self-inductance (150 um trace)
M  = Mmut(100e-6, min(4000e-6, 150e-6))# mutual over the 150 um overlap at d = 100 um

def simulate(R2, C2):
    """RK4 solve of x=[I1,Vc1,I2,Vc2]; returns (times, victim voltage across C2)."""
    det = La*Lv - M*M
    def f(x):
        I1,Vc1,I2,Vc2 = x
        b1 = Vagg - R1*I1 - Vc1                 # primary branch
        b2 =       - R2*I2 - Vc2                # secondary branch (no source)
        return (( Lv*b1 - M*b2)/det, I1/C1, (-M*b1 + La*b2)/det, I2/C2)
    dt = min(La/R1, Lv/R2, R2*C2, math.sqrt(Lv*C2))/25
    T  = max(8*R2*C2, 40*La/R1, 3e-9); N = min(60000, math.ceil(T/dt)); dt = T/N
    x=[0.0]*4; ts=[0.0]; vs=[0.0]
    for i in range(N):
        k1=f(x); x2=[x[j]+.5*dt*k1[j] for j in range(4)]
        k2=f(x2); x3=[x[j]+.5*dt*k2[j] for j in range(4)]
        k3=f(x3); x4=[x[j]+dt*k3[j] for j in range(4)]; k4=f(x4)
        for j in range(4): x[j]+=dt/6*(k1[j]+2*k2[j]+2*k3[j]+k4[j])
        ts.append((i+1)*dt); vs.append(x[3])
    return ts, vs

def _paint(chars, cols):
    """chars: list of characters; cols: list of None/'r'/'g' -> ANSI string (run-length grouped)."""
    out=""; i=0; n=len(chars)
    while i<n:
        c=cols[i]
        if c is None: out+=chars[i]; i+=1
        else:
            j=i
            while j<n and cols[j]==c: j+=1
            out+=(RED if c=='r' else GREEN)+"".join(chars[i:j])+RESET; i=j
    return out

def ascii_plot(ts, vs, title, Verr, W=60, H=17):
    """ASCII waveform with a Verr spec line; trace beyond +/-Verr is red, within is green."""
    ip = max(range(len(vs)), key=lambda i: abs(vs[i])); vpk, tpk = vs[ip], ts[ip]
    Tplot = min(max(9*tpk, 300e-12), ts[-1])
    lo = min(0.0, min(vs), -Verr*1.15); hi = max(0.0, max(vs), Verr*1.15)  # keep +/-Verr in view
    row = lambda v: min(H-1, max(0, int(round((hi-v)/(hi-lo)*(H-1)))))
    col = lambda t: min(W-1, max(0, int(round(t/Tplot*(W-1)))))
    grid=[[' ']*W for _ in range(H)]; cols=[[None]*W for _ in range(H)]
    rSp, rSm, r0 = row(Verr), row(-Verr), row(0.0)
    for c in range(W):                                   # dashed spec lines at +/-Verr
        if c%3==0:
            if grid[rSp][c]==' ': grid[rSp][c]='.'
            if grid[rSm][c]==' ': grid[rSm][c]='.'
    for c in range(W): grid[r0][c]='-'                   # zero axis
    pkc=col(tpk); j=0
    for c in range(W):                                   # curve (on top), colored by spec
        tt=c/(W-1)*Tplot
        while j+1<len(ts) and ts[j+1]<tt: j+=1
        r=row(vs[j]); beyond = abs(vs[j])>Verr
        grid[r][c] = '*' if c==pkc else 'o'
        cols[r][c] = 'r' if beyond else 'g'
    ann=' <- peak = %.3f mV'%(vpk*1e3); rp=row(vpk); s=max(0,min(pkc+2,W-len(ann)-1))
    for i,ch in enumerate(ann):                          # peak label (color = peak's spec state)
        if s+i<W and grid[rp][s+i]==' ':
            grid[rp][s+i]=ch; cols[rp][s+i] = 'r' if abs(vpk)>Verr else 'g'
    slab=' spec (Verr)'                                  # spec-line tag on the -Verr row
    s2=W-len(slab)
    for i,ch in enumerate(slab):
        if 0<=s2+i<W and grid[rSm][s2+i] in ' .': grid[rSm][s2+i]=ch
    ylab={0:'%7.3f'%(hi*1e3), r0:'%7.3f'%0.0, rSm:'%7.3f'%(-Verr*1e3), H-1:'%7.3f'%(lo*1e3)}
    out=[title]
    for ri in range(H):
        out.append(ylab.get(ri,'       ')+' |'+_paint(grid[ri], cols[ri]))
    out.append('        +'+'-'*W)
    out.append('        0'+' '*(W//2-4)+'time [ps]'+' '*(W//2-13)+'%d'%round(Tplot*1e12))
    return '\n'.join(out), vpk

print("La=%.3f nH   Lv=%.4f nH   M=%.2f pH   (Vagg=%.0f V, R1=%.0f ohm, Cin=1 pF)\n"
      % (La*1e9, Lv*1e9, M*1e12, Vagg, R1))
Verr = 1e-3
for name, Rin in [("ISEN_CS1", 200.0), ("ISEN_CS2", 1000.0)]:
    ts, vs = simulate(Rin, 1e-12)
    plot, vpk = ascii_plot(ts, vs, "PWM_HDRV -> %s  (Rin=%.0f ohm)  [victim V across C2, mV]"%(name,Rin), Verr)
    ok = abs(vpk) <= Verr
    print(plot)
    print("        Verr = %.3f mV   ->   %s%s%s\n" % (Verr*1e3, GREEN if ok else RED, "PASS" if ok else "FAIL", RESET))

La=4.640 nH   Lv=0.0772 nH   M=19.79 pH   (Vagg=5 V, R1=100 ohm, Cin=1 pF)

PWM_HDRV -> ISEN_CS1  (Rin=200 ohm)  [victim V across C2, mV]
  1.150 |                                                            
        |.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  
        |                                                            
        |                                                            
  0.000 |o-------------------------------------------------------oooo
        |                                         ooooooooooooooo    
        |                                  ooooooo                   
        |                             ooooo                          
 -1.000 |.  .  .  .  .  .  .  .  .oooo .  .  .  .  .  .   spec (Verr)
        | o                    ooo                                   
        |                    oo                                      
        |                  oo                                        
        |  o             oo                                          
        |              oo                                            
        |   o        oo                                              
        |    o    ooo                                                
 -3.176 |     oo*o <- peak = -3.176 mV                               
        +------------------------------------------------------------
        0                          time [ps]                 792
        Verr = 1.000 mV   ->   FAIL

PWM_HDRV -> ISEN_CS2  (Rin=1000 ohm)  [victim V across C2, mV]
  1.150 |                                                            
        |.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  
        |                                                            
        |                                                            
        |                                                            
        |                                                            
        |                                                            
        |                                                            
  0.000 |o-----------------------------------------------------------
        |                                                           o
        |                                          ooooooooooooooooo 
        | o                            oooooooooooo                  
        |  o                 oooooooooo                              
        |   o        oooooooo                                        
        |    ooo*oooo peak = -0.846 mV                               
 -1.000 |.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   spec (Verr)
 -1.150 |                                                            
        +------------------------------------------------------------
        0                          time [ps]                 1310
        Verr = 1.000 mV   ->   PASS

11. Sources

1. [1] R. Blaymore, EELE 461/561 — Module 05: Crosstalk, Montana State University — V_M=L_MdI/dt, coupled-line model. montana.edu/…/eele461_module_05.pdf
2. [2] Teledyne LeCroy, Inductively Coupled Noise … SI/PI Tutorial (2023). blog.teledynelecroy.com/…
3. [3] Rosa/Grover partial-inductance formulas (rectangular self-L, parallel-filament mutual), in Johnson & Graham, High-Speed Digital Design. background: protoexpress.com/…
4. [4] ClimbTheLadder, 20 Signal Integrity Interview Questions. climbtheladder.com/signal-integrity-interview-questions
5. [5] Interviews.chat, Top 10 Signal Integrity Engineer Interview Questions. interviews.chat/…/signal-integrity-engineer

12. Design question asked in an interview