diff --git a/Draft1.asc b/Draft1.asc
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+++ b/Draft1.asc
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+Version 4.1
+SHEET 1 1556 680
+WIRE 176 144 -48 144
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+WIRE 976 592 880 592
+FLAG 112 304 0
+FLAG 1344 144 VL+
+FLAG 1344 288 VL-
+FLAG 432 144 V0
+FLAG 704 144 V1
+FLAG 976 144 V2
+FLAG 1232 144 V3
+FLAG 224 144 Vs
+SYMBOL voltage -48 144 R0
+WINDOW 3 -47 -23 Left 2
+WINDOW 123 0 0 Left 0
+WINDOW 39 0 0 Left 0
+SYMATTR Value PULSE(0 3 0 1f 1f 1)
+SYMATTR InstName V1
+SYMBOL res 400 128 R90
+WINDOW 0 0 56 VBottom 2
+WINDOW 3 32 56 VTop 2
+SYMATTR InstName RS
+SYMATTR Value 50
+SYMBOL cap 416 192 R0
+SYMATTR InstName C1
+SYMATTR Value 1p
+SYMBOL res 576 128 R90
+WINDOW 0 0 56 VBottom 2
+WINDOW 3 32 56 VTop 2
+SYMATTR InstName R2
+SYMATTR Value 0.02
+SYMBOL ind 576 160 R270
+WINDOW 0 32 56 VTop 2
+WINDOW 3 5 56 VBottom 2
+SYMATTR InstName L1
+SYMATTR Value 8n
+SYMBOL cap 688 192 R0
+SYMATTR InstName C2
+SYMATTR Value 10f
+SYMBOL res 848 128 R90
+WINDOW 0 0 56 VBottom 2
+WINDOW 3 32 56 VTop 2
+SYMATTR InstName R3
+SYMATTR Value 0.02
+SYMBOL ind 848 160 R270
+WINDOW 0 32 56 VTop 2
+WINDOW 3 5 56 VBottom 2
+SYMATTR InstName L2
+SYMATTR Value 40n
+SYMBOL res 1104 128 R90
+WINDOW 0 0 56 VBottom 2
+WINDOW 3 32 56 VTop 2
+SYMATTR InstName R4
+SYMATTR Value 0.02
+SYMBOL ind 1104 160 R270
+WINDOW 0 32 56 VTop 2
+WINDOW 3 5 56 VBottom 2
+SYMATTR InstName L4
+SYMATTR Value 8n
+SYMBOL cap 1216 192 R0
+SYMATTR InstName C4
+SYMATTR Value 1p
+SYMBOL cap 960 192 R0
+SYMATTR InstName C5
+SYMATTR Value 10f
+SYMBOL res 1328 160 R0
+SYMATTR InstName RL
+SYMATTR Value 100k
+SYMBOL res 1472 160 R0
+SYMATTR InstName R6
+SYMATTR Value 120
+SYMBOL ind 736 512 R270
+WINDOW 0 32 56 VTop 2
+WINDOW 3 5 56 VBottom 2
+SYMATTR InstName L5
+SYMATTR Value 500n
+SYMBOL cap 944 480 R90
+WINDOW 0 0 32 VBottom 2
+WINDOW 3 32 32 VTop 2
+SYMATTR InstName C6
+SYMATTR Value 100f
+SYMBOL res 896 576 R90
+WINDOW 0 0 56 VBottom 2
+WINDOW 3 32 56 VTop 2
+SYMATTR InstName R7
+SYMATTR Value 0.1
+SYMBOL ind 736 304 R270
+WINDOW 0 32 56 VTop 2
+WINDOW 3 5 56 VBottom 2
+SYMATTR InstName L3
+SYMATTR Value 50n
+SYMBOL cap 944 272 R90
+WINDOW 0 0 32 VBottom 2
+WINDOW 3 32 32 VTop 2
+SYMATTR InstName C3
+SYMATTR Value 50p
+SYMBOL res 896 368 R90
+WINDOW 0 0 56 VBottom 2
+WINDOW 3 32 56 VTop 2
+SYMATTR InstName R1
+SYMATTR Value 0.1
+SYMBOL res 160 160 R0
+SYMATTR InstName R5
+SYMATTR Value 120
+TEXT 72 344 Left 2 !.tran 0 20n 0 1p
diff --git a/Draft1.fft b/Draft1.fft
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diff --git a/Draft1.log b/Draft1.log
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+LTspice 24.1.9 for Windows
+Circuit: C:\Users\Judson\Desktop\Nextcloud\Miscellaneous\Reflection_Study\Draft1.net
+Start Time: Sat Nov  1 20:11:50 2025
+solver = Normal
+Maximum thread count: 4
+tnom = 27
+temp = 27
+method = trap
+WARNING: Node n007 is floating.
+
+.OP point found by inspection.
+Total elapsed time: 2.406 seconds.
+
+Files loaded:
+C:\Users\Judson\Desktop\Nextcloud\Miscellaneous\Reflection_Study\Draft1.net
+
diff --git a/Draft1.op.raw b/Draft1.op.raw
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diff --git a/Draft1.plt b/Draft1.plt
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+[Transient Analysis]
+{
+   Npanes: 1
+   {
+      traces: 5 {524290,0,"V(vl+)-V(vl-)"} {268959747,0,"V(v0)"} {268959748,0,"V(v1)"} {268959749,0,"V(v2)"} {268959750,0,"V(vl+)"}
+      X: ('n',1,7e-10,7e-10,8.4e-09)
+      Y[0]: (' ',1,2,0.2,4)
+      Y[1]: ('_',0,1e+308,0,-1e+308)
+      Volts: (' ',0,0,1,2,0.2,4)
+      Log: 0 0 0
+      GridStyle: 1
+   }
+}
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diff --git a/python/gnd_plane_reflection/reflection_wave.png b/python/gnd_plane_reflection/reflection_wave.png
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diff --git a/python/gnd_plane_reflection/reflection_wave_N24.gif b/python/gnd_plane_reflection/reflection_wave_N24.gif
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diff --git a/python/impedance_mismatch_no_gnd_plane_split/Figure_1.png b/python/impedance_mismatch_no_gnd_plane_split/Figure_1.png
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diff --git a/python/impedance_mismatch_no_gnd_plane_split/reflection_wave.png b/python/impedance_mismatch_no_gnd_plane_split/reflection_wave.png
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diff --git a/python/impedance_mismatch_no_gnd_plane_split/reflection_wave_N24.gif b/python/impedance_mismatch_no_gnd_plane_split/reflection_wave_N24.gif
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diff --git a/python/main.py b/python/main.py
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+++ b/python/main.py
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+import numpy as np
+
+class Simulation:
+    def __init__(self):
+        self.node_ids = {}
+        self.next_node_id = 0
+
+        self.branches = []
+        self.shunt_caps = []
+
+        self.branch_count = 0
+        self.cstate_count = 0
+
+        self.dirichlet = []
+
+    def add_node(self, label:str="") -> int:
+        if label in self.node_ids:
+            raise ValueError(f"Node {label} already exists")
+        idx = self.next_node_id
+        self.node_ids[label] = idx
+        self.next_node_id += 1
+        return idx
+    
+    def _get_node_idx(self, label: str) -> int:
+        if label not in self.node_ids:
+            raise KeyError(f"Unknown node '{label}'")
+        return self.node_ids[label]
+
+
+    def add_branch(self, node1:str, node2:str, R:float=0.0, L:float=0.0, C:float=0.0, label:str="") -> int:
+        branch_idx = self.branch_count
+        self.branch_count += 1
+
+        if label == "":
+            label = f"b_{branch_idx}"
+            
+        a, b = self._get_node_idx(node1), self._get_node_idx(node2)
+
+        c_idx = None
+        if C and C > 0.0:
+            c_idx = self.cstate_count
+            self.cstate_count += 1
+        self.branches.append((a, b, R, L, C, branch_idx, c_idx, label))
+        
+        return branch_idx, c_idx
+    
+    def add_shunt_C(self, node1: str, node2: str, C: float):
+        a, b = self._get_node_idx(node1), self._get_node_idx(node2)
+        self.shunt_caps.append((a, b, float(C)))
+
+    def build_matrices(self):
+        Nv = self.next_node_id
+        Ni = self.branch_count
+        Nc = self.cstate_count
+        N = Nv + Ni + Nc
+
+        E = np.zeros((N,N), dtype=float)
+        A = np.zeros((N,N), dtype=float)
+
+        for (a, b, R, L, C, branch_id, cap_id, _) in self.branches:
+            i = Nv + branch_id      # Index of this branch's current
+
+            # KCL for the nodes that have this current. Current leaves a and enters b
+            A[a, i] += 1.0
+            A[b, i] -= 1.0
+
+            # KVL for branch: v(a)-v(b) - R*i - L di/dt - v_c = 0
+            A[i, a] += 1.0
+            A[i, b] -= 1.0
+            if R: A[i, i] += -R
+            if L: E[i, i] += L
+            if C and cap_id is not None:
+                vc = Nv + Ni + cap_id
+                A[i, vc] += 1.0        # subtract v_c in KVL
+                E[vc, vc] += C         # C * d v_c/dt = i
+                A[vc, i] -= 1.0
+
+        # Shunt capacitors into E(v,v) block
+        for (a, b, C) in self.shunt_caps:
+            if a != b:
+                E[a, a] -= C;  E[a, b] += C
+                E[b, a] += C;  E[b, b] -= C
+
+        return E, A, N
+        
+
+    def step_BE(self, E, A, x_n, h, dirichlet_now):
+        # dirichlet_now: dict label->value at t_{n+1}
+        Nv = self.next_node_id
+        Ni = self.branch_count
+        Nc = self.cstate_count
+        N  = Nv + Ni + Nc
+
+        # fixed voltage indices and values
+        fixed_v_idx = np.array([self._get_node_idx(lbl) for lbl in dirichlet_now.keys()], dtype=int)
+        v_d = np.array([float(dirichlet_now[lbl]) for lbl in dirichlet_now.keys()], dtype=float)
+
+        # free sets
+        all_v_idx = np.arange(Nv, dtype=int)
+        free_v_idx = np.array(sorted(set(all_v_idx) - set(fixed_v_idx)), dtype=int)
+        free_i_idx = np.arange(Nv, Nv+Ni+Nc, dtype=int)
+        free_idx   = np.concatenate([free_v_idx, free_i_idx])
+
+        # Partitioned matrices
+        Ef_f = E[np.ix_(free_idx, free_idx)]
+        Af_f = A[np.ix_(free_idx, free_idx)]
+        Ef_d = E[np.ix_(free_idx, fixed_v_idx)]
+        Af_d = A[np.ix_(free_idx, fixed_v_idx)]
+
+        # Left matrix and RHS for BE
+        LHS = Ef_f - h * Af_f
+        # print(np.linalg.cond(LHS))
+        rhs = (E[np.ix_(free_idx, np.arange(N))] @ x_n) - (Ef_d - h * Af_d) @ v_d
+
+        x_free_next = np.linalg.solve(LHS, rhs)
+        x_next = np.zeros_like(x_n)
+        x_next[free_idx]   = x_free_next
+        x_next[fixed_v_idx]= v_d
+        return x_next
+
+def build_trace_with_split(sim, N=20, Rseg=0.05, Lseg=8e-9, Csh0=1e-12, Cshi=1e-14,
+                           Rsrc=10.0, Rs_gnd=120.0, Rload=1e5,
+                           Rsplit=0.3, Lsplit=50e-9, Cgap=100e-12,Rsplit_parallel=0.1):
+    # nodes: Vs, V0..V{N}, GNDL, GNDR
+    sim.add_node("Vs")
+    for k in range(N+1):
+        sim.add_node(f"V{k}")
+    sim.add_node("GNDL"); sim.add_node("GNDR")
+
+    # source
+    sim.add_branch("Vs","V0", R=Rsrc)
+    sim.add_branch("Vs","GNDL", R=Rs_gnd)
+
+    # series ladder
+    for k in range(N):
+        Lk = Lseg if (k==0 or k==N-1) else (Lseg*(N/(N-1)))  # tweak if you want end-sections different
+        sim.add_branch(f"V{k}", f"V{k+1}", R=Rseg, L=Lk)
+
+    # shunt caps to local grounds
+    sim.add_shunt_C("V0","GNDL", Csh0)
+    for k in range(1, N):
+        g = "GNDL" if k <= N//2 else "GNDR"
+        sim.add_shunt_C(f"V{k}", g, Cshi)
+    sim.add_shunt_C(f"V{N}", "GNDR", Csh0)
+
+    # load
+    sim.add_branch(f"V{N}", "GNDR", R=Rload)
+
+    # ground tie: lossy via in parallel with gap capacitance
+    sim.add_branch("GNDL","GNDR", R=Rsplit, L=Lsplit)    # via
+    sim.add_shunt_C("GNDL","GNDR", Cgap)                 # gap
+    
+    sim.add_branch("GNDL","GNDR", R=Rsplit_parallel) # Parallel across the gap
+
+if __name__ == "__main__":
+    import matplotlib.pyplot as plt
+
+    N = 50
+    sim = Simulation()
+    # build_trace_with_split(sim, N=N, Rseg=0.02, Lseg=8e-9, Csh0=1e-12, Cshi=10e-15,
+    #                     Rsrc=50.0, Rs_gnd=120.0, Rload=120,
+    #                     Rsplit=0.5, Lsplit=50e-9, Cgap=50e-12, Rsplit_parallel=0.1)
+    build_trace_with_split(sim, N=N, Rseg=0.02, Lseg=8e-9, Csh0=1e-12, Cshi=1e-12,
+                        Rsrc=50.0, Rs_gnd=120.0, Rload=120,
+                        Rsplit=0.0, Lsplit=0.0, Cgap=0.0, Rsplit_parallel=0.1)
+
+    E,A,NN = sim.build_matrices()
+    dt = 1e-12            # 1 ps
+    t_final = 50e-9         # 50 ns
+    steps = int(t_final / dt)
+    x = np.zeros(NN)
+    drive = {"Vs":3.0, "GNDL":0.0}
+
+    nodes = ["Vs"]+[f"V{k}" for k in range(N+1)]+["GNDL","GNDR"]
+    idx   = [sim._get_node_idx(s) for s in nodes]
+    tvec  = np.arange(steps+1)*dt
+    traces = np.zeros((len(nodes), steps+1))
+    traces[:,0] = x[idx]
+
+    for n in range(steps):
+        x = sim.step_BE(E, A, x, dt, drive)
+        traces[:,n+1] = x[idx]
+
+    # local-ground spatial profile
+    def spatial_profile(n):
+        v = np.zeros(N+1)
+        for k in range(N+1):
+            if k <= N//2:
+                v[k] = traces[nodes.index(f"V{k}"), n] - traces[nodes.index("GNDL"), n]
+            else:
+                v[k] = traces[nodes.index(f"V{k}"), n] - traces[nodes.index("GNDR"), n]
+        return v
+    
+    # time plots for 4 evenly spaced taps (include V0 and VN vs local ground) ---
+    # pick indices: 0, ~N/3, ~2N/3, N
+    tap_idx = sorted(set([0, N//3, (2*N)//3, N]))
+    tap_labels = [f"V{k}" for k in tap_idx]
+
+    def v_local(k, n):
+        # local ground reference by side
+        if k <= N//2:
+            return traces[nodes.index(f"V{k}"), n] - traces[nodes.index("GNDL"), n]
+        else:
+            return traces[nodes.index(f"V{k}"), n] - traces[nodes.index("GNDR"), n]
+
+    Vt = {k: np.array([v_local(k, n) for n in range(traces.shape[1])]) for k in tap_idx}
+
+    plt.figure(figsize=(8,4))
+    for k in tap_idx:
+        plt.plot(tvec*1e9, Vt[k], label=f"{'V0' if k==0 else ('V_N' if k==N else f'V{k}')}")
+    plt.xlabel("Time [ns]")
+    plt.ylabel("Voltage vs local ground [V]")
+    plt.title("Selected nodes over time")
+    plt.xlim(0, 10)
+    plt.grid(True)
+    plt.legend()
+    plt.tight_layout()
+    plt.savefig("reflection_wave.png")
+    plt.show()
+
+    # GIF
+    from PIL import Image
+    import io, numpy as np, matplotlib.pyplot as plt
+
+    xpos = np.arange(N+1)
+    Nt = traces.shape[1]
+    target_fps = 5
+    max_frames = 300
+    stride = max(1, Nt // max_frames)
+    frame_idx = np.arange(0, Nt, stride)
+    vals = np.vstack([spatial_profile(i) for i in frame_idx])
+    ymin, ymax = 1.1*vals.min(), 1.1*vals.max()
+    duration_ms = int(1000/target_fps)
+
+    frames = []
+    for i in frame_idx:
+        v = spatial_profile(i)
+        fig, ax = plt.subplots(figsize=(6,3.5))
+        ax.plot(xpos, v, marker=None)
+        ax.set_xlim(0, N)
+        ax.set_ylim(ymin, ymax)
+        ax.set_xlabel("Position index (0 … N)")
+        ax.set_ylabel("V vs local ground [V]")
+        ax.set_title(f"t = {tvec[i]*1e9:.2f} ns")
+        ax.grid(True)
+        buf = io.BytesIO()
+        plt.tight_layout()
+        fig.savefig(buf, format='png', dpi=120)
+        plt.close(fig)
+        buf.seek(0)
+        frames.append(Image.open(buf).convert("P"))
+
+    out_path = "reflection_wave_N24.gif"
+    frames[0].save(out_path, save_all=True, append_images=frames[1:], duration=duration_ms, loop=0, optimize=False)
+    print("Saved", out_path)
\ No newline at end of file
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