{
  "format": "mathssim",
  "version": 1,
  "title": "02 - Complex Numbers, Exponentials and Phasors",
  "computeMode": "standard",
  "functions": [],
  "notation": {
    "zr": "z_r",
    "zi": "z_i",
    "zmag": "\\left|z\\right|",
    "zphase": "\\arg z",
    "zconj": "z^{*}",
    "mag_sq_check": "\\left|z\\right|^2",
    "theta": "\\theta",
    "unitCircle": "e^{j\\theta}",
    "V0": "V_0",
    "phi_deg": "\\phi_{\\mathrm{deg}}",
    "phi": "\\phi",
    "Vphasor": "\\tilde{V}",
    "tphase": "\\omega t",
    "vtime": "v(t)",
    "V1": "\\tilde{V}_1",
    "V2": "\\tilde{V}_2",
    "Vsum": "\\tilde{V}",
    "Vsum_mag": "\\left|\\tilde{V}\\right|",
    "Vsum_phase_deg": "\\angle\\tilde{V}",
    "Rohm": "R",
    "Lhenry": "L",
    "freq": "f",
    "omega": "\\omega",
    "Zseries": "Z",
    "Zseries_mag": "\\left|Z\\right|",
    "Zseries_deg": "\\angle Z",
    "Vrms": "V_{\\mathrm{rms}}",
    "Irms": "I_{\\mathrm{rms}}",
    "Scomplex": "\\tilde{S}",
    "Pwatts": "P",
    "Qvars": "Q",
    "alpha": "\\alpha",
    "beta": "\\beta",
    "zaxis": "z",
    "Ephasor_z": "\\tilde{E}(z)",
    "Emag_z": "\\left|\\tilde{E}(z)\\right|",
    "Ereal_z": "E(z)"
  },
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      "w": 990,
      "content": "02 - Complex Numbers, Exponentials and Phasors",
      "heading": true
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      "y": 140,
      "w": 990,
      "content": "Interactive M²S companion to EM Course Document 02.\nComplex plane, Euler identity, phasors, impedance, complex power and propagation exponentials.\nEdit the input Math regions, then Calculate. M²S values are unitless, so SI units are stated explicitly in labels and notes.",
      "heading": false
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    {
      "id": "ch02-head-003",
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      "content": "1. Rectangular and polar complex forms",
      "heading": true
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      "content": "2. Euler rotation and sinusoidal reconstruction",
      "heading": true
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      "expr": "exp(i*theta)"
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      "id": "ch02-plot-015",
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      "title": "Euler identity on the complex plane",
      "xexpr": "real(unitCircle)",
      "yexpr": "Imaginary := imag(unitCircle)",
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      "title": "Real sinusoid reconstructed from a phasor",
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      "xlabel": "ωt (rad)",
      "ylabel": "v(t) (V)",
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      "id": "ch02-head-023",
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      "y": 4076,
      "w": 990,
      "content": "3. Phasor addition",
      "heading": true
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      "id": "ch02-math-024",
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      "id": "ch02-math-026",
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      "title": "10∠0° + 10∠90°",
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      "content": "4. Impedance: the course 50 Ω + 10 nH example",
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      "target": "Zseries",
      "expr": "Rohm+i*omega*Lhenry"
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      "title": "Series R-L impedance at 1 GHz",
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      "content": "5. Complex power and conjugation",
      "heading": true
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      "id": "ch02-math-040",
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      "id": "ch02-math-041",
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      "target": "Irms",
      "expr": "polar(0.1,-60*pi/180)"
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      "id": "ch02-math-042",
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      "title": "RMS complex-power worked example",
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      "content": "6. Propagation exponential: attenuation and phase",
      "heading": true
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      "target": "alpha",
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      "target": "zaxis",
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    {
      "id": "ch02-math-052",
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      "y": 8906,
      "w": 990,
      "target": "Ereal_z",
      "expr": "real(Ephasor_z)"
    },
    {
      "id": "ch02-plot-053",
      "type": "plot",
      "kind": "line",
      "title": "Propagation factor exp(-(α+jβ)z)",
      "xexpr": "zaxis",
      "yexpr": "Real part := Ereal_z\nMagnitude [y2] := Emag_z",
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      "ylabel": "Re{E}",
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    {
      "id": "ch02-out-054",
      "type": "output",
      "kind": "csv",
      "title": "Propagation samples",
      "filename": "ch02-propagation.csv",
      "sources": "z_m := zaxis\nphasor := Ephasor_z\nmagnitude := Emag_z\nreal_part := Ereal_z",
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      "complexFormat": "abi"
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      "y": 9831,
      "w": 990,
      "content": "Checkpoint / exploration",
      "heading": true
    },
    {
      "id": "ch02-note-056",
      "type": "text",
      "x": 65,
      "y": 9936,
      "w": 990,
      "content": "Keep the course convention explicit: these examples use exp(+jωt). Change alpha and beta separately to see attenuation and spatial phase play different roles.",
      "heading": false
    }
  ]
}