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Carrier harmonics through L and LCL filters

This tutorial follows the ideal PWM pole-voltage error through the physical AC filter. It demonstrates why converter-side current, grid-side current, midpoint- capacitor current, and neutral current must be reported separately for an LCL converter.

The calculation is a local carrier-harmonic audit around a solved fundamental operating point. It is more detailed than an RMS allowance, but it is not EMT or a controller impedance scan.

1. Create an unbalanced four-wire operating point

using PowerOptLab
using BMOPFTools: parse_bmopf

network = parse_bmopf("""
{"bus":{
  "grid":{"terminal_names":["a","b","c","n"],
          "perfectly_grounded_terminals":["n"]},
  "poc":{"terminal_names":["a","b","c","n"],
         "perfectly_grounded_terminals":["n"],
         "v_min":[180.0,180.0,180.0],"v_max":[270.0,270.0,270.0]}},
 "voltage_source":{"vs":{"bus":"grid","terminal_map":["a","b","c"],
   "v_magnitude":[245.0,215.0,230.0],"v_angle":[0.05,-2.15,2.0]}},
 "linecode":{"lc":{"R_series_1_1":0.05,"R_series_2_2":0.05,
   "R_series_3_3":0.05,"R_series_4_4":0.05}},
 "line":{"l1":{"bus_from":"grid","bus_to":"poc",
   "terminal_map_from":["a","b","c","n"],
   "terminal_map_to":["a","b","c","n"],
   "linecode":"lc","length":1.0}}}
"""; from_string=true)

2. Compare a reduced inductor with an explicit LCL circuit

The two series reactances sum to the reduced model's reactance. Therefore, with no midpoint capacitor the two representations are identical.

common = (
    id="inverter", bus="poc",
    phase_terminals=["a","b","c"], neutral="n",
    topology=:FOUR_LEG,
    s_max=20e3, i_max=45.0, i_grid_max=45.0, In_max=100.0,
    v_dc=700.0, c_dc=1.1e-3, i_cap_max=50.0,
    f_sw=10e3, pwm_strategy=:CENTERED, pwm_ac_ripple=true,
    pwm_fundamental_samples=36, pwm_carrier_samples=128,
    pwm_ac_harmonics=64,
)

reduced = AdvancedInverter(; r_filter=0.05, x_filter=0.15, common...)
series_only = AdvancedInverter(; r_filter=0.02, x_filter=0.05,
    r_filter_grid=0.03, x_filter_grid=0.10, common...)
lcl = AdvancedInverter(; r_filter=0.02, x_filter=0.05,
    r_filter_grid=0.03, x_filter_grid=0.10,
    c_filter_mid=20e-6, r_filter_damping=1.0, common...)

r_reduced = solve_advanced_inverter(network, reduced)
r_series = solve_advanced_inverter(network, series_only)
r_lcl = solve_advanced_inverter(network, lcl)
@assert all(r.termination_status in ("LOCALLY_SOLVED", "OPTIMAL")
            for r in (r_reduced, r_series, r_lcl))

@show r_reduced.i_ac_switching_rms r_reduced.i_grid_switching_rms
@show r_series.i_ac_switching_rms r_series.i_grid_switching_rms
@show r_lcl.i_ac_switching_rms r_lcl.i_grid_switching_rms
@show r_lcl.i_filter_shunt_switching_rms

Check the limiting case numerically:

@assert all(isapprox.(r_reduced.i_ac_switching_rms,
                      r_series.i_ac_switching_rms; rtol=1e-6))
@assert all(isapprox.(r_reduced.i_grid_switching_rms,
                      r_series.i_grid_switching_rms; rtol=1e-6))

With the midpoint capacitor present, converter current splits between the shunt branch and grid arm. A low grid-side ripple does not imply low semiconductor or filter-capacitor stress.

3. Compare fundamental, switching, and total RMS currents

for phase in eachindex(r_lcl.i_mag)
    @assert isapprox(r_lcl.i_ac_total_rms[phase],
        hypot(r_lcl.i_mag[phase], r_lcl.i_ac_switching_rms[phase]);
        rtol=1e-10)
    @assert isapprox(r_lcl.i_grid_total_rms[phase],
        hypot(r_lcl.i_grid_mag[phase], r_lcl.i_grid_switching_rms[phase]);
        rtol=1e-10)
end

@show r_lcl.i_mag r_lcl.i_ac_switching_rms r_lcl.i_ac_total_rms
@show r_lcl.i_grid_mag r_lcl.i_grid_switching_rms r_lcl.i_grid_total_rms
@show r_lcl.i_ac_switching_reserved r_lcl.i_grid_switching_reserved

The sequential closure allocates the predicted carrier RMS inside i_max, i_grid_max, and In_max. Confirm that each *_reserved diagnostic covers its prediction before publishing the point.

4. Show why the neutral inductor matters

no_neutral_l = AdvancedInverter(; r_filter=0.05, x_filter=0.15,
    x_filter_neutral=0.0, common...)
with_neutral_l = AdvancedInverter(; r_filter=0.05, x_filter=0.15,
    x_filter_neutral=0.30, common...)

r_no_ln = solve_advanced_inverter(network, no_neutral_l)
r_with_ln = solve_advanced_inverter(network, with_neutral_l)

@show r_no_ln.i_neutral_switching_rms r_with_ln.i_neutral_switching_rms
@show r_no_ln.i_neutral_total_rms r_with_ln.i_neutral_total_rms

The fourth-wire impedance participates in a shared return path, so it changes all phase-to-neutral harmonic equations. It must not be added as an independent scalar correction after solving three uncoupled phase circuits.

For coupled or unequal conductors, provide the complete primitive matrices in a,b,c,n order:

Xprimitive = [0.15 0.01 0.01 0.02;
              0.01 0.15 0.01 0.02;
              0.01 0.01 0.15 0.02;
              0.02 0.02 0.02 0.30]

Use x_filter_matrix=Xprimitive instead of the scalar phase/neutral reactances; the two parameterisations cannot be mixed.

5. Refine the retained harmonic series

for (samples, harmonics) in ((128,32), (128,64), (256,128))
    parameters = merge(common, (r_filter=0.02, x_filter=0.05,
        r_filter_grid=0.03, x_filter_grid=0.10,
        c_filter_mid=20e-6, r_filter_damping=1.0,
        pwm_carrier_samples=samples, pwm_ac_harmonics=harmonics,
    ))
    device = AdvancedInverter(; parameters...)
    r = solve_advanced_inverter(network, device)
    @assert r.termination_status in ("LOCALLY_SOLVED", "OPTIMAL")
    println((samples=samples, harmonics=harmonics,
        converter=maximum(r.i_ac_switching_rms),
        grid=maximum(r.i_grid_switching_rms),
        shunt=maximum(r.i_filter_shunt_switching_rms),
        peak_to_peak=maximum(r.i_ac_switching_pp)))
end

RMS quantities normally converge sooner than peak-to-peak reconstructions. Refine both fundamental-angle and carrier grids when an operating point lies close to a physical rating.

6. Interpret the resonance diagnostic correctly

filter_resonance_hz is the undamped scalar LCL natural-frequency estimate. It helps catch implausible component selections, but it is not a stability margin. The carrier audit reuses constant R/L/C primitives at harmonic frequencies and assumes a stiff high-frequency POC. It omits:

  • controller and digital-delay impedance;
  • frequency-dependent winding resistance, core loss, and saturation;
  • capacitor ESR/ESL unless represented separately in a future model;
  • common-mode capacitance and protective-earth paths; and
  • switching dead time and device capacitances.

Use an impedance scan or EMT model when those mechanisms determine the result.

7. Publication checks

Report converter, grid, shunt, and neutral paths separately; demonstrate harmonic-grid convergence; verify reserve coverage and total-RMS ratings; state the assumed high-frequency grid boundary; and avoid describing the scalar resonance estimate as a closed-loop stability result.