Technology / Technical guide
LCL Filter States, Free Oscillation & Boundary Conditions
An LCL network contains two inductive current states and one capacitive voltage state. Their interaction differs from a single isolated LC pair. Understanding its free response requires declared terminal boundaries: the familiar LCL resonance expression is a result of one ideal model, not a formula for every connected filter.
Three storage states do not mean three independent resonance frequencies.
Specify both end conditions before interpreting the internal oscillation and its frequency.
01 / Specified network
Two series inductors meet at one shunt capacitor node.
Consider an ideal lumped network with input-side inductance L1, output-side inductance L2 and a capacitor C from their intermediate node to a common reference.
Consider an ideal lumped network with input-side inductance L1, output-side inductance L2 and a capacitor C from their intermediate node to a common reference. Let i1 enter that node through L1, i2 leave through L2, and vc be capacitor voltage.
Both ends connect to ideal voltage sources with voltages v1 and v2. Inductances and capacitance are constant and positive; resistance and other losses are absent. This is an abstract theoretical network, not a connection diagram or an actual MARSTON filter.
02 / Three states
Each storage quantity has its own derivative.
Under the stated signs, L1 di1/dt = v1 − vc, L2 di2/dt = vc − v2, and C dvc/dt = i1 − i2.
Under the stated signs, L1 di1/dt = v1 − vc, L2 di2/dt = vc − v2, and C dvc/dt = i1 − i2. The capacitor responds to the difference between the two inductor currents, rather than to an assumed single current throughout the network.
These equations expose three state variables. They also explain why treating the two inductors as one simple series element can lose relevant internal behaviour: charging the intermediate capacitor allows the two currents to differ during a transient.
03 / Free-mode boundary
Hold both source perturbations at zero.
For the internal free response around a fixed operating condition, the ideal sources have zero voltage perturbations.
For the internal free response around a fixed operating condition, the ideal sources have zero voltage perturbations. Differentiating the capacitor equation and substituting the inductor equations gives d²(δvc)/dt² + [(1/L1 + 1/L2)/C]δvc = 0.
The resulting oscillatory angular frequency satisfies ωr² = (L1 + L2)/(L1L2C), with fr = ωr/(2π). This is the ideal lossless fixed-voltage-end model. The cited original LCL research discusses network resonance; its device values and controller results are not adopted.
04 / Mode interpretation
The current states also permit a nonoscillatory component.
In the free ideal model, equal constant current perturbations in both inductors with zero capacitor-voltage perturbation satisfy the state equations.
In the free ideal model, equal constant current perturbations in both inductors with zero capacitor-voltage perturbation satisfy the state equations. This zero-frequency component accompanies the oscillatory pair. Three states therefore do not imply three different nonzero resonance frequencies.
As a restricted algebraic comparison, L1 = L2 = L gives ωr = √(2/(LC)). The comparison identifies a mathematical consequence of these end conditions. It is not a general ranking of LCL against LC filters with different loads or terminal arrangements.
05 / Connected response
Additional impedances alter the represented dynamics.
External inductance, load impedance, source impedance and losses can change the combined network.
External inductance, load impedance, source impedance and losses can change the combined network. If an external inductance is folded into an equivalent path, its definition must be updated consistently. The original fixed-boundary expression cannot silently describe every extended arrangement.
A free natural mode, a driven-response peak and a defined attenuation point are different quantities. Damping and implemented feedback can influence a connected response, but this explanation gives no damping element, feedback gain or universal cutoff frequency.
06 / Model scope
A network name does not establish actual converter functions.
The LC guide explains a simpler oscillator boundary.
The LC guide explains a simpler oscillator boundary. The energy-accounting guide explains how the three storage terms can change even when the network contains no ideal dissipative element.
No MARSTON LCL topology, L/C values, resonance, active damping or grid-interface capability is assigned. These state equations are a theoretical explanation without filter construction, parameter selection, controller tuning or a claim of output-quality or compliance performance.
Related technical guides: Converter Instantaneous Power, Storage & Dissipation · Parasitic Inductance, Current Slew & Switching Overshoot · Thermal Resistance, Heat Capacity & Transient RC Models.
General background: Souri, Mehrizi-Sani and Tehrani: original LCL resonance study; selected network discussion. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Inductor states | i1 and i2 in A | Their currents can differ during capacitor charging. |
| Capacitor state | vc in V | C dvc/dt = i1 − i2 under stated signs. |
| Free boundaries | Zero perturbation at both ideal voltage sources | Lossless constant positive L1/L2/C. |
| Oscillatory mode | ωr² = (L1 + L2)/(L1L2C) | Only the declared fixed-voltage-end model. |
| Other mode | Equal constant current perturbations | Three states are not three nonzero resonant frequencies. |
Engineering questions.
Can the two inductors always be combined into one series L?
Not for the complete internal transient, because the intermediate capacitor allows unequal currents.
Is the stated resonance a universal LCL cutoff?
No. It belongs to the specified ideal free-mode boundary.
Do three storage states give three nonzero resonances?
No. This ideal model has an oscillatory pair and a zero-frequency component.
Can added external inductance change the response?
Yes. The combined path and boundary model change.
Is active damping assumed in this model?
No. The free-mode derivation is ideal and lossless.
Are actual MARSTON filter values or settings supplied?
No topology, component values or operational adjustments are assigned.
Technical enquiry
Discuss the technical question.
An LCL network contains two inductive current states and one capacitive voltage state. Share the model and applicable operating or design information for an engineering discussion.