Ideal Parallel LC Resonator

The intrinsic dynamics, admittance zero, energy scale, and one-port observable layers of a lossless lumped parallel LC tank.

Node role: Concept.

Reader task: Separate an ideal parallel tank’s intrinsic resonance from raw port-loaded and wave observables.

Why this is the first resonator to understand

A capacitor and an inductor connected in parallel are the smallest circuit that can store electric and magnetic energy in one oscillatory mode. The model is simple; its observable is not. An isolated ideal tank has zero admittance and infinite input impedance at resonance, while a solver that inserts a finite port shunt can report a finite raw impedance at the same frequency.

That distinction is the point of this node. Before trusting a resonance plot, identify which of these layers the trace represents:

Intrinsic tank — \(Y_{LC}\)

What does the ideal \(L\parallel C\) component do?

Raw solver network — \(Y_{\mathrm{raw}}\) or \(Z_{\mathrm{raw}}\)

What did the compiled circuit, including declared port elements, solve?

Port-compensated network — \(Y_{\mathrm{PTC}}\)

What remains after a proven artificial shunt is removed?

Wave observable — \(S_{11}\)

What reflection is defined at the chosen reference plane and impedance?

The result to remember

For the isolated, lossless parallel tank, resonance is an admittance zero and an impedance pole. A directly connected ideal one-port therefore has \(S_{11}=+1\) at resonance, not a reflection-magnitude dip.

Circuit and conventions

The model has one node voltage \(v(t)\) relative to ground. Both ideal elements connect from that node to ground:

flowchart TB
    Port["tank node, voltage v(t)"] --> C["capacitor C to ground"]
    Port --> L["inductor L to ground"]
    C --> GND["ground"]
    L --> GND

We use the canonical \(e^{-i\omega t}\) convention from Dynamics Conventions and the Flux-to-Mode Symbol Bridge and passive branch currents from the tank node toward ground. Symbols and SI units are:

Symbol	Meaning and SI unit
\(C\)	ideal shunt capacitance, F
\(L\)	ideal shunt inductance, H
\(\omega\)	angular frequency, rad/s
\(f=\omega/(2\pi)\)	ordinary frequency, Hz
\(Y\)	admittance, S
\(Z\)	impedance, \(\Omega\)
\(Z_0\)	real wave-reference impedance at the chosen plane, \(\Omega\)
\(R_{\mathrm{port}}\)	explicit resistive port shunt when present, \(\Omega\)

With the opposite \(e^{+i\omega t}\) convention, every susceptance sign below reverses. The resonance frequency and physical conclusions do not.

Intrinsic dynamics and natural frequency

The intuition is energy exchange: capacitor voltage determines electric energy, and inductor current determines magnetic energy. Node flux combines them in one coordinate. The circuit-energy starting model follows (Vool and Devoret 2017); the KCL-to-frequency algebra below is derived here for this ideal parallel topology, with no attached environment.

Node flux and node voltage obey

\[ \Phi(t)\equiv\int^t v(t')\,dt', \qquad v(t)=\dot\Phi(t). \]

LaTeX source

\Phi(t)\equiv\int^t v(t')\,dt',
\qquad
v(t)=\dot\Phi(t).

Kirchhoff’s current law with no external current gives

\[ C\ddot\Phi+\frac{\Phi}{L}=0. \]

LaTeX source

C\ddot\Phi+\frac{\Phi}{L}=0.

This is the harmonic-oscillator equation. Its natural angular frequency and ordinary frequency are

\[ \omega_0=\frac{1}{\sqrt{LC}}, \qquad f_0=\frac{1}{2\pi\sqrt{LC}}. \]

LaTeX source

\omega_0=\frac{1}{\sqrt{LC}},
\qquad
f_0=\frac{1}{2\pi\sqrt{LC}}.

The conserved energy can be written as

\[ E=\frac{C\dot\Phi^2}{2}+\frac{\Phi^2}{2L} =\frac{Cv^2}{2}+\frac{Li_L^2}{2}. \]

LaTeX source

E=\frac{C\dot\Phi^2}{2}+\frac{\Phi^2}{2L}
 =\frac{Cv^2}{2}+\frac{Li_L^2}{2}.

The capacitor and inductor exchange that energy every quarter-cycle. In this ideal model there is no dissipative term, no finite linewidth, and no energy decay; its ideal quality factor is unbounded. Once loss or coupling is added, use Resonator Decay, Linewidth, and Quality Factor rather than assigning a finite \(Q\) to the isolated tank.

The same conservative mode has Hamiltonian

\[ H=\frac{Q^2}{2C}+\frac{\Phi^2}{2L}, \qquad Q=C\dot\Phi, \]

LaTeX source

H=\frac{Q^2}{2C}+\frac{\Phi^2}{2L},
\qquad Q=C\dot\Phi,

which is the linear starting point for circuit quantization. The quantization rules and their rank checks belong to Circuit Lagrangian, Hamiltonian, and Canonical Quantization.

One LC in every language

Changing variables does not create quantum mechanics. This section carries the same classical energy through canonical scaling before promoting the variables to operators. Canonical circuit quantization provides the starting rule (Vool and Devoret 2017); the explicit scalar substitutions below are this page’s derivation.

The two factors of one half in

\[ H_{\rm cl}=\frac{Q^2}{2C}+\frac{\Phi^2}{2L} \]

LaTeX source

H_{\rm cl}=\frac{Q^2}{2C}+\frac{\Phi^2}{2L}

are classical electric and magnetic energy factors. They are not the quantum zero-point one half. With

\[ \omega_0=\frac1{\sqrt{LC}}, \qquad Z=\sqrt{\frac LC}, \qquad X=\frac\Phi{\sqrt Z}, \qquad P=\sqrt ZQ, \]

LaTeX source

\omega_0=\frac1{\sqrt{LC}},
\qquad Z=\sqrt{\frac LC},
\qquad X=\frac\Phi{\sqrt Z},
\qquad P=\sqrt ZQ,

the same classical energy is

\[ H_{\rm cl}=\frac{\omega_0}{2}(X^2+P^2), \qquad A=\frac{X+iP}{\sqrt2}, \qquad H_{\rm cl}=\omega_0|A|^2. \]

LaTeX source

H_{\rm cl}=\frac{\omega_0}{2}(X^2+P^2),
\qquad A=\frac{X+iP}{\sqrt2},
\qquad H_{\rm cl}=\omega_0|A|^2.

Writing \(\alpha=A/\sqrt\hbar\) gives

\[ H_{\rm cl}=\hbar\omega_0|\alpha|^2, \]

LaTeX source

H_{\rm cl}=\hbar\omega_0|\alpha|^2,

but \(|\alpha|^2\) is still a classical amplitude norm, not a number operator. Quantization promotes the canonical variables to noncommuting operators and yields

\[ \hat H=\hbar\omega_0\left(\hat a^\dagger\hat a+\frac12\right). \]

LaTeX source

\hat H=\hbar\omega_0\left(\hat a^\dagger\hat a+\frac12\right).

This final \(+1/2\) is the quantum zero-point energy and follows from the commutator. It is unrelated to the two classical quadratic-energy factors above, and also unrelated to the \(\kappa/2\) amplitude-decay convention used after an environment is attached.

For the matching local response normalization, see Normalization Across Circuit, Response, and Optimization.

Intrinsic admittance and antiresonance

The same resonance can be recognized without a time-domain solve: the capacitor and inductor currents cancel at their shared terminal. The following admittance is this page’s phasor-domain derivation of the same KCL equation. A zero of admittance, rather than a peak in a chosen plot, is the intrinsic statement.

In the frequency domain,

\[ I_C=-i\omega CV, \qquad I_L=\frac{V}{-i\omega L}. \]

LaTeX source

I_C=-i\omega CV,
\qquad
I_L=\frac{V}{-i\omega L}.

The intrinsic tank admittance is therefore

\[ Y_{LC}(\omega) =\frac{I_C+I_L}{V} =-i\left(\omega C-\frac{1}{\omega L}\right). \]

LaTeX source

Y_{LC}(\omega)
=\frac{I_C+I_L}{V}
=-i\left(\omega C-\frac{1}{\omega L}\right).

Its expected sign is an immediate audit check:

below \(\omega_0\), \(\operatorname{Im}Y_{LC}>0\) and the tank is inductive;
at \(\omega_0\), \(Y_{LC}=0\) and \(Z_{LC,\mathrm{in}}=1/Y_{LC}\rightarrow\infty\);
above \(\omega_0\), \(\operatorname{Im}Y_{LC}<0\) and the tank is capacitive.

The impedance scale

\[ Z_{\mathrm{mode}}=\sqrt{\frac{L}{C}} \]

LaTeX source

Z_{\mathrm{mode}}=\sqrt{\frac{L}{C}}

relates the mode’s voltage and current scales. It is not the frequency- dependent input impedance \(1/Y_{LC}\), and it is not automatically a port reference impedance such as \(50\,\Omega\).

From the tank to a one-port reflection

Now attach a wave-coordinate description, not a physical resistor. The real-positive power-wave relation is supported by (Kurokawa 1965); inserting the derived tank admittance gives the reflection below. This extra step is why an intrinsic resonance does not by itself predict a reflection-magnitude dip.

If the reference plane sees the intrinsic tank directly and uses a real wave-reference impedance \(Z_0\), its reflection coefficient is

\[ S_{11}=\Gamma =\frac{Z_{LC,\mathrm{in}}-Z_0}{Z_{LC,\mathrm{in}}+Z_0} =\frac{1-Z_0Y_{LC}}{1+Z_0Y_{LC}}. \]

LaTeX source

S_{11}=\Gamma
=\frac{Z_{LC,\mathrm{in}}-Z_0}{Z_{LC,\mathrm{in}}+Z_0}
=\frac{1-Z_0Y_{LC}}{1+Z_0Y_{LC}}.

Because the ideal load is purely reactive, \(|S_{11}|=1\). At resonance the tank is an open circuit, so \(S_{11}=+1\). The useful feature is the susceptance zero, impedance pole, and reflection phase evolution—not a magnitude dip.

Do not transfer this signature to another topology

A capacitively coupled hanger, a feedline-coupled distributed resonator, and a lossy or hybridized resonator can show a notch or a finite linewidth. Those are different networks. The formula \(1/(2\pi\sqrt{LC})\) remains a possible starting estimate, not proof of the measured or loaded resonance.

Historical Workbench mapping and numerical check

This section preserves the 2026-07-10 implementation observation and notebook example, not current capability certification. At that snapshot the Workbench’s compile_to_josephson path lowers an ExternalPort into both a JosephsonCircuits port row and an explicit shunt resistor R_port_1. For Notebook 00, the current fixture explicitly treats that resistor as solver scaffold for the intrinsic-tank comparison. The compiled one-port network is therefore

\[ Y_{\mathrm{raw}}=\frac{1}{R_{\mathrm{port}}}+Y_{LC}, \qquad Z_{\mathrm{raw}}=\frac{1}{Y_{\mathrm{raw}}}. \]

LaTeX source

Y_{\mathrm{raw}}=\frac{1}{R_{\mathrm{port}}}+Y_{LC},
\qquad
Z_{\mathrm{raw}}=\frac{1}{Y_{\mathrm{raw}}}.

The design declaration says why this shunt may be removed; the compiled rows and component value prove what is removed. Both conditions are required:

\[ Y_{\mathrm{PTC}} =\frac{1}{Z_{\mathrm{raw}}}-\frac{1}{R_{\mathrm{port}}} =Y_{LC}. \]

LaTeX source

Y_{\mathrm{PTC}}
=\frac{1}{Z_{\mathrm{raw}}}-\frac{1}{R_{\mathrm{port}}}
=Y_{LC}.

For this fixture, the solver port contract uses the same numerical value for the wave reference and explicit shunt, \(Z_0=R_{\mathrm{port}}=50\,\Omega\). Under that implementation-specific equality, the returned zero-mode reflection also satisfies

\[ S_{11}=\frac{2Z_{\mathrm{raw}}}{R_{\mathrm{port}}}-1 =\frac{1-Z_0Y_{\mathrm{PTC}}}{1+Z_0Y_{\mathrm{PTC}}}. \]

LaTeX source

S_{11}=\frac{2Z_{\mathrm{raw}}}{R_{\mathrm{port}}}-1
=\frac{1-Z_0Y_{\mathrm{PTC}}}{1+Z_0Y_{\mathrm{PTC}}}.

This is an implementation-specific mapping verified against the compiled rows and a real solver run. It is not permission to subtract every port reference from every artifact. Port-Termination Compensation requires explicit shunt evidence, and Port Reference Impedance Semantics defines the broader solver boundary.

flowchart LR
    Tank["intrinsic tank<br/>Y_LC"] --> Add["compiler adds<br/>1/R_port"]
    Add --> Raw["raw trace<br/>Y_raw = 1/Z_raw"]
    Raw --> Remove["declared + verified PTC<br/>subtract 1/R_port"]
    Remove --> Check["intrinsic check<br/>Y_PTC = Y_LC"]
    Raw --> Wave["solver wave trace<br/>S11"]
    Check --> Wave

flowchart TB
    Tank["intrinsic tank<br/>Y_LC"] --> Add["compiler adds<br/>1/R_port"]
    Add --> Raw["raw trace<br/>Y_raw = 1/Z_raw"]
    Raw --> Remove["declared + verified PTC<br/>subtract 1/R_port"]
    Remove --> Check["intrinsic check<br/>Y_PTC = Y_LC"]
    Raw --> Wave["solver wave trace<br/>S11"]
    Check --> Wave

Reproducible numerical check

The recorded Notebook 00 example used \(L=21.5\,\mathrm{nH}\), \(C=58.2\,\mathrm{fF}\), and \(Z_0=R_{\mathrm{port}}=50\,\Omega\) under the recorded fixture contract. These give

\[ f_0=4.4992425826\ \mathrm{GHz}, \qquad Z_{\mathrm{mode}}=607.7959\ \Omega. \]

LaTeX source

f_0=4.4992425826\ \mathrm{GHz},
\qquad
Z_{\mathrm{mode}}=607.7959\ \Omega.

The recorded three-point, pump-off JosephsonCircuits run gave the following check values. The compensated admittance agrees with the analytic intrinsic tank; the table does not replace the solver trace.

\(0.8f_0\)

\(Y_{\mathrm{PTC}}=-j\,7.40380\times10^{-4}\ \mathrm{S}\)
\(Z_{\mathrm{raw}}=49.9316+j1.84842\ \Omega\)
\(S_{11}=0.997263+j0.0739367\)

\(f_0\)

\(Y_{\mathrm{PTC}}\approx0\)
\(Z_{\mathrm{raw}}=50.0000\ \Omega\)
\(S_{11}=+1\)

\(1.2f_0\)

\(Y_{\mathrm{PTC}}=+j\,6.03273\times10^{-4}\ \mathrm{S}\)
\(Z_{\mathrm{raw}}=49.9545-j1.50681\ \Omega\)
\(S_{11}=0.998182-j0.0602724\)

The \(50\,\Omega\) raw result at \(f_0\) is not the tank’s intrinsic input impedance. It is the exact signature expected when the explicit solver shunt remains in \(Z_{\mathrm{raw}}\).

Review checklist

Is the circuit truly one ideal \(L\) and one ideal \(C\) in parallel to the same reference node?
Is the phasor convention stated before interpreting the sign of \(\operatorname{Im}Y\)?
Are hertz and radians per second related by \(\omega=2\pi f\) exactly once?
Is \(\sqrt{L/C}\) labeled as a mode impedance scale rather than input or port impedance?
Does the reported trace say intrinsic, raw, compensated, or wave observable?
If a shunt is removed, does the compiled artifact prove its identity and value?
At the analytic \(f_0\), does \(\operatorname{Im}Y_{\mathrm{PTC}}\approx0\) and does this direct ideal one-port give \(S_{11}\approx+1\)?
Are loss, coupling, parasitics, nonlinearity, and distributed behavior either modeled explicitly or named as exclusions?

Limits and next nodes

This node assumes ideal, linear, lumped, lossless elements and a single reference node. It does not cover ESR, dielectric or conductor loss, self-resonance, mutual coupling, feedline coupling, Josephson nonlinearity, distributed transmission-line modes, hybridization, or external loading.

Network Trace Views explains when \(S\), \(Y\), or \(Z\) best exposes a modeling question.
Resonator Decay, Linewidth, and Quality Factor adds loss and coupling semantics.
Harmonic Balance: Periodic Steady State and Mode Semantics explains why the Workbench may use one solver workflow even for this linear, pump-off fixture.

References

U. Vool and M. H. Devoret, “Introduction to Quantum Electromagnetic Circuits,” International Journal of Circuit Theory and Applications 45, 897–934 (2017), SCQ_Design source page, arXiv:1610.03438.
MIT OpenCourseWare, Physics II: Electricity and Magnetism, Chapter 12: AC Circuits.
Keysight Technologies, Fundamentals of RF and Microwave Power Measurements, Part 3, reflection-coefficient reference.

Review Links

Field	Value
Status	Artifact-backed
Semantic role	Separate the intrinsic ideal tank from raw port-loaded solver matrices, verified PTC, and one-port wave observables.
Physics evidence	Source-backed by Introduction to Quantum Electromagnetic Circuits and the authoritative references above.
Workbench mapping evidence	Locally artifact-backed on 2026-07-10 against compiler revision `b689fc1f`. The compiler revision is immutable; the Notebook edits were working-tree evidence at the recorded date, not a currently certified release.
Validation record	`julia --startup-file=no -e 'include("notebooks/pluto/00_parallel_lc_resonator.jl"); println(sanity)'` returned all ten gates `true`; the current path uses the shared evidence-backed PTC helper, which verifies exact compiled port/resistor colocation and derives the shunt value before subtraction.
Used by	Knowledge Base learning path
Mutable implementation entrypoints	Notebook 00, `add_parallel_lc_resonator!`, and `build_parallel_lc_resonator_example`. These are audit entrypoints, not immutable evidence.
Human review gate	Can a new designer predict all four \(f_0\) results—\(Y_{LC}=0\), \(Y_{\mathrm{raw}}=1/R_{\mathrm{port}}\), \(Z_{\mathrm{raw}}=R_{\mathrm{port}}\), and \(S_{11}=+1\)—without confusing \(\sqrt{L/C}\), \(Z_0\), and the raw impedance trace?

Bibliography

Kurokawa, K. 1965. “Power Waves and the Scattering Matrix.” IEEE Transactions on Microwave Theory and Techniques 13 (2): 194–202. https://doi.org/10.1109/TMTT.1965.1125964.

Vool, U., and M. H. Devoret. 2017. “Introduction to Quantum Electromagnetic Circuits.” International Journal of Circuit Theory and Applications 45: 897–934. https://doi.org/10.1002/cta.2359.