CPW Quarter-Wave Resonator: RLGC, Phase Velocity, and Equivalent LC
Node role: Procedure.
Reader task: Convert one declared scalar CPW mode into a quarter-wave length and a local equivalent parallel-LC model, then identify which parameters remain adjustable after the CPW specification is fixed.
Read the scalar mode into its wave impedance and velocity, impose the short/open quarter-wave boundary, then match the root and local admittance slope to an LC. That last matching is local, not a replacement for the line at all frequencies.
From CPW physics to RLGC
Reorganized explanation. The scalar distributed RLGC starting model and wave relations follow transmission-line theory (Pozar 2012). The selected mode, quasi-TEM interpretation and low-loss limit are visible assumptions, not properties of every CPW geometry.
A CPW cross-section, material stack, superconducting film model, and selected propagation mode determine the per-unit-length quantities
\[ R'(\omega),\quad L'(\omega),\quad G'(\omega),\quad C'(\omega). \]
They are not four arbitrary fit knobs after the CPW specification is fixed. They summarize conductor loss, magnetic and kinetic energy, dielectric loss, and electric energy per unit length for that mode.
Under the \(e^{-i\omega t}\) convention,
\[ \boxed{ \gamma(\omega) =\sqrt{\bigl(R'-i\omega L'\bigr) \bigl(G'-i\omega C'\bigr)} =\alpha-i\beta, } \]
\[ \boxed{ Z_c(\omega) =\sqrt{\frac{R'-i\omega L'}{G'-i\omega C'}}. } \]
The phase and group velocities are different definitions:
\[ v_p(\omega)=\frac{\omega}{\beta(\omega)}, \qquad v_g(\omega)=\left(\frac{\mathrm d\beta}{\mathrm d\omega}\right)^{-1}. \]
For a lossless, nondispersive scalar mode,
\[ \boxed{ Z_0=\sqrt{\frac{L'}{C'}}, \qquad v_p=\frac{1}{\sqrt{L'C'}}. } \]
The inverse relations are
\[ \boxed{ L'=\frac{Z_0}{v_p}, \qquad C'=\frac{1}{Z_0v_p}. } \]
These equations give the useful physical picture:
- the product \(L'C'\) fixes propagation speed;
- the ratio \(L'/C'\) fixes characteristic impedance;
- \(Z_0\) and \(v_p\) together determine \(L'\) and \(C'\) uniquely for the lossless scalar model.
For a quasi-TEM CPW without strong dispersion,
\[ v_p\approx\frac{c_0}{\sqrt{\varepsilon_{\mathrm{eff}}}}. \]
This is an interpretation, not a universal material identity. Kinetic inductance, frequency-dependent dielectrics, conductor loss, radiation, and multimode behavior can make \(Z_c\), \(v_p\), and the inferred \(\varepsilon_{\mathrm{eff}}\) frequency dependent.
Do not insert RLGC matrices into these scalar square roots. Declare conductor order, reference conductor, and the mode being realized. The matrix semantics live in Multiconductor RLGC Matrices: Basis, Modes, and Physical Meaning.
Quarter-wave frequency and first length
Reorganized explanation. A uniform lossless shorted-line input impedance has a quarter-wave open-end voltage resonance (Pozar 2012). Coupling or end loading changes this declared boundary problem.
For a shorted line observed from its open end,
\[ Z_{\mathrm{in}}(\omega) =-iZ_0\tan\!\bigl(\beta(\omega)\ell\bigr), \qquad Y_{\mathrm{in}}(\omega) =\frac{i}{Z_0}\cot\!\bigl(\beta(\omega)\ell\bigr). \]
The fundamental open-end voltage resonance obeys
\[ \boxed{ \beta(\omega_0)\ell=\frac{\pi}{2}. } \]
For a nondispersive line,
\[ \boxed{ f_0=\frac{v_p}{4\ell}, \qquad \ell_{\lambda/4}=\frac{v_p}{4f_0}. } \]
This is a first matched-reference length. Couplers, end capacitance, bends, impedance steps, packaging, and fabrication shift the final length.
Local parallel-LC model
This-page derivation. Expand the stated lossless nondispersive line admittance at the quarter-wave root, then match the parallel-LC root and slope. The equivalent element values below follow from this matching, not from summing all distributed elements.
The distributed resonator and a single LC oscillator are not globally identical. They can be matched locally around one resonance.
Let \(\Delta\omega=\omega-\omega_0\). Expanding the line admittance around the quarter-wave root gives
\[ Y_{\mathrm{in}}(\omega) \approx -i\frac{\ell}{Z_0v_p}\Delta\omega. \]
A parallel LC has
\[ Y_{LC}(\omega) =-i\left(\omega C_r-\frac{1}{\omega L_r}\right), \qquad \omega_0=\frac{1}{\sqrt{L_rC_r}}, \]
and therefore
\[ Y_{LC}(\omega) \approx -2iC_r\Delta\omega. \]
Matching the root and slope yields
\[ \boxed{ C_r=\frac{C'\ell}{2} =\frac{\pi}{4\omega_0Z_0}, } \]
\[ \boxed{ L_r=\frac{8L'\ell}{\pi^2} =\frac{4Z_0}{\pi\omega_0}, } \]
and
\[ \boxed{ Z_r\equiv\sqrt{\frac{L_r}{C_r}} =\frac{4Z_0}{\pi}. } \]
The factors \(1/2\) and \(8/\pi^2\) come from the standing-wave energy profile. Consequently, \(C_r\ne C'\ell\) and \(L_r\ne L'\ell\). Simply summing the total distributed capacitance and inductance does not produce the locally matched resonator oscillator.
What remains adjustable after the CPW specification is fixed
Fixing the CPW width, gap, thickness, material stack, environment, and selected mode fixes the functions \(R',L',G',C'\) to the accuracy of the cross-section model. Near one low-loss frequency, this also fixes \(Z_0\) and \(v_p\).
| Change | First-order effect | What it cannot do alone |
|---|---|---|
| Change only \(\ell\) | Moves \(f_0\approx v_p/(4\ell)\); \(L_r\) and \(C_r\) both scale with \(\ell\). | Choose an arbitrary \(Z_r\) while preserving the CPW cross-section. |
| Change CPW cross-section or environment | Changes \(L'\), \(C'\), \(Z_0\), and usually \(v_p\). | Guarantee only one of those quantities changes. |
| Add end or coupling capacitance | Changes loading, coupling, frequency, and the local equivalent circuit. | Preserve the isolated quarter-wave formulas exactly. |
| Add impedance steps or multiple sections | Adds degrees of freedom to the distributed realization. | Remain a one-section uniform CPW model. |
Thus, at fixed \(\omega_0\) and fixed \(Z_0\), the locally equivalent \(L_r\) and \(C_r\) are already constrained. An upstream equivalent circuit requesting a different oscillator impedance must feed back to the CPW specification or to the distributed topology.
Approximation boundary
The parallel LC reproduces the fundamental root and local admittance slope. It does not reproduce propagation phase far from resonance, higher odd harmonics, frequency-dependent loss, discontinuities, or other line modes. Use the exact distributed section or a converged ladder when those features matter. The continuum-to-ladder derivation is in Telegrapher Model: From RLGC Continuum to a Finite LC Ladder.
Engineering illustration. The existing numerical inputs below illustrate the scalar formulas; they are not a material default or design target.
For the public illustrative values
\[ L'=529.7435\ \mathrm{nH/m}, \qquad C'=134.5187\ \mathrm{pF/m}, \qquad f_0=6\ \mathrm{GHz}, \]
the lossless scalar formulas give
| Quantity | Value |
|---|---|
| \(Z_0\) | \(62.754\ \Omega\) |
| \(v_p\) | \(1.18461\times10^8\ \mathrm{m/s}\) |
| \(v_p/c_0\) | \(0.395\) |
| \(\varepsilon_{\mathrm{eff}}\approx(c_0/v_p)^2\) | \(6.405\) |
| \(\lambda=v_p/f_0\) | \(19.7435\ \mathrm{mm}\) |
| \(\ell_{\lambda/4}\) | \(4.93588\ \mathrm{mm}\) |
| \(C'\ell\) | \(0.663968\ \mathrm{pF}\) |
| \(L'\ell\) | \(2.61475\ \mathrm{nH}\) |
| \(C_r=C'\ell/2\) | \(0.331984\ \mathrm{pF}\) |
| \(L_r=8L'\ell/\pi^2\) | \(2.11944\ \mathrm{nH}\) |
| \(Z_r=4Z_0/\pi\) | \(79.901\ \Omega\) |
These values are an initializer for one ideal uniform mode, not a final loaded resonator specification.
| Field | Value |
|---|---|
| Status | Seed |
| Review need | Confirm the selected CPW modal RLGC and loaded-boundary correction before promotion. |