Telegrapher Model: From RLGC Continuum to a Finite LC Ladder
Node role: Method.
Reader task: Start from one declared scalar transmission-line mode and construct either its exact distributed response or a convergent finite ladder.
The route is continuum first, exact section second, finite approximation third. Read the approximation boundary before using section refinement as evidence: more sections cannot restore a missing cross-sectional mode.
The model hierarchy
Reorganized explanation. The distributed series/shunt line model is the standard transmission-line starting point (Pozar 2012). The selected phasor sign is a project convention, not a choice mandated by that source.
A transmission line is not born as a ladder. The ladder is a spatial discretization of a continuous line:
\[ \boxed{ \text{cross-section and materials} \longrightarrow (R',L',G',C') \longrightarrow \text{Telegrapher continuum} \longrightarrow \text{finite ladder}. } \]
The prime means per unit length. For one selected mode,
\[ R'\,[\Omega/\mathrm m],\qquad L'\,[\mathrm H/\mathrm m],\qquad G'\,[\mathrm S/\mathrm m],\qquad C'\,[\mathrm F/\mathrm m]. \]
The cross-section solve is the physical model. RLGC is its one-dimensional modal reduction. The Telegrapher equations are the continuous distributed circuit. A finite ladder is only a numerical or circuit-level realization of that continuum.
This page uses the \(e^{-i\omega t}\) convention. With
\[ Z'(\omega)=R'-i\omega L', \qquad Y'(\omega)=G'-i\omega C', \]
the frequency-domain Telegrapher equations are
\[ \boxed{ -\frac{\mathrm dV}{\mathrm dz}=Z'I, \qquad -\frac{\mathrm dI}{\mathrm dz}=Y'V. } \]
This-page derivation. Differentiate the stated uniform-line equations to obtain propagation and impedance; the forward-wave branch follows the declared attenuation and orientation.
Differentiating once more gives
\[ \frac{\mathrm d^2V}{\mathrm dz^2}=\gamma^2V, \qquad \frac{\mathrm d^2I}{\mathrm dz^2}=\gamma^2I, \]
where
\[ \boxed{ \gamma=\sqrt{Z'Y'}=\alpha-i\beta, \qquad Z_c=\sqrt{\frac{Z'}{Y'}}. } \]
The passive forward-wave branch is chosen so that \(\alpha\ge0\) and \(\beta\ge0\). A forward wave therefore has
\[ V^+(z,t)\propto e^{-\alpha z}e^{i\beta z}e^{-i\omega t}. \]
Here \(\alpha\) is attenuation per length, \(\beta\) is phase accumulated per length, and \(Z_c=V^+/I^+\) is the traveling-wave voltage/current ratio.
The square roots above are scalar formulas. Do not apply them elementwise to RLGC matrices. A multiconductor line first needs a declared terminal basis, reference conductor, and modal reduction. See Multiconductor RLGC Matrices: Basis, Modes, and Physical Meaning.
Exact distributed section
Reorganized explanation. The uniform-line chain matrix follows from the forward/backward wave solution (Pozar 2012), translated consistently to this page’s phasor sign and same-direction current convention.
A uniform section of length \(\Delta z\) has the exact chain matrix
\[ \boxed{ \begin{pmatrix} V(z)\\ I(z) \end{pmatrix} = \begin{pmatrix} \cosh(\gamma\Delta z)&Z_c\sinh(\gamma\Delta z)\\ Z_c^{-1}\sinh(\gamma\Delta z)&\cosh(\gamma\Delta z) \end{pmatrix} \begin{pmatrix} V(z+\Delta z)\\ I(z+\Delta z) \end{pmatrix}. } \]
This expression retains propagation phase, attenuation, and every resonance created by the declared boundaries. It is the distributed model against which a finite ladder is checked.
Finite symmetric \(\pi\) section
This-page derivation. Cascade half-shunt, series and half-shunt matrices, then compare their Taylor series with the exact section. The error-order statement assumes a smooth uniform line; it does not choose a section count.
Divide a line of length \(\ell\) into \(N\) sections,
\[ \Delta z=\frac{\ell}{N}. \]
One symmetric \(\pi\) section uses
\[ Z_s=Z'\Delta z, \qquad Y_p=\frac{Y'\Delta z}{2}, \]
or, in element language,
\[ R_{\mathrm{sec}}=R'\Delta z, \quad L_{\mathrm{sec}}=L'\Delta z, \quad G_{\mathrm{half}}=\frac{G'\Delta z}{2}, \quad C_{\mathrm{half}}=\frac{C'\Delta z}{2}. \]
Its chain matrix is
\[ \mathbf T_{\pi} = \begin{pmatrix}1&0\\Y_p&1\end{pmatrix} \begin{pmatrix}1&Z_s\\0&1\end{pmatrix} \begin{pmatrix}1&0\\Y_p&1\end{pmatrix} = \begin{pmatrix} 1+Z_sY_p&Z_s\\ 2Y_p+Y_p^2Z_s&1+Z_sY_p \end{pmatrix}. \]
For \(|\gamma\Delta z|\ll1\), the exact section expands as
\[ \begin{aligned} \cosh(\gamma\Delta z) &=1+\frac{Z'Y'\Delta z^2}{2}+O(\Delta z^4),\\ Z_c\sinh(\gamma\Delta z) &=Z'\Delta z+\frac{Z'^2Y'\Delta z^3}{6}+O(\Delta z^5),\\ Z_c^{-1}\sinh(\gamma\Delta z) &=Y'\Delta z+\frac{Z'Y'^2\Delta z^3}{6}+O(\Delta z^5). \end{aligned} \]
The symmetric \(\pi\) section matches the diagonal terms through \(O(\Delta z^2)\) and differs first at \(O(\Delta z^3)\) in the off-diagonal terms. Cascading \(N\) uniform sections therefore gives second-order global convergence as \(\Delta z\rightarrow0\) under the usual smooth, uniform-line assumptions.
This is the precise meaning of “the LC ladder approaches the distributed line.” It is not enough for one ladder to look plausible.
Refinement and intended use
Engineering choice. The intended observable, band and any tolerances must come from the receiving task. The comparisons below diagnose discretization; this page does not prescribe a numerical threshold or universal \(N\).
Choose \(N\) from the highest frequency and narrowest feature that the model must resolve, then increase \(N\) until the required observables stop moving.
| Check | Required comparison |
|---|---|
| Electrical section size | \(\max_{\omega\in\Omega}|\gamma(\omega)\Delta z|\ll1\) over the declared band \(\Omega\). |
| Complex response | The ladder and exact distributed line agree in calibrated complex \(S/Y/Z\) on identical ports and loads. |
| Resonant features | Pole, zero, residue, frequency, and linewidth changes remain below declared tolerances under \(N\rightarrow2N\). |
| Passivity and stability | Refinement does not introduce active behavior or unstable poles. |
A ladder that matches only one resonant frequency has not established distributed-model closure. Phase, impedance, linewidth, and nearby poles may still be wrong.
Where the one-dimensional model stops
The RLGC continuum assumes that the selected cross-sectional mode remains a useful local description. Bends, steps, launches, couplers, air bridges, package transitions, and strong multimode regions may require their own network blocks or a full-wave model. Refining \(N\) cannot repair missing physics in \(R',L',G',C'\) or in the topology.
For a lossless multiconductor matrix construction, continue with Lossless MTL Matrix to Finite \(\pi\) Ladder. For the CPW resonator relations that follow after selecting one scalar mode, continue with CPW Quarter-Wave Resonator: RLGC, Phase Velocity, and Equivalent LC.
| Field | Value |
|---|---|
| Status | Seed |
| Used by | CPW Quarter-Wave Resonator |
| Review need | Confirm notation and convergence language against the transmission-line implementation used for release artifacts. |