From lumped circuits to distributed lines

In an ideal lumped circuit, all points on a node are assumed to share one potential at the same model time, and a branch current is treated as a single quantity through the element. That approximation is excellent when propagation time across the physical interconnect is much shorter than the timescale over which the signal changes.

A line of length ℓ with propagation speed v_p has one-way delay τ_d=ℓ/v_p. For a sinusoid, the same idea appears as electrical length βℓ=2πℓ/λ_g. The distributed regime is reached gradually: there is no universal magic frequency. The relevant comparison is geometry and delay versus signal wavelength or rise/fall time.

Rules such as 'shorter than λ/10' are engineering heuristics, not Maxwell laws. A stricter phase-error or timing budget may demand a much shorter interconnect; a tolerant application may allow more.

For a two-conductor line in a quasi-transverse electromagnetic (quasi-TEM) regime, define a local voltage v(z,t) between the conductors and a local current i(z,t) flowing along one conductor with return on the other. They are LOCAL variables: v(z,t) need not equal v(z+Δz,t), and the same is true for current.

AoN defines positive current toward +z on the upper conductor. The voltage arrow points from the lower negative reference conductor to the upper positive conductor, so v=V_top−V_bottom. These references determine the signs of the telegrapher equations; changing a reference is allowed, but it must be changed consistently.

The quasi-TEM reduction assumes the transverse field pattern adapts locally while longitudinal field components remain small enough that one voltage/current pair represents the dominant mode. Coaxial and ideal two-conductor TEM lines satisfy the stronger TEM case; microstrip is commonly treated as quasi-TEM over a useful frequency range.

Take a uniform line and inspect a short length dz. Series resistance per unit length R′ accounts for conductor loss, series inductance per unit length L′ represents magnetic energy storage, shunt conductance per unit length G′ represents dielectric leakage/loss, and shunt capacitance per unit length C′ represents electric energy storage.

Their units are Ω/m, H/m, S/m and F/m. Multiplication by dz produces the familiar local element values R′dz, L′dz, G′dz and C′dz. The prime means 'per unit length'; it is not a derivative.

In a lossless TEM line, electric energy per unit length is (1/2)C′v² and magnetic energy per unit length is (1/2)L′i². This is the field-theory content hidden inside the circuit-looking symbols.

Apply Kirchhoff's voltage law (KVL) around a slice from z to z+dz using the declared current direction. The input voltage equals the output voltage plus the series resistive drop and the inductive drop. To first order in dz, current inside those drops may be evaluated at z.

Rearrange, divide by dz and let dz→0. The finite difference becomes a spatial derivative. The minus sign means voltage decreases along +z when positive series drop is present.

This is not claiming electromagnetic action is instantaneous inside the line. KVL is being applied to an infinitesimal quasi-TEM slice whose distributed parameters already encode local electric/magnetic storage; the resulting continuum equation explicitly retains propagation.

Apply Kirchhoff's current law (KCL) at the slice: current entering at z equals current leaving at z+dz plus current diverted through the shunt conductance and capacitance. The capacitive current is C′dz·∂v/∂t and the conductive leakage is G′dz·v.

After rearranging and taking dz→0, current variation along the line is tied to shunt leakage and time-varying electric storage. This is the one-dimensional circuit expression of local charge continuity.

Together, the two first-order partial differential equations are the time-domain telegrapher equations. Chapter 02 will solve them rather than merely state their frequency-domain answer.

Every term in the voltage equation must have units V/m: R′i gives (Ω/m)A=V/m, and L′∂i/∂t gives (H/m)(A/s)=V/m. Every term in the current equation must have A/m: G′v gives (S/m)V=A/m, and C′∂v/∂t gives (F/m)(V/s)=A/m.

If R′=G′=0 the line is lossless but still distributed: inductive and capacitive storage remain. If L′ and C′ were also zero, the model would lose the very mechanism that supports finite-delay wave propagation.

In a static DC limit, time derivatives vanish. A real line can still show voltage drop through R′ and leakage through G′, so 'DC' does not automatically make a long lossy line spatially uniform.

Multiply the voltage equation by i and the current equation by v, then add them. The left side becomes ∂(vi)/∂z by the product rule. On the right, R′i² and G′v² are local dissipative powers per unit length; the L′ and C′ terms combine into the time derivative of stored magnetic and electric energy per unit length.

The result is a one-dimensional energy-continuity equation. A decrease of forward line power vi with z is accounted for by local dissipation and/or increasing stored energy. This is the distributed-circuit counterpart of the Poynting theorem studied in EM-03/EM-04.

For instantaneous real v and i under the declared passive line references, p(z,t)=v(z,t)i(z,t) is the local power crossing the transverse section in the +z bookkeeping direction.

Take a pedagogical 10 cm quasi-TEM interconnect with effective relative permittivity ε_eff≈2.8 and neglect dispersion for this estimate. Then v_p≈c/√ε_eff≈1.79×10^8 m/s and one-way delay is about 0.56 ns.

At 100 MHz the electrical length is about 20.1°, already not infinitesimal for precision phase work. At 1 GHz the same physical trace is about 201° long: a lumped-wire picture is completely inadequate.

The time-domain view gives the same warning. A 0.56 ns line delay is small compared with a 100 ns edge but comparable with a 1 ns edge. Broadband digital and pulsed systems must therefore compare interconnect delay with edge times, not only with a clock repetition frequency.

1. A 2 cm line has v_p=2.0×10^8 m/s. Compute one-way delay and electrical length at 500 MHz.
2. Explain why a fast 1 ns edge can require transmission-line treatment even if the repetition rate is only 10 MHz.
3. State the AoN voltage and current references in Figure 1.1 and predict which sign in the voltage telegrapher equation would change if the voltage arrow were reversed.
4. Convert R′=0.8 Ω/m, L′=250 nH/m, G′=2 µS/m, C′=100 pF/m into the four element values of a dz=5 mm slice.
5. From w_e′=(1/2)C′v², verify dimensionally that C′ has units F/m.
6. Starting from the incremental KVL equation, derive ∂v/∂z=−R′i−L′∂i/∂t without skipping the dz→0 step.
7. Starting from incremental KCL, derive ∂i/∂z=−G′v−C′∂v/∂t.
8. For a lossless line, reduce the two telegrapher equations and identify which term stores electric energy and which stores magnetic energy.
9. Derive the local line power-balance equation by multiplying the two telegrapher equations by i and v respectively.
10. Why is η=E/H of a bulk plane wave not automatically the same quantity as the characteristic impedance Z₀ of a transmission line?

Chapter 02 will solve the telegrapher equations in the harmonic regime using AoN phasors. From the coupled first-order equations we will derive the wave equations, propagation constant γ and forward/backward voltage/current waves before introducing characteristic impedance in chapter 03.