Time Domain Reflectometry (TDR)

Simply put, it's a measurement method that injects a step wave—a pulse with a steep rise—into a transmission line and observes the time-domain waveform of the reflected wave. Since signals reflect at locations with impedance discontinuities, analyzing the amplitude and arrival time of the reflected wave reveals "where" and "what kind" of discontinuity exists.

It's essential for design verification of vias and connectors in high-speed boards. For example, in PCIe Gen5/6 or DDR5 routing, the impedance discontinuity of a single via can ruin signal quality. The biggest benefit is being able to predict TDR waveforms with 3D FEM analysis and discover issues before prototyping.

Typical applications include:

• PCB Trace Characteristic Impedance Verification: Checking if it stays within ±5% of design values like 50 Ω or 100 Ω differential
• Via Discontinuity Evaluation: Optimizing stub length and anti-pad diameter of through-hole vias
• Connector Impedance Profile: Identifying capacitive discontinuities at solder joints
• Locating Break Points in Cables/Harnesses: Diagnostics for large-scale harnesses in automotive or aerospace applications

When the load impedance is $Z_L$ relative to the transmission line's characteristic impedance $Z_0$, the reflection coefficient $\rho$ (Greek letter "rho") is defined as:

Let's organize what the value of $\rho$ means:

• $\rho = 0$: Perfect match ($Z_L = Z_0$). No reflection. Ideal state.
• $\rho = +1$: Open circuit ($Z_L = \infty$). Total reflection in-phase with the incident wave.
• $\rho = -1$: Short circuit ($Z_L = 0$). Total reflection out-of-phase with the incident wave.
• $0 < \rho < 1$: Inductive discontinuity. TDR waveform steps upward.
• $-1 < \rho < 0$: Capacitive discontinuity. TDR waveform steps downward.

Absolutely. If you read the reflection coefficient $\rho(t)$ at time $t$ from the reflection waveform, the impedance at that location can be reconstructed using the following formula:

Here, $\rho(t)$ is obtained empirically as the ratio of the reflected wave $V_r$ to the incident wave $V_i$:

For example, when measuring with a $Z_0 = 50\,\Omega$ coaxial cable and obtaining $\rho = 0.2$ at a certain location:

$$Z = 50 \times \frac{1 + 0.2}{1 - 0.2} = 50 \times \frac{1.2}{0.8} = 75\,\Omega$$

This tells you there is a 75 Ω discontinuity. This is a common pattern seen in automotive antenna cables in practice. It instantly reveals connection errors between 50 Ω and 75 Ω systems.

Exactly. The distance $d$ to the discontinuity point is calculated from the propagation velocity $v_p$ and the round-trip time $\Delta t$:

Here, $c$ is the speed of light, and $\varepsilon_{\text{eff}}$ is the effective permittivity. For an FR4 substrate, $\varepsilon_{\text{eff}} \approx 3.8$, so the propagation speed is about 51% of the speed of light. A 1 ns round-trip delay corresponds to a distance of about 76 mm.

Transmission lines are described by Telegrapher's equations. Using per-unit-length inductance $L$, capacitance $C$, resistance $R$, and conductance $G$:

In the lossless case ($R = G = 0$), this system of equations becomes a wave equation propagating at velocity $v_p = 1/\sqrt{LC}$. The characteristic impedance is:

TDR utilizes the phenomenon of this wave reflecting at discontinuity points. In CAE simulation, the mainstream approach is to directly solve Maxwell's equations with 3D FEM/FDTD to obtain S-parameters, then calculate the TDR waveform from them. Telegrapher's equations are a 1D approximation, but 3D structures like vias and connectors cannot be represented by 1D models, so full 3D analysis is necessary.

Sharp intuition. The spatial resolution $\Delta d$ of TDR is determined by the step pulse's rise time $t_r$ (10% to 90%):

Let's look at some concrete examples:

Verifying PCIe Gen5 (32 GT/s) routing requires $t_r \leq 20$ ps. Resolving via stub lengths (around 0.5–2 mm) requires resolution in this class. In simulation, a major advantage over measurement is the ability to freely set the rise time as a design parameter.