Why Unterminated Transmission Lines Cause Signal Ringing

This article explains why an unterminated transmission line experiences severe underdamped ringing when a signal's rise time is shorter than the line's propagation delay. It covers the fundamental physics of impedance mismatch, how fast rise times expose the distributed inductance and capacitance of a circuit trace, and the resulting signal reflections that bounce back and forth to create voltage oscillations.

The Critical Ratio: Rise Time vs. Propagation Delay

In digital electronics, a PCB trace or cable acts as a simple wire only when the signal changes slowly. However, when the signal's rise time (\(t_r\)) is shorter than the time it takes for the signal to travel down the line—known as the propagation delay (\(t_{pd}\))—the wire stops acting like a simple conductor and begins acting like a transmission line.

Generally, if the propagation delay is more than half of the signal's rise time (\(t_{pd} > 0.5 \times t_r\)), the physical length of the trace is longer than the spatial length of the rising signal edge. Because the entire trace does not experience the voltage change simultaneously, the signal must propagate down the line as an electromagnetic wave.

Impedance Mismatch and Total Reflection

As the fast-rising signal wave travels down the transmission line, it encounters the line's characteristic impedance (\(Z_0\)), which is determined by the physical geometry and materials of the trace (typically 50 to 100 ohms).

An unterminated transmission line ends at a high-impedance load, such as the input of a CMOS logic gate, which behaves essentially as an open circuit (\(Z_{load} \approx \infty\)). When the traveling wave reaches this open end, the current has nowhere to go. Because energy cannot simply disappear, the impedance mismatch forces the signal to reflect back toward the source.

The severity of this reflection is dictated by the reflection coefficient (\(\Gamma\)):

\[\Gamma = \frac{Z_{load} - Z_0}{Z_{load} + Z_0}\]

For an open circuit where \(Z_{load}\) approaches infinity, the reflection coefficient equals \(+1\). This means 100% of the incident voltage wave is reflected back toward the driver, doubling the voltage at the receiver end upon arrival.

The Bounce Effect at the Source

The reflected wave travels back down the line toward the signal source (the driver). Most digital drivers have a very low output impedance (\(Z_{source}\)), often just a few ohms, which is much lower than the characteristic impedance of the line (\(Z_0\)).

When the returning wave hits this low-impedance source, it encounters another severe impedance mismatch. The reflection coefficient at the source is negative because \(Z_{source} < Z_0\). Consequently, the wave reflects again, this time inverted in polarity, and travels back toward the open load.

Because the rise time is shorter than the propagation delay, these reflections are completely separated in time from the initial transition. Instead of blending seamlessly into the rising edge (as they would with a slow rise time), each reflection arrives as a distinct step wave, causing the voltage at the receiver to overshoot, undershoot, and oscillate wildly.

Distributed LC Parameters and Underdamping

From a circuit theory perspective, a transmission line is a distributed network of series inductance (\(L\)) and parallel capacitance (\(C\)), along with a very small amount of series resistance (\(R\)).

In an unterminated line, the absence of a termination resistor means there is almost no resistive element to dissipate the energy stored in the line's magnetic (inductive) and electric (capacitive) fields. For an RLC circuit to be critically damped or overdamped, it requires sufficient resistance to absorb energy. Because the resistance of a copper trace is negligible compared to its reactance at high frequencies, the damping factor is exceptionally low.

A fast rise time contains high-frequency harmonics that excite the natural resonant frequency of this distributed LC network. With no termination resistor to absorb the energy, the energy simply swaps back and forth between the inductance and capacitance of the trace. This manifests as severe underdamped ringing that only slowly decays over time due to the minimal resistive losses of the copper and the driver's internal resistance.