Cable Impedance in EWIS Applications

Energy transmitted down a wire will progressively lose energy. In low frequency applications, we call it “voltage drop” or “ohmic heating”, and for high frequency applications, we call it “attenuation”. For short lengths of wire/cable, the impacts of signal attenuations are often minimal, but as EWIS designers seek to optimize systems (particularly for weight), efforts are made to balance the system performance requirements with weight savings from utilizing small gauge conductors. In this article, we discuss the factors impacting cable impedance, its impact on attenuation, and useful factors for selecting the right EWIS components for the application.

Background

Impedance is the measure of opposition to alternating current (AC) in a circuit, combining the effects of a component’s resistance, capacitance, and inductance. Every cable and interconnect/termination in a circuit presents some impedance, and understanding this impedance is critical for electrical systems. In the context of aircraft Electrical Wiring Interconnect Systems (EWIS), cable impedance is a key parameter that affects how signals and power are transmitted through wiring harnesses. A cable’s impedance influences the voltage and current distribution along its length, and whether signals are cleanly delivered or distorted by reflections. In high-frequency or fast-rise-time applications (e.g., Pulse Width Modulated Power), cables behave as transmission lines with a characteristic impedance. If the cable’s impedance is not compatible with the source or load (e.g., a cable’s impedance is 50 Ohms and the load’s impedance is 100 Ohms), a portion of the signal can be reflected, leading to reduced delivered voltage or interference with the original signal. This is why those selecting cables for EWIS applications must pay close attention to impedance to ensure signal integrity and power transfer efficiency.

The cable impedance’s significance can be demonstrated with a simple scenario: When a signal source drives a cable and load, the source voltage is divided between the source’s internal impedance, the cable, and the load and depends on the impedance values of the components. For maximum power transfer and predictable signal voltage, these impedances should be matched. If the ‘transmission line” and load impedance are mismatched, the voltage on the load may be lower than expected and/or can oscillate due to reflections. In extreme cases (such as high-power RF transmission), impedance mismatches create standing wave patterns with voltage peaks that risk damaging the cable insulation (a particular concern for high voltage systems).

Characteristic Impedance

While impedance in general refers to AC opposition in any component, characteristic impedance (often denoted \(Z_{0}\)) refers specifically to the impedance of a uniform cable or transmission line. It is defined as the ratio of the voltage to the current of a single wave propagating along the line, with no reflections present (idealized model). Equivalently, it is the input impedance of an infinitely long line, and for any finite line it is the impedance seen at the input when the line is terminated in \(Z_{0}\) itself (consideration of S-parameters further breaks down these factors). This parameter is determined entirely by the cable’s geometry and materials (conductors and dielectric) and is independent of the line’s length.

Mathematically, the characteristic impedance is given by:

\(Z_{0}=\sqrt{\frac{R+jwL}{G+jwC}}\)

where

• \(R\) is the per-unit-length resistance,
• \(L\) is the per-unit-length inductance,
• \(G\) is the per-unit-length conductance of the dielectric,
• \(C\) is the per-unit-length capacitance of the cable, and
• \(ω\) is the angular frequency of the signal.

For most practical cables at typical operating frequencies, the loss components \(R\) and \(G\) are relatively small. In a low-loss cable (or at high frequencies where reactive terms dominate), the expression simplifies and the characteristic impedance is approximately the constant value:

\(Z_{0}\approx \sqrt{\frac{L}{C}}\)

Frequency Independent

In other words, a cable with fixed inductance and capacitance per unit length has an essentially frequency-independent characteristic impedance, and energy from a source will propagate down the cable without being dissipated in the cable; reflections at terminations or splices are considered separately. For example, a typical coaxial cable might have \(L\) and \(C\) chosen such that \(Z_{0}\) is 50 Ω, which remains roughly constant over a wide frequency range.

It is important to note that \(Z_{0}\) being frequency-independent is an idealized model. In the so-called “RC region” where inductive effects are negligible and resistive and capacitive effects dominate, the effective impedance can become frequency-dependent (e.g. approximately \(Z_{0}\approx \sqrt{\frac{R}{j\omega C’}}\), which decreases with frequency). Fortunately, in most EWIS applications, cables operate in a regime where \(Z_{0}\) can be treated as a constant for signal integrity purposes. The key point for engineers is that each cable type has a characteristic impedance determined by its construction – and matching that impedance in the system is vital for optimal performance.

When the impedance is not controlled or there is an impedance mismatch at the terminations, part of the signal is reflected. To help quantify the importance of impedance matching, the voltage reflection coefficient (\(Γ\)) at the load is defined as the ratio of the reflected voltage wave to the incident (forward) voltage wave at the interface. For a transmission line with characteristic impedance \(Z_{0}\) terminated by a load with impedance \(Z_{L}\), \(Γ\) is given by:

\(\Gamma = \frac{Z_{L}-Z_{0}}{Z_{L}+Z_{0}}\)

In the case of a 50 Ω line and source driving a 100 Ω load, \(Γ\) would yield 33%. In practical terms, if the incident signal has a peak voltage of 1V, the reflected signal will have an amplitude of 0.33V. If the equipment is sensitive to signal disruptions, such a high reflection could impact device performance.

Conclusion

For those that took a graduate level E&M course, the description of waveguides was abstracted to the point where it was almost impossible to understand a real-world application. Thankfully, the theoretical implications of cable impedance have been distilled to be accessible to engineers seeking to pick the right components for their applications. A cable’s impedance is measurable property that may change based on the physical installation and environmental operating conditions (not to mention potential impact of material aging). These will be further assessed in future Lectromec articles. For those seeking support assessing cables for current or upcoming projects, contact Lectromec; our ISO 17025:2017 accredited EWIS test lab is ready to help.