5 Transmission Lines
In EMC, it is essential to understand transmission lines and when to apply the concept of transmission lines. Why? Because once a signal interconnection line exceeds a critical length lcritical [m], impedance matching (Zsource, Zline, Zload) MUST be considered to prevent reflections and ringing, thus preventing unwanted radiated emissions and bad signal quality - a signal integrity topic [5.2].
This chapter introduces you to transmission lines and their most important properties.
What is a transmission line?
A transmission line is a series of conductors, often but not necessarily two, used to guide electromagnetic energy from one place to the other [5.3]. It's that simple. The more complicated part is the math behind it (Maxwell's equations) because we do no longer consider a conductor as a lumped element (e.g. a simple R-L series circuit to emulate an electrical interconnection). Rather, we consider the signal conductor as a transmission line through which an electromagnetic field is moved from one point to another. Transmission lines are characterized or described by their characteristic impedance Z0.
Here are some common transmission line geometries [5.3]:
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Coax. The electromagnetic energy is propagating through the dielectric (mostly PTFE, because of its low loss and stable εr for many frequencies) between the center conductor and the inside surface of the outer conductor (shield) of a coaxial cable.
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Microstrip. Transmission line where the signal conductor is on the top or bottom layer of a PCB with an adjusted return path conductor (e.g. ground plane or power supply plane).
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Stripline. Transmission line where the signal conductor is embedded between two signal return path conductors (e.g. ground or power supply plane) on a PCB.
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Balanced line. Two conductors of the same size and shape with equal impedance to ground and all other conductors (e.g. ethernet cable).
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Waveguide. A waveguide consists of a single hollow conductor used to guide the electromagnetic energy. Waveguides are used in the gigahertz frequency range and they cannot pass direct currents (DC) signals.
When to consider a signal path as a transmission line?
Every signal interconnection is a transmission line. However, it is not necessary to threat every signal path as a transmission line. Rules of thumb - if a conductor should be treated as a transmission line or not - are mentioned below: one rule of thumb for the frequency domain and one for the time domain.
Frequency domain. A common rule of thumb, when working in the frequency domain, is the following [5.3]:
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Interconnection length llong ≥ λ/10. Consider the signal path as a transmission line, in order to minimize signal distortions and ringing due to reflections and in order to minimize radiated emissions and electromagnetic interference (EMI).
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Interconnection length lshort < λ/10. Consider the signal path as a simple conductor. If an interconnection length l [m] is short with respect to the signal wavelength λ [m], it is good practice that the interconnection is considered as a simple conductor with lumped-element parameters (e.g. resistor R [Ω] in series with inductance L [H]).
When determining the shortest wavelength λmin [m] in a digital signal (e.g. clock), it is necessary to know the maximum frequency fmax [Hz] of the signal by considering the rising- and falling-times (rather than the fundamental frequency). The rule of thumb for calculating the bandwidth [Hz] - or the highest significant sine wave frequency - of a rectangular digital signal is [5.1]:
Where t10%-90% is the rising- or fall-time (whichever is smaller) from 10% to 90% of the slope of a digital signal in [sec]. The corresponding minimum wavelength λmin [m] is
Where v [m/sec] is the propagation velocity and fmax [Hz] is the highest significant sinusoidal frequency in a digital signal. The frequency domain approximation for the critical length lcritical-fd [m] can be calculated with respect to the digital signal rise-/fall-time t10%-90%:
Where c [m/sec] is the speed of light (3E8m/sec), fmax [Hz] is the highest significant sinusoidal frequency in the digital signal, t10%-90% [sec] is the rising- or fall-time (whichever is smaller) from 10% to 90% of the slope of the digital signal and εreff [1] is the effective dielectric constant (relative permittivity) through which the electromagnetic wave is propagating.
Time domain. There is also a rule of thumb for the time domain [5.3]:
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t10-90% ≤ 2*tpd. If the rise- or fall-time t10-90% [sec] of a digital signal is smaller than twice the propagation delay tpd [sec] (along the signal line), then the signal path should be considered as transmission line, in order to minimize signal distortions and ringing due to reflections and in order to minimize radiated emissions and electromagnetic interference (EMI).
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t10-90% > 2*tpd. If the rise- and fall-time t10-90% [sec] of a digital signal are bigger than twice the propagation-delay tpd [sec] of the signal across the conductor, the signal path may be considered as simple conductor.
The time domain approximation for the critical length lcritical-td [m] can be calculated with respect to the digital signal rise-/fall-time t10%-90%:
Where c [m/sec] is the speed of light (3E8m/sec), t10%-90% [sec] is the rising- or fall-time (whichever is smaller) from 10% to 90% of the slope of the digital signal and εreff [1] is the effective dielectric constant (relative permittivity) through which the electromagnetic wave is propagating.
Critical length lcritical. The two rules above lead to similar values for lcritical [m], where the frequency domain lcritical-fd is smaller than the time domain lcritical-td. It is therefore recommended to go with the rule of thumb of the frequency domain: Consider an interconnection as transmission line if it is longer than λ/10.
Characteristic Impedance.
The characteristic impedance Z0, also called the wave impedance, is an important property of a transmission line. First, have a look at the equivalent circuit of a transmission line (the distribution parameter model of a two-conductor transmission line) [5.3]:
The picture shows transmission line segments of an infinitely small length dz in [m]. The parameters are defined per unit length (e.g [m]):
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R' = Resistance per unit length [Ω/m]
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L' = Inductance per unit length [H/m], mostly in [mH/m] or [nH/m]
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C' = Capacitance per unit length [F/m], mostly in [pF/m]
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G' = Conductance per unit length [S/m]
The formula for the characteristic impedance Z0 of a transmission line is defined as [5.3]:
where ω is the angular frequency [rad/sec] and all other parameters are explained above. In practice, it is often adequate to describe transmission lines as lossless (R' = 0, G' = 0). In that case, the equivalent circuit is simplified to this:
The formula for the characteristic impedance Z0 of lossless transmission lines is defined as [5.3]:
Enough theory, let's think about how to calculate the characteristic impedance for your projects.
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Cables. In case your cable contains signals which require impedance matching (e.g. CAN or Ethernet), you should only use cables that match with the required impedance (e.g. 100Ω or 50Ω). In general, it is not necessary to calculate the characteristic impedance of cables, this is already done for you by the cable manufacturer.
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PCBs. Because every PCB design is individually configurable (type of dielectric material, thickness of copper, PCB trace geometry, etc.), it is usually necessary to calculate the characteristic impedance by yourself. Therefore, you should know the following data:
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Dielectric media. Thickness, dielectric constant εr and loss tangents tan(δ) of the PCB substrate. Have a look at our PCB stackup and dielectric material collection here.
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Trace. PCB trace width and geometry, copper thickness, height above reference planes, distance to copper-pour (and separation distance for differential lines).
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Solder resist. For PCB traces at top or bottom, you must know the thickness and dielectric constant of the solder resist.
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There are plenty of free online and offline tools available for calculating the characteristic impedance of arbitrary PCB traces or signal pairs. However, be aware that these free tools usually use approximations and the results are only accurate in a certain parameter range. If you need accurate values use a solver, which calculates the transmission line impedance based on Maxwell's Equations (e.g. HyperLynx or Si8000). To help you even more, we wrote down some information on PCB stackups and properties of PCB materials.
Propagation Constant.
The propagation constant γ [1/m] is an important property of a transmission line. It describes the attenuation and phase shift of the signal as it propagates through the transmission line. Let's imagine a sinusoidal voltage, current, electric field, or magnetic field which propagates in the direction of the z-axis and which has an amplitude of A0 at its source and amplitude of A(z) at the distance z [m] from the source. Then A(z) can be written as:
where A(z) is the complex phasor term of a sinusoidal voltage [V], current [A], electric field [V/m], or magnetic field [A/m] at the distance z [m] away from the source and γ [1/m] is the complex propagation constant. Propagation constant γ is calculated like this [5.3]:
where R'=resistance per unit length [Ω/m], L'=inductance per unit length [H/m], C'=capacitance per unit length [F/m], G'=conductance per unit length [S/m], ω=angular frequency [rad] of the signal, μ=μrμ0, μ0=4πE-7 [H/m] μr=relative permeability [1], ε=εrε0, ε0=8.85E-12 [F/m] εr=relative permittivity [1].
Propagation constant γ in can also be written as [5.3]:
where the real part α in [1/m] is the attenuation constant and the imaginary part β in [rad/m] is the phase constant. α and β are calculated like this [5.3]:
For a lossless line (R'=0, G'=0, dielectric losses = 0) we get:
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No attenuation α [1/m]:
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Phase shift β [rad/m]:
A sinusoidal signal that travels along a transmission line with attenuation α > 0 [1/m] has a signal form along the z-axis that looks like this:
High-Frequency Losses.
The lossless line model is often accurate enough for frequencies up to 100MHz. However, above 100MHz, the high-frequency losses may not be neglected anymore, and therefore the attenuation factor α [1/m] cannot be assumed to be zero. High-frequency losses are a result of [5.3]:
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Ohmic Loss. Resulting from the resistance of the conductors. Ohmic losses are a function of the frequency f [Hz] as well as of the geometry of the conductor (skin-effect!).
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Dielectric Loss. Resulting from the dielectric material absorbing energy from the propagating electric field (heating the material). Dielectric losses are a function of frequency f, the dissipation factor (tan(𝛿)) and the dielectric constant (εr ).
NOTE: Dielectric losses do NOT depend on the geometry of the transmission line, only on the dielectric material.
Balanced vs. Unbalanced Transmission Lines.
EMC design engineers should be well aware of the concept and benefits of balanced transmission lines. Here are the differences between balanced and unbalanced transmission lines:
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Balanced. A balanced transmission line consists of two conductors which have the same impedance along their line and the same impedance to ground and all other conductors. Differential signals should be transmitted over balanced transmission lines.
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Unbalanced. For an unbalanced transmission line, the impedances of the forward and return current lines to ground are unequal. Single-ended signals are signals which are referenced to ground and should therefore be sent over unbalanced transmission lines.
Balanced transmission lines are very robust against common-mode noise because common-mode signals will be canceled out at the receiver's side. However, to prevent differential-mode noise coupling, the two signal conductors must be routed close to each other (e.g. by twisting them or by routing them as differential-pair on a PCB). The pictures below illustrate the common-mode noise suppression of a balanced transmission line compared to an unbalanced transmission line.
Generally, single-ended interfaces (CMOS, TTL, I2C, SPI) should be sent over unbalanced transmission lines, and differential signal interfaces (LVDS, USB, Ethernet, CAN, HDMI) should be sent over balanced transmission lines. Examples of balanced and unbalanced transmission lines are:
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Unbalanced. Suitable for single-ended signal interfaces.
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PCB data lines. Microstrip lines, striplines coplanar waveguides.
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Cables and wires. Coaxial cables, multilayer flat-ribbon or flat-flex cables with at least one solid ground plane
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Balanced. Suitable for differential and pseudo-differential signal interfaces.
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PCB data lines. Microstrip lines and stripline routed as differential pairs.
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Cables and wires. Twisted pair, twin-lead cables, flat-ribbon or flat-flex cables (without ground plane).
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References:
[5.2] Eric Bogatin. Signal and Power Integrity - Simplified. Prentice Hall Signal Integrity Library, 3rd Edition, 2018, p.78
[5.3] Henry W. Ott, Electromagnetic Compatibility Engineering, John Wiley & Sons, Inc., 2009, pp. 215-223