This article covers in-depth the topic of losses in PCB transmission lines. We will talk about conductor loss, signal trace resistance, dielectric loss, loss tangent/dissipation factor of a dielectric, and total insertion loss.
In our previous PCB Transmission Line series, we gave you the characteristic impedance of a transmission line:
R = Resistance of the line conductors per unit length (pul)
L = Inductance of the line conductors loop pul
G = Conductance (due to dielectric material) between the signal and return paths pul
C = Capacitance between the signal and return paths pul (it increases with the Dk of the dielectric)
For a uniform transmission line, R, L, G, C are the same at every point on it and therefore Zc has the same value at every point on the transmission line.
For a sinusoidal signal of frequencytraveling in the direction over the line, expressions for voltage and current at various points and time are given by:
Where α and β are the real and imaginary components of , which is given by:
At our frequencies of interest, R << ωL and G << ωC, so that:
This represents a wave which travels with a propagation delay per unit length and which is attenuating as it travels along the line.
The signal attenuation factor of a transmission line of length l is given by:
The attenuation or signal loss factor is usually expressed in dBs.
Thus the dB loss is directly proportional to the line length. Hence, we can express the above as dB loss per unit length as:
We usually omit the minus sign, keeping in mind that it is a dB loss – always to be subtracted from the signal strength in dB.
The above is also called total insertion loss per unit length of the transmission line and written as:
Now R/Z0 component of the loss is proportional to R, the resistance of the length per unit length and is called the conductor loss and is due to the resistance of the conductors forming the transmission line. It is represented by ‘alfa’C. GZ0 part of the loss is proportional to G – the conductance of the dielectric material and is called the dielectric loss – denoted by ‘alfa’ d.
Where R is the resistance of conductors per inch.
Now there are two conductors in a PCB transmission line – the signal trace and the return path.
Usually the return path is a plane, however the return current is not uniformly distributed over the plane – it can be demonstrated that most of the current is concentrated on a strip of width three times the width of the signal trace and just underneath the signal trace.
It is possible to approximate:
Signal trace resistance
Does the entire cross-sectional area of the signal trace participate in the signal current equally? The answer is: not always – it depends on the frequency of the signal.
At very low frequencies – until about 1 MHz, we can assume that the entire conductor participates in the signal current and hence Rsig is the same as the ‘alfa’ C resistance of the signal trace, which is given by:
ρ = Copper resistively in ohm-inch
W = Trace width in inches (example: a 5-mil ie. 0.005” trace for 50 ohms)
T = Trace thickness in inches (usually ½ oz to 10 oz ie. 0.0007” to 0.0014”)
For example, for a 5-mil wide trace:
For our purpose, we are interested in A/C resistance at frequency f. Here, the skin effect comes into the picture. As per skin effect, the current at a frequency f travels only up to a certain depth called the skin depth of the conductor which is given by:
The following table gives the value of the skin depth at various frequencies:
We see from above that at 4 MH, the skin depth equals the 1 oz copper thickness and at 15 MHz, it equals ½ oz copper thickness. Beyond 15 MHz, the signal current travels only in-depth less than 0.7 mil and it keeps decreasing as the frequency increases.
As we are concerned here with high-frequency behavior, we can safely assume that T is greater than the skin depth at frequencies of our interest, hence instead of using T in the formula for signal resistance, we will use the skin depth. So we now have:
We use 2δ instead of δ since the current uses all the periphery of the conductor – technically 2W can be replaced by 2(W+T).
The return signal travels in only one thickness δ along the surface nearest to the signal trace and its resistance can be approximated by:
Increase in conductor loss due to copper surface roughness at the conductor – dielectric interface:
It is important to know that in a circuit board, the ‘copper conductor – dielectric interface’ is never smooth (if it were smooth, the copper conductor would easily peel off the dielectric surface); it is roughened into a teeth-like structure to increase the peel strength of conductors on the circuit board.
For a typical copper clad laminate, the interface will look like:
hz = Peak to peak height of teeth
hz is a measure of surface roughness.
Typically, hz varies from one foil type to another and the typical values are:
If the roughness hz is smaller than the skin depth (which will be the case at very high frequencies), this will cause additional conductor losses. This increase has been observed experimentally by making test circuit boards with different foils having different hz.
Losses in case of VLF foil have been found lower than in case of the usual HTE foil.
For RF/microwave boards at frequencies greater than 1 GHz, these conductor losses due to roughness can become significant over long signal lines.
At low frequencies, it will still be:
Use the higher of the equations above for R.
At high frequencies:
If f were in GHz and W and T in mils, we would get:
Let’s calculate it for a 5-mil, 1-oz, 50-ohm and for a 4-mil, 0.5-oz and 50-ohm line:
The important point to note is that at frequencies greater than 50 MHz, conductor loss is proportional to square root of the frequency:
It is not easy to predict the additional loss due to copper roughness – simple formulae do not exist.
As was mentioned earlier, the dielectric loss in units of dB per unit length in a transmission line is given by:
G = Conductance pul of the dielectric material
Z0 = Impedance of the transmission line is about ≈√L/C
The PCB dielectric materials are characterized by two properties:
1. Dielectric constant – Dk or Er – also called relative permittivity.
2. Dissipation factor – Df – also called tanδ.
Values of Er and Df are published by the PCB material manufacturers.
We will now find the relation between G and Er, Df.
Loss tangent/dissipation factor of a dielectric
A dielectric layer between two conductors can be modeled as a conductance G in parallel with a capacitance C:
The A/C voltage and current of frequency across this conductor is given by:
IG is the current through G and IC is the current through the capacitor.
tanδ is also called as dissipation factor Df ≡ tanδ.
If σ is the effective conductivity of the dielectric material, then:
It has been experimentally observed that tanδ or Df varies very little with frequency and may be considered as a value independent of the frequency for all practical purposes:
The above equation indicates that conductivity σ and thus the conductance G of the dielectric increases with the frequency – this is to be expected as greater the frequency, more will the heat be dissipated in mechanical motion of dielectric dipoles in their effort to align with the alternating electric field across the dielectric. (This is called ‘damping of the vibrating dipole moments’.)
We now have:
Recall that √LC gives the propagation delay per unit length – Pd – of the transmission line.
Now Pd is given by:
Therefore, we get:
We see from above that the dielectric loss is proportional to the frequency.
To get some idea of its magnitude, let’s consider the PCB materials Isola 370HR and I-Speed and I-Meta:
Total insertion loss
It is the sum of conductor loss – ‘alfa’ C – and the dielectric loss: ‘alfa’ d.
This is the value of the loss that is measured. (It is not easy to measure the conductor and dielectric losses separately.)
If we measure the insertion loss for sinusoidal signals at varying frequencies – say from 1 GHz to 10 GHz – we can use the above equation to separate the two types of losses:
If we now plot ‘alfa’ ins / √f vs √f, we expect a linear plot, from which we can determine A1 and A2.
Keep reading about PCB transmission lines: