Estimated reading time: 6 minutes

Transmission Lines – a Voyage From DC – No, Not Washington, Part 1

In this two-part article I'd like to join you on a voyage through a transmission line from DC onwards and upwards through the frequency spectrum. Step by step exploring the characteristics through the dizzying rise from very low to ultra high frequencies – but rest assured, I won't go beyond what is of practical use to you right now. In Part 1 we trace the impedance from infinity at DC to the GHz region where it reaches the steady state value of its characteristic impedance.

Starting with a confession to long time followers of Polar's website – many are process and chemical engineers where electronics is not their specialised subject. If you have read Polar applications notes or been to our seminars you have probably heard us say time and again that impedance is frequency independent. Well, just like your elementary school teacher who taught you that the molecule was the smallest element and then you went on to high school only to find that was the partial truth and atoms were smaller, a couple of years on you realise actually atoms comprise protons, neutrons and electrons and then before university suddenly entering the realms of quarks, muons and mesons, strangeness and charm, etc., etc... well, our secret is that the story we taught at Polar was true – BUT – only under a certain range of conditions.

So now its time to tell you the whole story, or rather the whole story up to just above the frequencies that are relevant today.

PCB process engineers took our word for it that a 6 inch (150mm) coupon comprised of a narrow trace a fixed height above a wide ground plane had an impedance (often of 50 Ohms) but I am sure many only half believed us, as if you measure resistance between the trace and ground with a digital multi-meter – the circuit has infinite resistance! So what is really going on here? Another puzzle to new customers is that measuring the resistance of the trace from one end to the other shows a short circuit or almost short, NAD* (*Near As Dammit – as my math teacher used to say.)

“So, tell me the truth then!” I can hear you saying. Well, the truth is that at different frequencies all the transmission line characteristics have influence on line behaviour in differing ways, some of them surprising and some counter intuitive.

The transmission line at DC

Let’s begin at the start. Nice and easy – 6-inch-long line, FR4 substrate at DC. For ease of visualisation, I'll imagine a line designed for a 50 Ohm characteristic impedance on a 63 mil (1.6 mm) thick double sided board. Just take it as given that the trace needs to be around 130 mil (3 mm) wide to make a 50 Ohm characteristic impedance line. Measure the resistance between the trace and ground with a simple digital multi-meter – and it reads open circuit.

Let’s consider two ways of looking at the line: one by measuring the characteristics of the line when "looking into" the signal line and the ground line, the other by looking at the resistance of the signal trace itself – i.e. measure the resistance between the start (0 inches 0 mm) and the end of the line (6 inches ~150 mm.)

Using the dimensions of a typical trapezoidal profile PCB copper trace:

From the calculation above, Figure 1, you can see the end to end resistance of the trace itself will be about 27 milliohms – or small enough to be ignored. Just for interest – half way down the trace (line length 3 inches ~ 75 mm) the resistance is 13 milliohms, Figure 2, so there is a linear relationship between resistance and length of trace (that might seem obvious but keep this in mind for later.) So at DC, an unterminated impedance controlled trace with a characteristic impedance of 50 Ohms is actually an open circuit.

Increasing the frequency

Next step is to start ramping up the frequency and see what happens. Going back to physics, at low frequencies a length of copper over a ground plane separated by an insulator (the FR4 base material) would seem to be like a capacitor – and at low frequencies it behaves just in that way. As the frequency ramps up the impedance steadily falls from infinity like this...

Look closely at Figure 3 – in the KHz decade you can see the impedance magnitude falling rapidly from its DC value of infinity. However, you can also see that unlike a pure capacitor – whose impedance magnitude you would expect to fall to a negligible reading at higher frequencies – something else seems to be slowing the rate of decline in impedance magnitude. The picked data point on the graph at 1 MHz shows an impedance of 72 Ohms. So what could be causing the graph to flatten out?

Impedance frequency independent?

Back to basics here... for the past couple of decades at Polar seminars we have explained to fabricators that the PCB characteristic impedance for lossless lines is equal to Z₀ = √(L/C). Well, I'll let you into a secret – that information was true, but only true over a band of frequencies commonly used at the time. With ultra high speed serial transmission designers have become preoccupied with very high – and, somewhat counter intuitively, lower– frequencies. “Why lower?” you may ask yourself. Well, for example, ultra long streams of data comprised of all 0 or all 1 represent a much, much lower frequency than the clock rate. This is why bit error rate test (BERT) testers deploy very long data words with a pseudo random series of highs and lows to represent the broad band nature of high speed serial data transmission.

I digress though; let’s go back to the equations for a moment – for lossless lines operating from a few 10s of MHz to one or two GHz the above approximation worked just fine, and the lines operated in that region behaved reasonably independently of frequency.

However, the full equation for transmission line impedance is:

Z₀ = √((R+jωL)/(G+jωC))

Let’s look at the low frequency case first and have a think about these terms; we have already learned that at mid frequencies R is small enough to ignore and G (the conductance of the substrate) is also small enough to ignore. But as the frequency falls so does the inductive reactance, and gradually jωL becomes small compared with R; as this starts to happen the line impedance becomes frequency dependent and the impedance equation shifts to a point where jωL can be ignored but R cannot. The impedance in this low frequency region tends towards:

Z₀ = R/jωC

I like to take math to the extreme to understand phenomena, as if we understand the extremes it is easier to fit in the center regions (and the math often becomes simple at the limits – I like that…) So at DC the frequency represented by jω is ZERO. Z₀ at DC is then R (which is small but not zero) divided by zero – and, surprise, the characteristic impedance at DC is infinite.

Push the frequency higher and as jωL grows large compared to R then the line behaves in a frequency independent way as the jω top and bottom cancel; the instantaneous impedance reaches a reasonably steady state.

In Part 2 we’ll look at other mechanisms that introduce loss into the line, draining the signal of its energy as it transits the line.