# The capacitance of a transmission line: there is a fundamental relationship between characteristic impedance, time delay and total capacitance. (No Myths Allowed).

IN ANY UNIFORM transmission line there is a simple relationship between its characteristic impedance, its time delay and its total capacitance. This applies for every line, whether it is microstrip, stripline, coplanar or even twisted pair. The only criterion is that it be of uniform cross-section, which is, of course, the definition of a controlled impedance interconnect. If we know any two of the following--time delay (TD), characteristic impedance ([Z.sub.0]) or total capacitance (C)--then we can find the third.

Every interconnect on every board is really a transmission line. It takes two conductors to make up a transmission line. One we label the signal path; the other we label the return path. Whether we explicitly provide a return path or not, the signal will find an adjacent conductor to provide the other leg of the current path. If the cross-sectional geometry between the two conductors is uniform, then it is a controlled impedance transmission line. The capacitance of the line is the capacitance between the signal conductor and the return conductor.

As a signal propagates down the transmission line, at each step along the way it is successively charging up each section of the line it encounters. The current flowing down the signal path and returning on the return path is the current required to charge the amount of line the signal encounters in each interval of time.

Since the signal travels down the line at a constant speed, v, and charges up the same amount of capacitance per time interval, there is a constant current flowing between the signal and return path. Increase the capacitance per length the signal sees and the current from the signal will increase.

The impedance the signal sees for each step along the line is the ratio of the voltage applied to the current through the line. Provided that the capacitance per length is constant and the speed of the signal is constant, the instantaneous impedance the signal sees will be constant down the line.

It is as though the uniform transmission line has one impedance that is characteristic of it, which we call its characteristic impedance, [Z.sub.0]. It is related to the capacitance per length, [C.sub.L], and the speed of the signal, v, by

[Z.sub.0] = 1/([C.sub.L] x v)

This is true for every uniform transmission line (1).

For a uniform line with a physical length, Len, there is a time delay for the signal to travel from the front of the line to the end of the line. This is shown simply as TD = Len/v. If we combine these two relationships, we end up with a simple connection between the total capacitance of the line and the characteristic impedance

[Z.sub.0] = TD/([C.sub.L] x Len) = TD/C or C = TD/[Z.sub.0] or [Z.sub.0] x C = TD

This is a fundamental relationship between the characteristic impedance, the time delay and the total capacitance of any uniform transmission line. For example, for a 50 [OMEGA] line that has a time delay of 1 nsec, the total capacitance between the signal and return path is 1 nsec/50 [OMEGA] = 0.02 nF = 20 pF. Every 50 [OMEGA] line will have the same 20 pF of capacitance if its time delay is 1 nsec.

If a stripline transmission line in FR-4 has a dielectric constant (Dk) of 4, then its speed will be 6 in./nsec, the TD = 1 nsec and the line is physically 6 in. long. This means that every 50 [OMEGA] line in FR-4 has a capacitance per length of 20 pF/6 in. = 3.3 pF/in. This is a good number to remember.

This is true if the line is 0.010" wide or 0.100" wide; as long as it has a characteristic impedance of 50 [OMEGA] and a Dk of 4, it will have 3.3 pF/in. of trace capacitance.

This relationship applies not just to the whole transmission line but to any part of it. Any section of a uniform line, with a time delay of TD, has a capacitance of C = TD/[Z.sub.0]. A section of a 40 [OMEGA] line, 0.1 nsec long (physically 0.1 nsec x 6 in./nsec = 0.6 in. long in FR-4) would have a capacitance in the region of C = 0.1 nsec/40 [OMEGA] = 2.5 pF. This is approximately the capacitance of a lead in a typical BGA package.

This simple relationship between characteristic impedance, total capacitance and time delay is as easy to remember as the RC time constant of a charging circuit. If you can remember RC = time constant, just replace R with [Z.sub.0] and time constant with TD.

Next month we'll show the relationship between the loop inductance and the characteristic impedance of a transmission line.

REFERENCES

(1.) Eric Bogatin, "What is Characteristic Impedance?" Printed Circuit Design, January 2000, pp. 18-20.

DR. ERIC BOGATIN is vice president and CTO of GigaTest Labs (www.gigatest.com) and a frequent speaker at the PCB Design Conferences. He can be reached at eric@gigatest.com.
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