What is the input impedance of a transmission line?
You can go to any electronics store and buy a length of RG58 coaxial cable. The description written on the shelf reads "50 Ω cable." But what does this really mean? If you take an ohm meter to the cable, will it read 50 Ω?
If you snipped off a three-foot length and touched one lead of the ohm meter to the center conductor and the other lead to the adjacent shield, what would the ohm meter read? If the ohm meter doesn't read 50 Ω, what does it mean to have a 50 Ω cable or a 50 Ω trace on a circuit board?
The 50 Ω refers to the impedance of the line, but if an ohm meter isn't going to read 50 Ω, what impedance is this referring to?
The resolution of this paradox stems from the confusion about the word impedance. There are three impedances, each with a different and distinct meaning, yet, we too often use the same word for each application, without the appropriate qualifier.
First there is the instantaneous impedance. This is the impedance a signal sees as it propagates down the line. At each step along the way, the signal is asking, "What is the ratio of the voltage to the current?" The voltage is the signal it is applying and the current is the current it sends into the signal conductor and back to the return conductor, each step of the way, to charge up each successive section of the distributed capacitance of the line.
If the cross-section of the line were uniform, the instantaneous impedance the signal saw each step of the way would be constant. In a uniform transmission line, there is just one value of the instantaneous impedance, which is "characteristic" of that transmission line.
The characteristic impedance of the line is what we call the one instantaneous impedance a signal would see on a uniform transmission line. Only a uniform transmission line has a characteristic impedance.
Given these two types of impedance - the instantaneous impedance and the characteristic impedance - what is the impedance a driver would see, looking into the front end of the line? This is the third impedance, the input impedance of a transmission line.
FIGURE 1. What is the input impedance of an RG58 cable that's so long that it stretches between the earth and the moon?
|
What is the input impedance of the three-foot length of RG58 cable? The ohm meter would measure an open. Is the input impedance of the cable really an open? Let's make it easier. Suppose the cable were so long that it stretched between the earth and the moon.
It would take a signal about 5 seconds to enter the cable, race to the moon, reflect and come back to the front. We have 5 seconds before we even know there is an end. What does the ohm meter read in the first 5 seconds?
The ohm meter works by applying a 1 v signal from a battery and measuring the current draw. The RG58 cable has a characteristic impedance of 50 Ω. This means that as the 1 v signal is launched into the cable, it sees an instantaneous impedance of 50 Ω. The current launched into the cable is 1v/50 ohms or 20 mAmps, which lasts for at least the first 5 seconds.
The ohm meter, looking into the front end of the cable, sees an input impedance of 1v/20 mA or 50 Ω. The input impedance of the cable is 50 Ω for the first 5 seconds. But what if we wait a day? Obviously, the ohm meter will read an open if it is connected all day.
So, what is the input impedance of the cable? Is it 50 Ω, or is it open? There is no one value; it is time-dependent, and it changes.
If we look for a time - short - compared to the round-trip time of flight, the input impedance of a transmission line is the characteristic impedance of the line. If we look for a time - long - compared to the round-trip time of flight, it may look open. If we look for a time comparable to the round-trip time of flight, we will see the impedance changing in a complicated way that cannot be easily calculated with pencil and paper, but can be simulated with a circuit simulator, or measured with a TDR. PCD&M
Dr. Eric Bogatin is the CTO at IDI and president of Bogatin Enterprises. Many of his papers are available on his Web site, www.BeTheSignal.com. He can be reached at This email address is being protected from spambots. You need JavaScript enabled to view it..
