Characteristic impedance

Updated April 25, 2008

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New for February 2007! We've added some discussion of the equations that convert Z0 and VP to L' and C'! See below.

Note to readers.. if you want more discussion on this topic we suggest you pick up a copy of the latest revision of Pozar's Microwave Engineering, look for it on our book page.

What is this thing called characteristic impedance, also known as "Z-zero"? Let's start with a common misconception. If you have a 50-ohm coax cable (or microstrip line or any other 50 ohm transmission line) that isn't connected to anything on either end, and connect an ohmmeter across it, you might expecting to read 50 ohms if you were new to microwave engineering. But you'll measure an open circuit (unless something is loading the opposite end which means you didn't follow our simple instructions...) nowhere near fifty ohms. What's going on?

But wait, Jens from Denmark wants you to know that the above statement is not 100% technically correct:

On the new page for characteristic impedance, you start out by claiming that you cannot measure anything if you connect an ordinary ohmmeter to a 50 ohm cable or microstrip. This is simply because the line is too short for the meter to get the reading. The meter may take about 1 second to get a stable reading. If you take a loss free transmission of say c x 1.5 (about 4.5 x exp(8) meters!) you will get a reading that lasts about 3 seconds. You may have some practical problems with this experiment though. The line will wind the equator about 11.25 times. And getting it lossless will also be difficult.

Thanks, and let us know when you have that cable in place so we can perform the measurement!

And now an opposing viewpoint from Raphael...

I believe Jens is somehow wrongly deriving is conclusion from the wave propagation phenomena. As I see it, assuming you could make the experiment, you would never see (let's say) 50 Ohm on your meter because traditional multimeters derive resistance from measured voltage and current to display the result of V / I and you would never have any current flowing between the two conductors.

The definition of resistance is "the opposition to the flow of electric current" whereas the definition of impedance is "The opposition to an alternating current". Meaning that a 50 Ohm "resistance" behaves the same (offer the same opposition to current) in both direct or alternating current. A 50 Ohm "impedance" on the other hand does not necessarily behave the same in direct or alternating current.

It's not because an impedance has only a real part that's it's necessarily a "resistance" ... it might just still be an "impedance".
Z = R + jX
Z = R

We're inclined to agree with Raphael on this, but we look forward to further discussion! And here comes some more discussion from Mark, who agrees with Jens...

Although a DC measurement will indeed show no current passing through an unterminated coaxial cable, in a short term view (think in nanoseconds) this is incorrect. When you touch your meter leads to the cable, those leads are at a different electrical potential when compared to the electrical potential between the center conductor and the shield. In order to equalize the potential between the test leads and the cable, current MUST flow. After all, you need charge to have an electrical potential and you need a current to create that charge. During the time where the initial energy applied to the cable is still propagating away from the meter, the meter will indeed read 50 ohms. The problem is, as Jens pointed out, you would either need a very fast multimeter or a very long cable to actually see 50 ohms on a display.

To look at it another way, a lossless cable can be approximated as an infinite series of "L" shaped series inductor, shunt capacitor pairs. When you think in terms of series inductors and shunt capacitors, you can easily imagine getting a different reading until all the L's and C's have stabilized at their steady state values.

To state that the meter would always read an open, requires information to travel faster than the speed of light. After all, how can the meter find out if a cable is terminated until it "queries" it with electrons? For the first 100 us, a 100 us long cable is going to look the same to a meter regardless of any open, short, or loaded conditions on the other end. To do otherwise would require information to travel faster than the wave propagating along the cable.

Characteristic impedance

Now back to the topic at hand...

Any media that can support a electromagnetic wave has a characteristic impedance associated with it. Although characteristic impedance units are in Ohms, it is not a "real" impedance you can measure using direct current equipment such as a DC Ohmmeter. And although transmission lines have real loss at microwave frequencies, this isn't what we're talking about either.

Let's add one caveat to the above paragraph.... waveguide must be treated differently from this whole discussion of characteristic impedance, and we'll get to that eventually. Technically, waveguide isn't really a transmission line, even though we often treat it as one!

The best way to think about characteristic impedance it envision an infinitely long transmission line, which means that there will be no reflections from the load. Placing an alternating current voltage V_in(t) will result in a current I_in(t). The impedance of the transmission line is then:

Sounds simple enough, but unless you are dealing with "free space", there is no transmission line that is infinitely long. But that equation is starting to look like a version of Ohm's law, where R=V/I.

Transmission line lumped element equivalent circuit

The transmission line model now has its own page.

Now let's look at the general equivalent circuit of an infinitesimally small piece of a transmission line. All circuits elements are normalized to length in transmission line models; in the metric system the units are Ohms/meter, Farads/meter, mhos/meter and Henries per meter, we will use the "prime" notation when we are discussing quantities that are normalized per unit length.

By the way, the transmission line model graphic can be downloaded in a Word document along with lots of other microwave schematic symbols, just visit our download page.

The T-line model is repeated infinite times along the length of a real transmission line. Hmm, this is starting to sound like calculus, which we have pledged to avoid on Microwaves101. And we will, so we'll stop with this one section. For microwave engineers, the general expression that defines characteristic impedance is:

Here R', G', L' and C' are normalized to length, the same as in the model. Note that in its general form, characteristic impedance can be a complex number. Also note that it only becomes complex if either R' or G' are non-zero, which will give you a headache if you think about it too long. In practice we try to achieve nearly lossless transmission lines. For a low-loss transmission line, the following relationships will occur:

Then for all practical purposes we can ignore the contributions of R' and G' from the equation and end up with a nice scalar quantity for characteristic impedance. For lossless transmission lines the transmission line model reduces to this:

and the more familiar equation for characteristic impedance is simply:

What are L' and C' to the lay person? L' is the tendency of a transmission line to oppose a change in current, while C' is the tendency of a transmission line to oppose a change in voltage. Characteristic impedance is a measure of the balance between the two. How do we calculate L' and C'? that depends on what the transmission line is. For example our page on coax give the coax equations.

Relationship of L' and C' to Z₀ and V_P

There are many situations where you need to know inductance per unit length and capacitance per unit length of a transmission line. Both can be calculated from the characteristic impedance and the propagation velocity of the wave in a transmission line. The key to solving these equations is that the propagation velocity of a transmission line is a very simple function of its capacitance and inductance per unit length:

Note than when you plug in inductance in Henries/meter and capacitance in Farads/meter, you are talking about one speedy wave, limited to be less than or equal to the velocity of light which is about 3x10E8 meters/second. From this equation and that for Z₀ (above) you can arrive at the following for L' and C':

Now the truth comes out... for a TEM transmission line such as stripline or coax, and for a given dielectric material (which would correspond to Keff in the above equations), the inductance and capacitance per unit length don't change when you scale the geometry up and down. So all semirigid, 50 ohm PTFE-filled cables (and PTFE-filled stripline!) will have 94.8 pF/meter (28.9 pF/foot) capacitance and 237 nH/meter (72.2 nH/foot) inductance! Let's state that as a Microwaves101 Rule of Thumb:

For coax and stripline 50 ohm transmission lines that employ PTFE dielectric (or any dielectric material with dielectric constant=2), the inductance per foot of is approximately 70 nH, and the capacitance per foot is about 30 pF.

Telegrapher's equations

This solution to the wave equation for transmission lines was developed by Oliver Heaviside a long time ago. We'll cover this later. You can look it up in Wikipedia for now!

Intrinsic impedance

Characteristic impedance does not even need a transmission line, there is a characteristic impedance associated with wave propagation in any uniform medium. In this case we use the Greek letter eta for impedance. The intrinsic impedance is a measure of the ratio of the electric field to the magnetic field.

Intrinsic impedance is calculated the same way as any transmission line. Assuming there are no "real" conductances or resistance in the medium, the equation is reduced to the simpler SQRT(L'/C') form. In this case the inductance per unit length reduces to the permeability of of the medium, and the capacitance per unit length reduces to the permittivity of medium.

Note that the approximation that ₀ is related to . There is no physical reason for this, it just happens that 1/36 gives better than 99.9% accuracy!

Impedance of free space

In space the terms for relative permeability and relative permittivity are each equal to unity, so the intrinsic impedance equation is simplified to the equation for characteristic impedance of free space:

Here's where the approximation involving 1/36 for permeability is what gives us that 120 value for free-space impedance (accurate to 99.9%!). Note that permeability and permittivity of the atmosphere on earth behave very close to free space (if it ain't raining!), so we use 377 ohms for the characteristic impedance of free space in most calculations involving atmospheric propagation.

Please feel free to forward comments or questions on this page, we want you to understand it!