Coaxial cables are an interesting piece of the signal measurement paraphernalia . They can virtually be found on every instrumentation table in labs and yet we pretty much overlook their function. They can be simply seen as a piece of wire or as a transmission line, all ultimately depending on the frequency of the signals flowing thru them.
Although high, the propagation speed of electric signals if finite, and in a coax with PFTE dielectric it is ~69% of speed of light. Thus, for instance, a coax is just a wire for a 50Hz signal, but a transmission line for a nanosecond-duration pulse. We can also say that the short pulse changes in time and in space but the 50Hz one changes in time only. This is a relative issue, though: look, if the coax is >4000km in length then, even a 50Hz signal changes in time but also in space. But, We better focus on the case of the short pulse traveling in a coax with lengths that you might have at hand!.
For an easier understanding of the propagation of a pulse in a coax a simple simulation can be of good help. For no particular reason, other than curiosity, I am using CST Microwave Studio for this example. As with any 3D modeling tool, the simulation comprises the construction of the geometry, assigning materials and setting the solver. Simulating a coax is something that will surely not rack your brains. The process is very easy with CST, plus tutorials are readily available to help start.
The model is as shown in the image below (also available to download here). A helpful remark is that one of the waveguide port (the red squares in the model) can be replaced by a discrete port. One would be interested in doing so if the purpose is to see the effect of a sudden change in characteristic impedance. You can set the impedance of the port to a value other than 50 Ohms. When the background box (black wire frame) encloses tightly the model geometry and you are using a discrete port then you can have the solver complaining like this:
“Staircasing failed for discrete edge port “2”: discrete edge port “2” is completely inside metal material.
Maybe geometry is too complex or mesh is too coarse in this region.
Please consider refining the geometry.”
Increasing in 1 mm or so the Z distance of the background should solve the problem.
… the input signal
A normalized Gaussian pulse is the excitation signal by default at the ports. The width and rise time are determined based on the frequency range entered when creating the CST project. In this case, from 0 to 1 GHz.
… the propagation of the pulse
Because the pulse travels at a certain speed then at a given instant of time the potential of the central conductor of the coax will have a particular value depending on the position along the Z axis (length of the coax). But then, how can we visualize such a propagation? Well, the electric and magnetic field inside the coax are well-known and easy to imagine. For the first, it can be represented as arrows pointing from the central conductor to the outer conductor (or sheath of the coax) with a decreasing magnitude obeying some exponential law. For the second, it can be drawn as arrows always perpendicular to the arrows representing the electric field and describing a trajectory that encloses the current. After all it is convenient that the coax has such a simple geometry.
The above figures recall basics of electromagnetism, that E and H field are always orthogonal. They are maybe not as interesting as to see the pulse actually propagating, although let us see that either E or H can represent the fly of the pulse. To get an animation of it, We must ask CST to save the values of E or H during the simulation time. In the tab Simulation, it is just a matter of clicking on Field Monitor and in the prompt window you tick the option E-field or H-field and Surface current in the Type group and tick the option Time in the Specification group.
Finally, We can have a dynamic representation of the propagation of the pulse in the coax!, available in the 2D/3D Results menu. After tweaking the cutting plane and choosing a proper cross section view, the propagation of the pulse can be seen as in the animation below.
Imagine for example that the animation is stopped at the middle of the total span of 4.6 ns. Then, You will be staring at a reddish area somewhere around the middle of the coax, say 50 cm. That reddish area represents the peak of H-field (or the current) and changes in color towards blue as the magnitude of the current decreases. The pulse, thus, travels along the coax and it is “absorbed” by the matched port 2.
… Characteristic Impedance
Above was mentioned that the propagation of the pulse could be observed resorting to either the E or the H field. Of course. In circuit quantities, E is equivalent to the voltage (voltage difference for accurateness), H is equivalent to the current and these two, V and I, are related by the characteristic impedance Zo: thus V = Zo*I.
We can probe V and I at different position to check the value of Zo. To do so, first, for the voltage, We need to draw a path whose ends are going to define V2 and V1. This because V=V2-V1. Since We are interested in the voltage in the conductor at a specific position, then an straight line starting from the inner (V1) to the outer conductor (V2=0) will do the job. Second, for the current, We need to draw a closed path that embraces the current-carrying conductor. The construction of 5 voltage monitors and 1 current monitor should look like in the image below.
If all goes well with the definition of the monitors, You must be able to see the the results under the menu 1D Results –> Voltage(Current) Monitors –> Signals.
On purpose I’ve set 5 voltage monitors 1/4th of the total length of the coax (200 mm) apart as an alternative to observe the propagation of the pulse. Notice then, that any two voltage peaks are 0.2428 ns apart, which corresponds to the expected propagation speed of 0.69 x (speed of light).
voltage3 is the voltage just at the middle point of the coax where the current monitor is also placed. Thus, the current peak occurs simultaneously with the voltage peak. The peak of this voltage is 7.1 V and the corresponding peak current is 0.14 A, which leads to a characteristic impedance Zo = 7.1 / 0.14 = 50.7 Ohms.
Quite a lot can be said about signals in coax cables. Here We simply exercised basics and got grips with CST!!!