HOW LOW CAN YOU GO? K

We measure S-units on our radios ’S-meters. But “Sparameters ” are different, andparticularly importantto understandwhen a signalisgoing to more than oneplace. K3MTexplains.

S Parameters Made Easy

BY MICHAEL TOIA, * K3MT

Consider this: A 17-meter full-wave loop, fed by 75-ohm coax threaded through a balun of several ferrite beads, produced a 4:1 SWR, and the rig could not drive full power into it. It needed a tuner of some sort. The SWR is just a clue: It doesn’t give enough data to do much. We need the Sparameter, S11, of the coax. Why? Read on …

So what is an S parameter, and how many are there? They’re scattering parameters of a network. Let’s demonstrate with a cable TV 1-to-4-way splitter that scatterscable TV signals to four rooms of a house. It’s five type F coax connectors (Figure 1) show why it’s called a five-port network. They’re labeled ports 1 to 5, and a 75-ohm load, the video system Zo, is connected to the output ports 2 to 5. Then port 1 presents 75 ohms to a signal generator.

The generator drives a voltage into port 1, Vin 1. This signal scatters about, some lost to heat, and the remainder going out the four remaining ports as voltages Vout 2, Vout 3, Vout 4, and Vout 5. These are usually not in phase with the input, because of a time delay in traversing the network.

The ratio Vout 2 / Vin 1 is called S21. S31, S41, and S51 are similarly defined. The first subscript refers to the output port, the second to the input port. These four are just ratios of two voltages which can differ in phase, so are complex numbers with a magnitude and a phase delay. Since the output power can’t exceed the input power, Vout is never greater than Vin. An S parameter is never greater than 1.

If the signal generator is connected differently, such as to port 2 (as shown in Figure 2), the same measurements yield more S parameters, S12, S32, S42, and S52. By moving the generator to any of the five ports, you obtain the splitter’s 20 S parameters. It has five more.

* <k3mt@jokalympress.com> <www.jokalympress.com>

A directional coupler connected between the generator and port 1 (Figure3) finds the signal reflected back out of the port, Vout 1, across its load resistor. The ratio Vout 1 / Vin 1 is the famous S11. Similarly, you may measure S22, S33, S44, and S55 by moving the generator and loads about. These are the splitter’s five reflection coefficients. So this five-port network has 25 S parameters: a network with N ports has N2 S parameters.

Back to Our Loop Antenna

The loop antenna has only one port, and a single S parameter, S11. If it’s zero, the reflected voltage is zero. All the input is lost in the antenna, some as

Figure 1. Five-port (4-way) cable TV signal splitter with a signal generator connected to the input(Port 1).

Figure2. Thesamesplitterwiththesignalgeneratorconnectedtooneoftheoutputports (see text for discussion).

heat in resistive losses, the rest to radiation into the Aether. The antenna is perfectly matched.

The coax has two ends, its ports: it’s a two-port network. If you know S11, you can calculate SWR, and design a tuning system, using a Smith chart, a result of plotting S11 on a graph as a vector with magnitude and phase angle (Figure 4).

We calculate the standing wave ratio, SWR, from the magnitude of S11 like this: SWR = [ 1 + |S11|] /[ 1 - |S11|]

If S11 = +1.0, the output voltage equals the input voltage. All input is reflected back, and all applied power heats the coupler’s load resistance. None actually enters the antenna. The antenna fitting is an open circuit. Similarly, if S11 = -1, all input voltage is again reflected, 180° out of phase, and all applied power is soaked up in the coupler’s load. The antenna fitting is shorted.

There are standing waves of both voltage and current on the coax. Too many hams insist on calling this ratio “VSWR. ” Please … it is simply “SWR. ”

A Few More Examples

How about a few more examples? An attenuator is a twoport network. S11 indicates the signal reflected to the coupler’s load resistor, and S21, the signal into the output load. A proper attenuator is matched in impedance, with S11 = 0.

S21 is its voltage attenuation. Attenuators are almost always bilateral, where S11 = S22 and S12 = S21. That is, “you can

Figure 3. Adding a three-way coupler to the network.

Figure 4. A Smith chart showing S11 on a graph as a vector with magnitude andphase angle.