Supporting docs:

The Smith Chart link above leads to a wealth of information, especially see "How Does a Smith Chart Work?" (Nelson).

Here we play with a Directional Coupler and see if it works as advertised. The device is a Minicircuits PDC-20-3 which has 3 ports, IN, OUT and CPL. I'm going to use it as a Reflectometer to check transmission line termination. Here is the setup:

The 8-pin coupler can is mounted on a piece of perf board with some SMA connectors and the -12db attenuation pad (3 resistors). All I'm using here is a signal generator and an oscilloscope watching the source signal from the splitter, and the return signal from the coupler. Since the source signal is way bigger than the return signal we use 20db PGA gain in the USRP to get them both visible on the scope again. Also, I notice the directional coupler inverts the reflected signal. This wasn't mentioned in the docs but is easily verified:

In the above I'm feeding from the splitter into the coupler IN port, a 50ohm load on the OUT port, and measuring the CPL port in channel 1&2. In the oscope, what is labled "channel 1" and "channel 2" is the signal on J19, or the CPL port, and channel 3 & 4 is the source signal from the splitter. Anyway, you can see that, for instance, ch1 & ch3 are 180 degress out of phase, as are ch2 & 4, they cross zero at the same spot. The signal undergoes -19db through the coupler, then +20db in the pga, and you can see it is slightly amplified.

All these experiments are done at frequencies less than 10Mhz. In fact, at 8Mhz, the smallest lumped inductor in my junk box, 1uH, has 50 ohms of reactance ( X = 2πfL ). Now, wired up as in the above diagram, first look at a normal 50 ohm termination load - it should have little reflected power:

and you can see the reflected power in channel 1 & 2 (from J19 above) is very small. That's with a 51ohm resistor - it can be adjusted even smaller. Just putting my fingers across it makes an even better match. This would be a point in the very center of the Smith chart.

Next, an OPEN termination, or R=∞. This should reflect 100% IN PHASE, or, since the coupler inverts the reflected signal, it should show up in the scope at max level and 180 deg out:

and indeed it is. This would be a point on the far right of the Smith chart. Next, a SHORTED termination - this should reflect 100% OUT OF PHASE, and appear as in phase through the directional coupler:

In a purely resistive system, as the load varies from 0 to 250 or so Ohms, with no reactance, the point on the Smith chart should move from the far left for zero ohms, short circuit, towards the center - that is, remaining 180 deg out of phase, but getting smaller and smaller in magnitude, until at 50ohms we reach the center. As the load resistance gets larger than 50ohms the reflected signal again gets larger and larger in magnitude but now is in phase. This behavior matches the formula for reflection coeffecient: Γ = (Zl - Zo)/(Zl + Zo) where Zl is the load impedance and Zo the transmission line impedance. If Zl is less than Zo, Γ is negative and tends toward -1 as Zl approaches zero. If Zl is greater than Zo, Γ is positive and tends toward 1 as Zl approaches infinity in a purely resistive system.

Also, we now know experimentally what the maximum reflected power is - looks like about 500 on the scope. Theoretically, the reflected signal, relative to the reference from the splitter, is going through -12db in the attenuator. 100% is being reflected in the open and short circuit termination, and -19db of the reflected power is fed to the CPL port, with the rest applied to the load side of the attenuation. The reflected power sample then gets +20db gain in the USRP pga. That's -12-19+20 = -11db. The source signal appears to be about 1800 on the oscope above. -11db of 1800 as a VOLTAGE ( using db = 20*log(v1/v2) is 452.

For the next experiment, I want a point at the top of the Smith chart, where R=0 but X = 1 Inductive ( Z = 0 + 1j). Since the system Z is 50, that would be an inductive reactance of 50 = 2 *

Above, we get the max R=0 amplitude corresponsing to the outer ring of the Smith chart, and looks like the +90 deg phase shift (advance) of the reflected power (remember, the reflected power is phase inverted - look at the open signal above and advance (move to the left) the reflected signal by 90 deg, to match this one).

Finally, just the opposite to collect all four corners of the compass, let X = 50 capacitive, or 50 = 1 / ( 2*

Which shows a nice, almost -90 deg delay (shift to the right) of the reflected power, at the R=0 amplitude. At this point, since we have shown the response to be as expected, or at least we know WHAT to expect for known values, we can now measure the phase and amplitude of a reflected signal and calculate the appearent impedance of any load.

More ideas - since we have X(l) and X(c) both = 50 we can try some combinations. Putting both in series we should have a Series Resonant circuit at f=3.9Mhz:

Which should appear as a short circuit, R=0, with the reactances cancelling out, like the scope of a short above. Well, you can see there is some error, the lines don't cross right at zero, and looks like there's not enough advance. To advance LEFT around the Smith chart, keeping R=0, we use LESS inductance. Look at the chart - more inductance gives you less phase shift, less inductance gives more. Makes sense when you remember that Z=∞ gives NO phase shift. So I put a 1.5uH in place of the 2uH:

Now that looks like the short circuit, altho R looks to be something more than zero with the diminished magnitude.

Then, we can try putting them in parallel, which should look like an open at f=3.9e6 (again using the 1.5uH):