To: file

From: Jon Hagen

Date: 4-24-03

Subject: Baseline ripple caused by an improperly terminated splitter

1. Ripple

The Downstairs IF/LO System has several 8-way amplifier units, allowing several back-ends to be left connected, whether or not more than one back-end is in use at any time. In each unit, one of the eight outputs feeds a front panel monitor port, which is often left unterminated. The other seven ports feed rear panel connectors, which are terminated by back-end processors or by 50- Ohm loads. It was noticed that the base line ripple, as measured by back-end processors was considerably worse when the front panel monitor port was left unterminated, even though there was an isolation amplifier between the splitter output and this front panel port. (Ripple attributable to the unterminated port can be made obvious by connecting the port to a long unterminated cable whose reflection coefficient, ei , changes in phase w.r.t. frequency, producing many ripples across the passband).

Analyzing this ripple (see next section) under the assumption that the splitter is a perfect (a tree of ideal hybrid junctions, each with its fourth port terminated) shows that

Maximum pk-to-pk power ripple in dB =

6 + splitter's one-way splitting loss in dB

+ 0.5 x return loss in dB of splitter's signal source

+ 0.5 x return loss in dB of the improper termination

as seen by the splitter. 1)

The factor 0.5 that multiples the return loss of the improperly terminated splitter port is due to the fact that the ripple power is proportional to the reflection coefficient of the improper termination instead of being proportional to the square of the reflection coefficient. This dependence on the return loss of the improper termination should be the same when the ripple is entirely or partially caused by imperfections in the 8-way splitter.

Phil Perillat and Lisa Wray looked at the passband in two cases. In both cases the monitor port was terminated with a narrow band filter which, outside its passband, provided 100% reflection with a frequency dependent phase (like a long unterminated cable).

In the first case, an 11dB amplifier isolated output Port 8 of the splitter from the front panel monitor jack. This amplifier had -17 dB of reverse 'gain'. This amplifier should reduce the reflection coefficient of the totally reflective open termination from 0dB to 11-17 = -6dB. (This simple calculation assumes that the amplifier's output VSWR is close to unity. See Appendix 1).

In the second case, the 11dB amplifier was replaced by an amplifier having 24dB of forward gain and -30 dB of reverse gain, together with a 10 dB pad at its input and a 3 dB pad at its output. This combination should reduce the reflection coefficient of the totally reflective termination from 0dB to (-3 +24 -10 -10 -30 -3)dB= -32dB.

Using Equation 1, the relative change in the ripple amplitude between the first and second cases should be 0.5(32-6)= 13dB. Given the simplifying assumption that the amplifiers had low output VSWR, this is in good agreement with the 12dB difference measured by Phil and Lisa.

2. Ripple Calculation

At the left side of Figure 1 is an N-way power splitter for N=8, a canonical tree of 3db hybrid junctions and loads. Note that power reflected back into any of the output ports (Ports 2 through N+1), will exit through the input port (Port 1), reduced by a factor N, but will not exit through any of the other output ports. The output ports will therefore be perfectly decoupled if power exiting Port 1 is not reflected back into Port 1, i.e. whatever drives Port 1 should ideally have an output VSWR of 1:1.

Figure 1 An N-way Power Splitter

Left: An 8-way Splitter realized with 3dB hybrids

Right: An abstract (S-parameter) representation

The right side of Figure 1 shows an N-way splitter drawn for S-parameter (voltage transfer coefficient) analysis. Each port is drawn as two lines: an input wave and an output wave. The output ports are given arbitrary terminations, specified by the reflection coefficients 2 through N+1. The source impedance is specified by the reflection coefficient 1 . The source is providing a unit input signal. For an ideal splitter, Snm = 0 when both n nor m differ from unity; the splitter's S-
matrix is all zeros except for the first row and first column. Except for S00, which is also zero, the remaining S parameters all have magnitude N-1/2. By inspection of Figure 1, we can write

Substituting for the y's, we have

Solving for x, we find

where we have used the fact that Sn,1 = S1,n. Suppose now that we are analyzing the spectrum seen at Port 2 and that Port 3 is connected to the improper reflective termination. The voltage V2, at Port 2 is given by

We will assume that all the n are small, so that

The power is proportional to V22 ,

From Equation 7) we see that the relative peak to peak ripple in the power caused by the improper termination, 3, will be

Note the linear dependence on 3.

3. Signal injected into monitor port

Phil also did the experiment of injecting a weak signal into the monitor port to see how much would leak into the analysis channel. By inspection of Figure 1, we see that the power injected at the monitor port will exit the analysis port, attenuated in dBs by

AttendB = A1 dB - reverse gain of isolation amp in dB + A2dB + 20 log|S1,3| + ( 1)dB +20 log|S2,1|

where A1 dB the attenuation of any attenuator between the amp and the monitor jack and A2dB is the attenuation of any attenuator between the amp and the splitter output. In Phil's two cases, the difference would have been

0 + 17 + 0 vs. 3 + 30 +10 or 17dB vs. 43dB

or a difference of 26dB, which seems to be just what Phil observed.

Appendix 1 Return loss of a load preceded by an attenuator, isolation amplifier, and attenuator.

This arrangement is shown schematically below in Figure 2.

Figure 2. Load isolated by attenuators and an amplifier

In Figure 2, a2 is the voltage gain of attenuator ATTEN2. (If the attenuation is 20dB, a2 is 1/10).

Same goes for a1. Inspection of the figure and a few steps of algebra leads to the result

Note added 9-2-03

Phil deemed the 12 dB ripple improvement to be insufficient; baselines were still noticably affected by poor terminations on the front panel output. However, an additional 13 dB can be obtained by using both the 11dB amplifier (Cougar AC559) and the 24 dB amplifier (Cougar AC582).

A suggested combination, starting from the power divider is:

1.	10dBpad

2.	Cougar AC582 (24dB gain, -30dB reverse gain)

3.	6 dB pad

4.	Cougar AC559 (11dB gain, -17dB reverse gain)

6.	6 dB pad

7.	Front panel jack

Memo to: file from: Jon Hagen Date: 4-24-03 Subject: Baseline ripple caused by an improperly terminated splitter