Tegam 1505A, Instruction And Service Manual, Page: 21 / 27

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4-7
Section IV – Operating Instructions
Noting that if we collect the two terms beginning with
𝜌
𝑀
2
𝜌
𝐹
2 , we get,
1 − 2 𝜌
𝑀
𝜌
𝐹
cos( 𝜙
𝑀
+ 𝜙
𝐹
) + 𝜌
𝑀
2
𝜌
𝐹
2
(cos
2
( 𝜙
𝑀
+ 𝜙
𝐹
) + sin
2
( 𝜙
𝑀
+ 𝜙
𝐹
))
The term,
cos
2
( 𝜙
𝑀
+ 𝜙
𝐹
) + sin
2
( 𝜙
𝑀
+ 𝜙
𝐹
)
is always identically equal to 1, so the final simplified equation
becomes,
1 − 2 𝜌
𝑀
𝜌
𝐹
cos( 𝜙
𝑀
+ 𝜙
𝐹
) +
𝜌
𝑀
2
𝜌
𝐹
2
Or combining this result with the general transfer equation,
𝑘
𝑀
= 𝑘
𝐹
𝑃
𝑆𝑆𝑆𝑀
𝑃
𝑆𝑆𝑆𝐹
(1
− 2 𝜌
𝑀
𝜌
𝐹
cos( 𝜙
𝑀
+ 𝜙
𝐹
) +
𝜌
𝑀
2
𝜌
𝐹
2
)
If we look at the scalar result of the mismatch term, the “1” part is what would happen if at least one
of the ports was “perfect”, or had no reflection. In that case, one of the
𝜌
is zero. The right-most
element has magnitude of
𝜌
4 , which is typically so much smaller than the middle term that it can be
ignored for most connections.
The middle part,
2 𝜌
𝑀
𝜌
𝐹
cos( 𝜙
𝑀
+ 𝜙
𝐹
)
, contains the bulk of the impact of port mismatch. Since it is
multiplied by k, the sensitivity to this change is equal to k, or about 1. In an UNCORRECTED transfer,
this part represents the probable error of the transfer. Since we can’t know the angles in an
uncorrected transfer, we let cos() take its limits of +/- 1, and say that the uncertainty of the
uncorrected transfer is
2 𝜌
𝑀
𝜌
𝐹
. That is a little pessimistic, because that is worst-case rather than
probable, but it’s what the industry usually does. Unless the rhos were measured on a scalar analyzer,
we have to use manufacturer’s worst-case values.
Some common power sensor calibration practices do not always use gamma correction. An argument
could probably be made that this was a reasonable practice at lower frequencies. We typically see this at
18 GHz and lower. A value of 0 would be inserted for Γ
Μ
making that portion of the formula “1”.
This would make the formula look like:
𝑘
𝑀
= 𝑘
𝐹
𝑃
𝑆𝑆𝑆𝑀
𝑃
𝑆𝑆𝑆𝐹
This revision of the original formula assumes many things that are not necessarily true. Things that may
not necessarily show up at lower frequencies but will certainly show up at higher frequencies where
connectors change from the very rugged N-type connector to the more sensitive 3.5 mm and 2.4 mm
connectors.
By looking at Figure 4.4 we can get a practical visualization of the relationship between calibration
factor, gamma and effective efficiency .