# Instrumentation & Measurement Magazine 25-7 - 27

```Fig. 1. Uncertainties specified by Keysight for the VNA: PNA Series: N3382A, using the Calibration Kit: 85032F. (a) Reflection uncertainty; (b) Transmission
uncertainty.
Using the Uncertainty (linear) = U(lin) specified in Fig. 1a,
the relation between the measured and the actual reflection coefficient
is:
11MA U lin
SS 
11

(3)
From (1), the expression of the measured normalized resistance
is:
r
M 
1
1
S M
S M
11
11
(4)
Substituting (3) into (4) and replacing the actual reflection coefficient
by its approached expression (2) yields:
rr
MA
U 
2
lin
(5)
For a given relative error objective /,rr the measured normalized
resistance is:
rr
MA 1

r
 

r
(6)
The minimum actual resistances RA(min) that can be measured
with a given relative error are attained when (5) and (6) are
identical, i.e.:
October 2022
RZ
A   C
min
2
Ulin
R
R



(7)
Following the same (but dual) calculation procedure, one can
find the maximum actual resistances that can be measured
with a desired relative error:
2
RZ
A   C U
max
R
lin



R
(8)
To give an insight, we computed the extreme resistances
that can be measured with two meaningful relative errors: 10%,
that can be considered as a practical error limit for a measurement
technique; and 100%, which yields the unacceptable sign
inversion error. Furthermore, knowing that the objective of
this work is to measure the ESR of a 100 pF ultra high-Q capacitor,
which is characterized by a first Self Resonant Frequency
(SRF1) of about 1 GHz, and an ESR lower than 0.1 Ω. One can
see on Fig. 1a, that for a frequency of 1 GHz and a magnitude of
the reflection coefficient close to 1 (obtained when measuring a
very low resistance), the reflection uncertainty is: U(lin)
Substituting this value and ZC = 50 Ω in (7), (8) yields the validity
domains of the reflection technique represented in Fig. 2.
≈ ± 0.02.
IEEE Instrumentation & Measurement Magazine
27
```

# Instrumentation & Measurement Magazine 25-7

## Table of Contents for the Digital Edition of Instrumentation & Measurement Magazine 25-7

Instrumentation & Measurement Magazine 25-7 - Cover1
Instrumentation & Measurement Magazine 25-7 - Cover2
Instrumentation & Measurement Magazine 25-7 - 1
Instrumentation & Measurement Magazine 25-7 - 2
Instrumentation & Measurement Magazine 25-7 - 3
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Instrumentation & Measurement Magazine 25-7 - 60
Instrumentation & Measurement Magazine 25-7 - 61
Instrumentation & Measurement Magazine 25-7 - Cover3
https://www.nxtbook.com/allen/iamm/26-6
https://www.nxtbook.com/allen/iamm/26-5
https://www.nxtbook.com/allen/iamm/26-4
https://www.nxtbook.com/allen/iamm/26-3
https://www.nxtbook.com/allen/iamm/26-2
https://www.nxtbook.com/allen/iamm/26-1
https://www.nxtbook.com/allen/iamm/25-9
https://www.nxtbook.com/allen/iamm/25-8
https://www.nxtbook.com/allen/iamm/25-7
https://www.nxtbook.com/allen/iamm/25-6
https://www.nxtbook.com/allen/iamm/25-5
https://www.nxtbook.com/allen/iamm/25-4
https://www.nxtbook.com/allen/iamm/25-3
https://www.nxtbook.com/allen/iamm/instrumentation-measurement-magazine-25-2
https://www.nxtbook.com/allen/iamm/25-1
https://www.nxtbook.com/allen/iamm/24-9
https://www.nxtbook.com/allen/iamm/24-7
https://www.nxtbook.com/allen/iamm/24-8
https://www.nxtbook.com/allen/iamm/24-6
https://www.nxtbook.com/allen/iamm/24-5
https://www.nxtbook.com/allen/iamm/24-4
https://www.nxtbook.com/allen/iamm/24-3
https://www.nxtbook.com/allen/iamm/24-2
https://www.nxtbook.com/allen/iamm/24-1
https://www.nxtbook.com/allen/iamm/23-9
https://www.nxtbook.com/allen/iamm/23-8
https://www.nxtbook.com/allen/iamm/23-6
https://www.nxtbook.com/allen/iamm/23-5
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