Instrumentation & Measurement Magazine 25-7 - 28

Fig. 1b shows on the one hand that a VNA has the ability to
measure very low magnitude reflected waves (bj), but on the
other hand, this uncertainty limits the impedance's ranges that
can be measured with a given accuracy. These ranges are computed
in the next sub-sections with a theoretical analysis that
is followed by an error analysis.
Theoretical Analysis
The theoretical expressions of the S21
Fig. 2. Validity domains of the reflection technique for measuring resistances
with a relative error lower than (a) 10% and (b) 100%.
Fig. 2b shows particularly that a sign inversion error (corresponding
to a measurement error of 100%) is encountered
when measuring positive resistances lower than 0.5 Ω. This
limit proves the inability of the reflection technique for measuring
an ESR lower than 0.1 Ω that typically characterizes a
100 pF ultra high-Q capacitor. Consequently, when measuring
an impedance in which the real part reaches one of the
limits given by (7), (8), one should avoid the One-Port reflection
technique and look for a more accurate solution such as
the Two-Port transmission technique that is addressed in the
next section.
The Two-Port Transmission Technique
Principle
Equation (2) shows that the sign inversion error that plagues
the One-Port reflection technique exists because, when measuring
extreme impedances, the magnitudes of the reflected
and the incident waves are quite identical ( ba ). This error
can be avoided with the Two-Port transmission technique if
the extreme impedances to be measured are connected in such
a way that the magnitudes of the reflected and the incident
waves are very different

ba . Hence, if the resistance to be
measured is very low, the shunt admittance insertion shown in
Fig. 3a must be employed; conversely, if the resistance is very
high, it is the series impedance insertion represented in Fig. 3b
that must be selected.
In both cases, the circuits in Fig. 3 are strong attenuators that
satisfy the condition: ji
ba . The transmission uncertainty in
parameters and the impedances
(or admittances) measured with the circuits shown in
Fig. 3 have been established by [5]. Written as a function of the
normalized admittance and impedance
y Y Y z ZZ ,

/ ,CC /

these expressions are defined for the shunt admittance insertion
represented in Fig. 3a:
S
2
21 
y YY
SS
 
21
21
11
22C
S
2
2  y
 21
21
SS
(9)
(10)
and for the series impedance insertion represented in Fig. 3b:
21

2  z
z ZZ
SS
 
21
21
11
22 C
 21
21
SS
(11)
(12)
A comparison of (9) and (10), the expressions specific to
the shunt admittance insertion, and (11) and (12), the ones
of the series impedance insertion, shows that they have the
same form since they are expressed with dual variables (admittances
and impedances). Therefore, the computations
can be reduced by a method that consists in deriving the
theoretical expressions for one insertion mode and then
extending the results to the other by an application of the
duality principle. Considering first the shunt admittance
insertion, in the particular case of a very high normalized
admittance, (9) shows that the corresponding S21
parameter
has a very small magnitude. Hence, (10) can be approximated
as:
y YY
SS
22
 
21
C
21
(13)
Fig. 3. Two-Port transmission measurement of extreme impedances. (a) Shunt admittance insertion for measuring low impedances; (b) Series impedance insertion
for measuring high impedances.
28
IEEE Instrumentation & Measurement Magazine
October 2022

Instrumentation & Measurement Magazine 25-7

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