VNA Concepts and Features
R&S
®
ZNL/ZNLE
230
User Manual 1178.5966.02 ─ 07
stability. This means that μ
1
and μ
2
provide direct insight into the degree of stability
or potential instability of linear circuits.
References: Marion Lee Edwards and Jeffrey H. Sinsky, "A New Criterion for Linear 2-
Port Stability Using a Single Geometrically Derived Parameter", IEEE Trans. MTT, vol.
40, No. 12, pp. 2303-2311, Dec. 1992.
8.3.8
Delay, Aperture, Electrical Length
The group delay τ
g
represents the propagation time of wave through a device. τ
g
is a
real quantity and is calculated as the negative of the derivative of its phase response.
A non-dispersive DUT shows a linear phase response, which produces a constant
delay (a constant ratio of phase difference to frequency difference).
The group delay is defined as:
df
d
d
d
deg
rad
g
360
where
Φ
rad
, Φ
deg
= phase response in radians or degrees
ω = angular velocity in radians/s
f = frequency in Hz
In practice, the analyzer calculates an approximation to the derivative of the phase
response, taking a small frequency interval Δf and determining the corresponding
phase change ΔΦ. The delay is thus computed as:
f
deg
meas
g
360
,
The aperture Δf must be adjusted to the conditions of the measurement.
If the delay is constant over the considered frequency range (non-dispersive DUT, e.g.
a cable), then τ
g
and τ
g,meas
are identical and:
c
L
t
f
d
t
f
d
mech
g
360
)
360
(
where Δt is the propagation time of the wave across the DUT, which often can be
expressed in terms of its mechanical length L
mech
, the permittivity ε, and the velocity of
light c. The product L
mech
· sqrt(ε) is termed the electrical length of the DUT and is
always larger or equal than the mechanical length (ε > 1 for all dielectrics and ε = 1 for
the vacuum).
Measurement Results
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