VNA concepts and features
R&S
®
ZNL/ZNLE
229
User Manual 1178.5966.02 ─ 19
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.
9.3.8
Group delay
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 group delay is computed as
f
deg
meas
g
360
,
where ΔΦ/Δf is the slope of the regression line through the frequency points of aper-
ture Δf.
Δf must be adjusted to the conditions of the measurement, e.g. it must be reduced if
phase slope fluctuates significantly over frequency. Otherwise group delay variations
are flattened out.
Note that the input value "Aperture Points" does not define the number of frequency
points, but the number of frequency steps between the points. I.e. "Aperture Points" is
always the number of frequency points involved minus 1. If "Aperture Points" is an odd
number, the part of the aperture towards lower frequencies contains one more fre-
quency step than the part towards larger frequencies.
If the group 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
Measurement results
