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The group delay introduced by a capacitive amplifier load is the negative angular frequency derivative of the extra wave phase produced by that load. For the resistive output model in reactive-load phase shift in a finite-output-impedance amplifier, divide the loaded transfer function by the unloaded one:The added group delay is thereforeThis subtraction of unloaded wave phase prevents the amplifier's intrinsic group delay from being attributed to the capacitor. The terminal model uses a Thévenin equivalent; the group delay definition differentiates an unwrapped wave phase, as in the SciPy signal-processing documentation. Its digital-filter function reports samples, whereas the analog derivative above uses radians per second and gives seconds.^[ref][ref]

For an original example, choose $R=200\Omega$ and $C=10\mathrm{nF}$, hence $\tau =2\mu \mathrm{s}$. At $\omega \tau =1$, corresponding to $f=1/(2\pi \tau )\approx 79.6\mathrm{kHz}$, the added group delay is $1\mu \mathrm{s}$. The same operating point has $-{\varphi }_{\mathrm{rel}}/\omega =\pi \tau /4\approx 1.571\mu \mathrm{s}$: a single sinusoid's phase displacement and the delay of a narrow frequency packet need not agree.

A practical test applies a weakly modulated carrier, varies its central frequency, and measures the output modulation displacement while comparing loaded and unloaded operation. The group delay describes that narrow-band envelope only when the frequency response varies slowly across its bandwidth. The corresponding causal impulse response begins at zero time, $h(t)=e^{-t/\tau }/\tau$ for $t\ge 0$, and vanishes for $t<0$; it is not a rigidly translated impulse. Thus group delay versus pulse-front arrival for an RC load matters when a measured pulse is broad enough to change shape.

Group delay versus pulse-front arrival for an RC load distinguishes a narrow-band modulation displacement from the earliest response to a newly applied input. For the relative amplifier-load model in group delay introduced by a capacitive amplifier load, the causal impulse response is $h(t)=e^{-t/\tau }/\tau$ for $t\ge 0$ and zero for $t<0$. Its convolution with an input that vanishes before zero also vanishes before zero. Nevertheless its group delay is $\tau /[1+(\omega \tau {)}^2]$, which is positive and frequency dependent. The phase-derivative definition concerns spectral components and their envelopes.^[ref]

An original finite-pulse calculation shows why a threshold measurement gives another time. For an initially relaxed load, apply a rectangular pulse of height $V_0$ and duration $T>0$. Convolution givesThe first mathematical response occurs immediately after zero. A rising threshold equal to fraction $a$ of the input pulse height is reached at $t_a=-\tau \log (1-a)$ only if the pulse lasts at least that long. This is the threshold-crossing time of a finite rectangular pulse through an RC load.

With $\tau =1\mu \mathrm{s}$ and $T=0.1\mu \mathrm{s}$, the largest output is only $(1-e^{-0.1})V_0\approx 0.0952V_0$: a $0.5V_0$ threshold is never reached. If $T=3\mu \mathrm{s}$ instead, that threshold is crossed at $0.693\mu \mathrm{s}$. Neither value is the $0.5\mu \mathrm{s}$ group delay evaluated at $\omega \tau =1$.

A pulse generator and oscilloscope can compare the threshold observation with a slow modulation on a carrier; the chosen threshold and modulation bandwidth must be recorded. MIT's laboratory likewise separates square-wave decay measurements from sinusoidal amplitude/phase measurements.^[ref] If wiring adds a transport delay $d$, shift the impulse response to $h(t-d)$: the front moves to $d$, while the modulation delay becomes $d+{\tau }_g$. The immediate onset of the unshifted lumped model is not a model of propagation along a finite-length cable.