 Group delay introduced by a capacitive amplifier load

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.