= Group delay introduced by a capacitive amplifier load
{title2=$\tau_g(\omega)$}
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:
$$
H_{\rm rel}(\omega)=\frac{1}{1+i\omega\tau},\qquad \tau=RC>0,\qquad \phi_{\rm rel}=-\arctan(\omega\tau).
$$
The added <group delay> is therefore
$$
\tau_g(\omega)=-\frac{d\phi_{\rm rel}}{d\omega}
=\frac{\tau}{1+(\omega\tau)^2}.
$$
This 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.<Thevenin and Norton Equivalents, University of Houston teaching notes>[Printed pp. 1–2, terminal equivalence]{ref}<SciPy 1.16.2 documentation>[scipy.signal.group_delay, definition and Parameters: fs; discrete-frequency units]{ref}
For an original example, choose $R=200\,\Omega$ and $C=10\,\mathrm{nF}$, hence $\tau=2\,\mathrm{\mu s}$. At $\omega\tau=1$, corresponding to $f=1/(2\pi\tau)\approx79.6\,\mathrm{kHz}$, the added <group delay> is $1\,\mathrm{\mu s}$. The same operating point has $-\phi_{\rm rel}/\omega=\pi\tau/4\approx1.571\,\mathrm{\mu s}$: 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 $t\ge0$, 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.
Back to article page
