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<Karl K. Berggren> is an <electrical engineer> whose research includes <superconductivity>-based <electrical circuits>, electronic devices and <single-photon detectors> for quantum applications. MIT's Research Laboratory of Electronics identifies him as a professor of electrical engineering and physics and as head of its quantum nanostructures and fabrication group.<Karl K. Berggren profile by MIT Research Laboratory of Electronics>[Biographical paragraph and research-focus paragraph]{ref} These research areas connect the behavior of very small devices to the circuits needed to read and control them.

MIT's identified https://www.rle.mit.edu/people/karl-k-berggren/[faculty profile] also links an externally hosted https://rledevs4.mit.edu/176702/photo%3Bsmall%3DFalse%3Bversion%3D20240416170829227[portrait of Karl K. Berggren].

His <6.200 Notes: Sinusoidal Steady State by Karl K. Berggren> explain why <complex numbers> simplify <linear differential equations> driven by sinusoidal inputs. For a series <RC circuit> observed across its <capacitor>, the equation $RC\,dv_C/dt+v_C=v_{\rm in}$ gives the late-time <frequency response>
$$
H(\omega)=\frac1{1+i\omega RC}.
$$
The notes obtain this result by solving for a <complex exponential> and taking its <real part>.<6.200 Notes: Sinusoidal Steady State by Karl K. Berggren>[Printed pp.10–11, Approach and Conclusion; p.3, decayed-transient assumption]{ref}

For an original illustration, $R=1\,\mathrm{k\Omega}$ and $C=1\,\mathrm{\mu F}$ give a <time constant> of $1\,\mathrm{ms}$. At $\omega=1000\,\mathrm{rad\,s^{-1}}$, $H=(1-i)/2$: a one-volt peak sinusoidal input produces a $1/\sqrt2$-volt peak output lagging by $\pi/4$. A separate term proportional to $e^{-t/(RC)}$ is the transient and depends on the starting state. Distinguishing that term from the forced sinusoid is what makes the steady-state calculation experimentally useful. This numerical illustration applies his teaching derivation; it is not a reported device measurement.


 Karl K. Berggren (source code)