 Karl K. Berggren

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.^[ref] These research areas connect the behavior of very small devices to the circuits needed to read and control them.

MIT's identified faculty profile also links an externally hosted 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 $RCd{v}_C/dt+v_C=v_{\mathrm{in}}$ gives the late-time frequency responseThe notes obtain this result by solving for a complex exponential and taking its real part.^[ref]

For an original illustration, $R=1\mathrm{k\Omega }$ and $C=1\mu \mathrm{F}$ give a time constant of $1\mathrm{ms}$. At $\omega =1000\mathrm{rad}{\mathrm{s}}^{-1}$, $H=(1-i)/2$: a one-volt peak sinusoidal input produces a $1/\sqrt{2}$-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.