The Light-Emitting Diode (LED) — Question 7

Question 7

Carrier storage limits the optical modulation bandwidth

Small excess population changes obey $d\delta N/dt+\delta N/\tau=\delta I/e$, with constant $\tau=10.0\,\mathrm{ns}$. Optical modulation is proportional to $\delta N$. Use phasors proportional to $e^{i\omega t}$.

1. Derive the normalized modulation transfer function and its minus-3-dB frequency.

2. At 30.0 MHz, calculate the amplitude ratio and phase lag.

Question 7 - Solution

Carrier storage limits the optical modulation bandwidth

(a) Method and calculation

$$(i\omega+1/\tau)\widetilde N=\widetilde I/e\Rightarrow\boxed{H(\omega)=1/(1+i\omega\tau)}.$$$$|H|=1/\sqrt{1+\omega^2\tau^2}\Rightarrow\boxed{f_{3\mathrm{dB}}=1/(2\pi\tau)=15.92\,\mathrm{MHz}}.$$

(b) Interpretation and further calculation

$$\omega\tau=2\pi(30\times10^6)(10^{-8})=1.885\Rightarrow\boxed{|H|=.4686}.$$$$\boxed{\arg H=-\tan^{-1}(1.885)=-62.05^\circ}.$$

Physical check

This first-order result neglects electrical parasitics and assumes linear recombination about the bias point. A nonlinear rate law requires the differential lifetime there.