Electric Current — Question 6

Question 6

A shaped pulse fixes its peak from a specified transported charge

A unidirectional pulse obeys $I(t)=I_p\sin(\pi t/T)$ for $0\le t\le T$ and is zero otherwise. It must deliver positive charge $Q_0$.

1. Find the required peak current.

2. Find the fraction delivered by $t=T/3$ and the time when half is delivered.

Question 6 - Solution

A shaped pulse fixes its peak from a specified transported charge

(a) Method and calculation

$$Q_0=\int_0^T I_p\sin(\pi t/T)\,dt=2I_pT/\pi.$$$$I_p=\pi Q_0/(2T).$$

(b) Interpretation and further calculation

$$Q(t)/Q_0=\tfrac12[1-\cos(\pi t/T)],\quad Q(T/3)/Q_0=1/4.$$$$Q(t_{1/2})/Q_0=1/2\Rightarrow t_{1/2}=T/2.$$

Physical check

The mean pulse current is smaller than its peak by the factor 2 divided by pi.