Sinusoids And Phasors
Sinusoids are defined as the mathematical waveforms that are used to describe the nature of periodic oscillations.
Circuit Theory
Electric circuits are a network that comprises of a closed-loop, which helps in providing a return path for the current through a switch. When the switch is activated, the load operates, and the current accepts a path to finish the circuit at a low potential level from the opposing high potential level. Electric circuits theory is a linear analysis that helps in establishing a linear relation of voltage and current for R (resistance), L (inductance), and C (capacitance).
The voltage source is at a constant level for a long time, which we approximate as infinite. This is the circuit:
The voltage drop across the capacitor rises from 0 to ℰ. Note that ℰ is never actually known in the measurement. In fact, the oscilloscope voltage is decalibrated, so that, whatever ℰ is, ℰ is at the top line while zero is at the bottom line. We don't measure voltage levels, but rather 1/2, 1/4, and 1/8 the maximum.
Kirchhoff's voltage law give: ℰ = IR + Q/C or the following:
dQdt=−1RC(Q−EC)dQdt=−1RC(Q−ℰC)
The solution for the capacitor voltage is
VC(t)=E(1−e−t/RC)VC(t)=ℰ(1−e−t/RC)
The voltage is zero at t = 0, t is the rising time, and you have to know when the rising begins.
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