Electronic – Finding the voltage and current at t = 0 using Step Response

For a shared current into a resistor and capacitor you might be tempted to say: -

\$V_{SUPPLY} = V_R + V_C\$ (incorrect)

This would not be true because the voltage across a capacitor does not rise and fall sinusoidally as the current rises and falls sinusoidally. For a capacitor the current and voltage looks like this: -

_{(source: electronics-tutorials.ws)}

In other words it is 90 degrees out of phase with voltage. This is because the basic formula for a capacitor is

\$I = C\dfrac{dV}{dt}\$

And, if V is a sinewave voltage then I has to be a cosine current.

If instead of real waveforms we drew them as phasors we would represent the voltages and current like so: -

So now if we want to "relate" Vsupply to the individual voltages of the capacitor and resistor we have to add them using pythagorous i.e.

\$V_{SUPPLY} = \sqrt{V_R^2 + V_C^2}\$.

It follows from this that impedances also add this way.

One way to go about solving circuits with switches is looking at the circuit with the switch on and off. Use superposition. Pretend like your working with two different circuits.

Solve the circuit with the switch off (no current will flow through R so you can remove (or ignore elements R and E)) You will end up with a capacitor that is charging up so for the first circuit you will have a circuit with initial conditions and a steady state (meaning after the capacitor is fully charged.

Then you have the other circuit with the switch on, you need to solve it with the cap fully charged for initial conditions of the second circuit, which was the steady state conditions of the first. I'll repeat that again, the cap will charge with the switch off, it will then have a voltage on it, the switch will turn on and then it will reach a different voltage value.

This is a great exercise because it wraps your mind around two different concepts at the same time and if studied enough should leave you with an intuitive feel for time varying circuits. Learn onward...