RC network solving for the exponential equation
1using Node parallel analysis and then KVL series analysis.
Shows why Node analysis is simple, while KVL series
analysis is more difficult, when writing the Initial equation
of the RC network.
Another way to do the Node analysis :
Assume currents that exit the node are +pos
while currents that enter the node are -neg.
write All Terms on the Right side of = symbol :
0 = V/R - - C dv/dt
to interpret the above Eq.
V/R exits the top node and is +pos ohm's law.
the first -neg sign says that Cap current is
entering the top node and this also suggests
current is moving C.W. direction entering the
Cap at it's lower side, the -neg side, thus
-neg passive sign for the cap.
Even if the Book really ment to do that,
they would most likelh NOT have shown the - -
instead , the book might just show :
0 = V/R + C dv/dt
and
once again obfuscate the inner details
of equation writing by NOT showing
how the +-signs consolidation.
The above allows the circuit diagram to have
only one current arrow i , shown in the C.W.
direction for this node analysis .
The Cap is the only source of voltage, thus
seems reasonable that Cap current Enters
the top node and then exits the top node
down through resistor R .
( if we want to view it that way )
this should have been noted.
Integrals that define Cap voltage:
1/C S i dt
where S is the integration symbol.
and the integrand i with dt
is
I * T = Q = Current * Time = S i dt
and Q/C = Vcap
thus
each of the integrals and limits of integration
defines a voltage of the capacitor.