32.5, example 32.6 is important.
The RLC circuit puts everything together, the phasor of the RLC circuit will
always look something like
V_R
is always in phase with current while V_L
leads and
V_C
lags.
Note that the length of these phasors are all the peak values at each circuit
element. The instantaneous voltage obey the equation
v_L+v_R+v_C=\mathcal{E}
Note that one of those must be positive when the other is negative since the
V_L and V_C phasor are 180 degrees shifted from one another.
To find the relationship between the peak values,
we add the phasors as vectors
and find the magnitude. The easiest way is to first subtract V_C from V_L (if
the capacitor is bigger, you would do the reverse).
The resulting addition of all the phasor will give the source phasor of length
\mathcal{E}_0
Whats the point of all this? Two things. What we are interested is to see the
lag (or lead) between the current in the circuit (aligned with V_R) and the
emf of the source. This is denoted \phi
in the figure. Second we want to know the size of the current.
Just like when we did DC circuit with Kirchhoff's law the goal is to find the
current. We keep things simpler here that we only look at a single loop, the
R,L and C are all in series. The complication is that we want both the
magnitude of the current and the phase that tells us how in sync or out of
sync the current is with the power source.
We define the phase \phi
when the current lags the emf. Negative if the current leads the emf.
Using Pythagoras and the known relationships between capacitive/inductive
reactances, one arrives at
I =\frac{\mathcal{E}_0 }{Z}
with the impedance of the RLC circuit given by
Z=\sqrt{R^2+(X_L-X_C)^2}
and the phase is
\tan \phi = \frac{X_L -X_C}{R}
Resonance
One can see that the impedance of an RLC circuit is minimized when the phasor
for the inductor and capacitor exactly cancel. This is called resonance and
the phase angle is zero. The source emf and the current are in phase.