Phasors and Reactances
32.1-32.4
An AC circuit is a circuit powered by a source whose EMF alternates with time.
Our convention will be to take a cosine.
\mathcal{E}=\mathcal{E}_0\cos(\omega t)
where \mathcal{E}_0 is the peak value.
The average value over time of the potential is zero. It goes between positive
and negative values.
To do AC circuit correctly, one needs to be careful with notation, we will use
- lower case for instantaneous values
- upper case for peak values.
For example, the current on a resistor at any given time will be denoted
i_R . That current could be zero, the max current
(denoted I_R ), the max current but in opposite
direction or anything in between.
Capacitor and inductor voltages are related to the integral/derivative of the
current (eq 32.8 inverted and eq 32.16). This has two main implications:
-
The derivative/integral changes the phase by
\pm \pi/2 between their voltage and their current. The
capacitor lags while the inductor leads the current.
-
The peak value of V_C and V_L
depends on angular frequency.
In terms of peak value (and only of peak value), the capacitor and the
inductor obey something that looks like Ohm's law
V_C = X_C I_C while
V_L = X_L I_L but the "reactances"
X_{C,L} depend on the frequencies unlike for a resistor
where it is just a constant R.
This dependence on the angular frequency allows one to build circuits for
low-pass and high pass filters. The idea is that you measure the voltage of a
circuit element and that voltage will be blocked (near zero) for low or high
frequencies depending on how you build the circuit.
Examples, 31.1-32.5 are all very good to practice.