# Phasor

A resistor, inductor and capacitor are connected in series to a time-dependent voltage source

A phasor is a tool in mathematics that is commonly used to describe AC circuits in which the relation between current and voltage exhibits a phase shift. The advantage of solving such problems with phasors is that it greatly simplifies the expressions involving integration and differentiation.

## Example: LRC series circuit

Suppose a time-dependent voltage source is connected in the series LRC circuit shown. The governing equations are:

${\displaystyle V_{0}\cos \omega t-{\frac {1}{C}}Q(t)-L{\frac {d}{dt}}I(t)-I(t)R=0\,,}$
${\displaystyle Q(t)=\int ^{t}I({\tilde {t}})d{\tilde {t}}\,,}$

where ${\displaystyle V_{0}}$ is a constant that represents the peak applied voltage, ${\displaystyle C}$ is capacitance, ${\displaystyle L}$ is inductance, and ${\displaystyle R}$ is resistance. The unknown functions of time ${\displaystyle Q=Q(t)}$ and ${\displaystyle I=I(t)}$ represent charge and current, respectively.

The absence of a lower limit on the integral relating current and charge does not need to be resolved because we are looking for only one solution to this equation.[1]

While these equations have well known solutions \ ${\displaystyle Q=Q(t)}$ and ${\displaystyle I=I(t)}$, the algebra is simpler if we instead solve this equation:

${\displaystyle V_{0}e^{i\omega t}-{\frac {1}{C}}Q(t)-L{\frac {d}{dt}}I(t)-I(t)R=0}$

where ${\displaystyle i={\sqrt {-1}}}$ is an imaginary number. As is customary with equations like this, we "try" a solution of the form:

${\displaystyle I(t)=I_{0}e^{i\omega t}}$
${\displaystyle I(t)=I_{0}e^{i\omega t}}$

The advantage of replacing the real sinusoidal function by complex exponential functions is that integration and differentiation are converted into algebraic factors. Neglecting the constant of integration allows us to calculate an important solution to these equations:

${\displaystyle Q(t)=\int ^{t}I({\tilde {t}})d{\tilde {t}}={\frac {1}{i\omega }}I_{0}e^{i\omega t}}$

After a bit of algebra we get something like this:

${\displaystyle I_{0}={\frac {1}{junk}}V_{0}}$ or better yet:

${\displaystyle V_{0}=I_{0}\left({\frac {1}{i\omega C}}+i\omega L+R\right)=I_{0}\left(Z_{C}+Z_{L}+Z_{R}\right)}$

I will find the "junk" after a short nap.

## Special case that R=0

If the resistance in an LRC circuit is zero (or almost zero) we obtain:

${\displaystyle I_{0}={\frac {constant}{\omega ^{2}-{\frac {1}{\sqrt {LC}}}}}V_{0}={\frac {constant}{\omega ^{2}-\omega _{0}^{2}}}V_{0}}$

Here, the net impedance can vanish, and this occurs at resonance. If R=0 it is possible for the two impedances to cancel each other, as if the net (series) resistance equals zero. A loop of "wire" with zero resistance could have any arbitrarily chosen current. And if a voltage is applied, the current would diverge to infinity.