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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[change | change source]
Suppose a time-dependent voltage source is connected in the series LRC circuit shown. The governing equations are:
where is a constant that represents the peak applied voltage, is capacitance, is inductance, and is resistance. The unknown functions of time and 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.
While these equations have well known solutions \ and , the algebra is simpler if we instead solve this equation:
where is an imaginary number. As is customary with equations like this, we "try" a solution of the form:
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:
After a bit of algebra we get something like this:
or better yet:
I will find the "junk" after a short nap.
Image to upload to commons[change | change source]
Special case that R=0[change | change source]
If the resistance in an LRC circuit is zero (or almost zero) we obtain:
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.
Footnotes and comments to editors as this page is constructed[change | change source]
References[change | change source]
- I will later link this to transient solutions and also define the terms