6.5: Phasor Relationships for Circuit Elements
Phasor representation is a powerful tool used to transform the voltage-current relationship for resistors, inductors, and capacitors from the time domain to the frequency domain. This transformation simplifies the analysis of alternating current (AC) circuits.
In the time domain, Ohm's law provides a fundamental relation between the current flowing through a resistor and the voltage across it:
where V is the voltage, I is the current, and R is the resistance. In phasor representation, this relationship holds true as well, with the voltage and current phasors being in phase and following Ohm's law.
For an inductor, the relationship between the voltage across it and the current flowing through it is given by the rate of change of current. The sinusoidal function representing this relationship can be converted into its phasor in polar format. When comparing the current and voltage phasors for an inductor, it can be observed that the current lags the voltage by 90 degrees. Using Euler's identity, a fundamental formula in complex analysis, the current-voltage relationship in the phasor domain can be obtained.
Similarly, when charging a capacitor, the current passing through it is determined by the rate of change of voltage across it. Again, the sinusoidal function representing this relationship can be converted into its phasor in polar form. In the case of a capacitor, the phasor representations indicate that the current leads the voltage by 90 degrees. The relationship between the current and voltage phasors for a capacitor can be obtained by using the time derivative of the voltage.
