peak value, () at 270° or . Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represents a scaled voltage or current value of a rotating vector which is "frozen" at some point in time, () and in our example above, this is at an angle of 30°. Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the alternating quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed in the waveform above that the waveform starts at time with a corresponding phase angle in either degrees or radians. But if a second waveform starts to the left or to the right of this zero point, or if we want to represent in phasor notation the relationship between the two waveforms, then we will need to take into account this phase difference, of the waveform. Consider the diagram below from the previous Phase Difference tutorial. Differentiation and integration The time derivative or integral of a phasor produces another phasor. For example: Therefore, in phasor representation, the time derivative of a sinusoid becomes just multiplication by the constant . Similarly, integrating a phasor corresponds to multiplication by The time-dependent factor, is unaffected. When we solve a linear differential equation with phasor arithmetic, we are merely factoring out of all terms of the equation, and reinserting it into the answer. For example, consider the following differential equation for the voltage across the capacitor in an RC circuit: When the voltage source in this circuit is sinusoidal: we may substitute where phasor and phasor is the unknown quantity to be determined. In the phasor shorthand notation, the differential equation reduces to: Solving for the phasor capacitor voltage gives: As we have seen, the factor multiplying represents differences of the amplitude and phase of relative to and In polar coordinate form, the first term of the last expression is: where . Therefore: Ratio of phasors A quantity called complex impedance is the ratio of two phasors, which is not a phasor, because it does not correspond to a sinusoidally varying function. Applications Circuit laws With phasors, the techniques for solving DC circuits can be applied to solve linear