Phasor

1. Complex Plane

In mathematics, a complex plane or sometimes called -plane, is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.
Any number corresponds to in complex plane.

As you can see in the picture, the point is the corresonding point on the complex plane for . Now we can deduce some useful facts.

1. Denoting , length of . We denote this value as .

2. Denoting the angle between and positive -axis as , another expression for is . Again in the viewpoint of , it is equal to
   

   
   by Euler's identity. In engineering, we write for convenience.

2. Well, then What is Phasor? - Definition

In physics or electrical engineering, there are multiple sinusoids all with the same frequency, but different amplitudes and phases. For instance, on AC current with frequecy , voltage, current, and impedance all have same angular frequency , connected by Ohm's Law . So what we have to consider is not the angular frequency, but their phases and amplitudes. Now, suppose we have a sinusoidal function of the form

where is amplitude, is angular frequency, and is the phase. Then the complex number satisfying

is called the phasor representation of such sinusoidal function. In many cases, only the time invariant factor is used, because multiple sinusoids share same angular frequency. So to sum up, a phasor representation is a corresonding map such that

Its counterpart sine function is nothing but so that

3. Arithmetic of Phasor

Here we assume that different sinusoids share common angular frequency . If not, all phasor arithmetic is invalid.

3-1. Constant Multiplication

Suppose we multiply a complex constant to a phasor . Then using multiplication rule for complex numbers,

is the new sinusoid. So multiplication scales the amplitude by along with phase shift .

3-2. Differentiation and Integration

Suppose we have a sinusoid of the form . Then differentiating respect to time gives . The corresponding phasor representation would be (by section 2) . So differentiation in function domain is nothing but multiplication of in phasor domain. Since integration is inverse operator of differentiation, it would be division of .

3-3. Addition

There is no concise and neat formula for addition, because in general, sum of two complex numbers is the vector sum on complex plane (not scaling or rotation). So, you should rewrite into form and use term by term addition (in real and imaginary parts respectively) and then re-express it as a phasor form. However, if the phases are same, then

holds.

4. The intuition inside Phasor

4-1. Ohm's Law

Phasor gives a lot of convenience especially in Electrical engineering. Consider Alternating voltage source . The impedance, consists of resistor and reactance ; satisfying . Then the sinusoidal current flowing through impedance can be directly calculated using Ohm's Law, which is then

4-2. Series of RLC Circuit

Current flowing in typical RLC circuit in the above figure can be directly calculated using phasor. If we denote

• Voltage source
• Current

Then the differential equation would be

with one more differentiation respect to time gives

Now, this differential equation turns into , or equivalently

Comparing amplitude gives

where are reactance of inductor and capacitor respectively.

Without any heavy computation involving differentiation of sinusoids and solving techniques for differential equation on domain , we can solve the differential equation very quickly only using algebraic operations on phasor.

5. Conclusion

1. If physical objects share same frequency but differ only in amplitude and phase, then phasor analysis is way more convenient then usual differential calculus in time domain.

2. We assume that frequency, amplitude, and phase does not change over time, (i.e are constants).

3. Differential equation turns into algebraic equations. In fact, a lot of transformations has this property; which we will discuss further in later talks (such as Fourier, Laplace, z-transform...)