Phasors

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Introduction to phasors

Many real world signals can be described by a time depending sinusoidal signals such as: $$x\left(t\right)=A\cos ({\omega }_{o}t+\varphi )$$ In this equation is $A$ represents the amplitude, from which the dimension is equal to the dimension of the signal $x\left(t\right)$, ${\omega }_0$ [rad/sec] the radian (or angular) frequency and $\varphi$ [rad] the phase.

By using the Euler equations and replacing $\theta$ by ${\omega }_{o}t+\varphi$ we obtain the following time depending complex exponential function $z\left(t\right)$: $$z\left(t\right)=A{e}^{j({\omega }_{o}t+\varphi )}=A\cos ({\omega }_{o}t+\varphi )+jA\sin ({\omega }_{o}t+\varphi )$$

Such time depending complex exponential, which is depicted as a time depending complex vector in Fig. 1, is called a phasor. The projection of the phasor on the real axis behaves like a cosine signal, while the projection on the imaginary axis behaves like a sine function. In other words we can generalize a time depending sinusoidal signal by a phasor, since it describes both a sine and cosine function at the same time. Alternatively we write the cosine and sine function in phasor notation as follows using the Euler equations: $$\begin{array}{ccc}& & A\cos ({\omega }_{o}t+\varphi )=Re\left\{A{e}^{j({\omega }_{o}t+\varphi )}\right\}=\frac{A{e}^{j({\omega }_{o}t+\varphi )}+A{e}^{-j({\omega }_{o}t+\varphi )}}{2}\\ & & A\sin ({\omega }_{o}t+\varphi )=Im\left\{A{e}^{j({\omega }_{o}t+\varphi )}\right\}=\frac{A{e}^{j({\omega }_{o}t+\varphi )}-A{e}^{-j({\omega }_{o}t+\varphi )}}{2j}\end{array}$$

Phasor addition rule

We can write a phasor as the following product: $$A{e}^{j({\omega }_{o}t+\varphi )}=A{e}^{j\varphi }\cdot e^{j{\omega }_{0}t}$$

The second part $e^{j{\omega }_{0}t}$ is the phasor component, containing the radian frequency ${\omega }_0$. The first part $A{e}^{j\varphi }$ is a complex number, in which $A$ represents the amplitude and $\varphi$ the phase of the phasor. By using this fact it becomes obvious that the addition of two phasors, $A_1{e}^{j({\omega }_{0}t+{\varphi }_1)}$ and $A_2{e}^{j({\omega }_{0}t+{\varphi }_2)}$, with the same radial frequency ${\omega }_0$ result in one new phasor $A{e}^{j({\omega }_{0}t+\varphi )}$ having the same frequency ${\omega }_0$. This can be shown as follows: $$A_1{e}^{j({\omega }_{0}t+{\varphi }_1)}+A_2{e}^{j({\omega }_{0}t+{\varphi }_2)}=A_1{e}^{j{\varphi }_1}\cdot e^{j{\omega }_0}+A_2{e}^{j{\varphi }_2}\cdot e^{j{\omega }_0}=(A_1{e}^{j{\varphi }_1}+A_2{e}^{j{\varphi }_2})\cdot e^{j{\omega }_0}$$

The amplitude $A$ and phase $\varphi$ of the resulting phasor $A{e}^{j({\omega }_{0}t+\varphi )}$ can be calculated by using the complex addition rule for the complex numbers $A_1{e}^{j{\varphi }_1}$ and $A_2{e}^{j{\varphi }_2}$ of the individual phasors as follows: $$A{e}^{j\varphi }=A_1{e}^{j{\varphi }_1}+A_2{e}^{j{\varphi }_2}$$

This phasor addition example is depicted in Fig. 2.

We can use this property when a signal $x\left(t\right)$ consists of the sum of two sinusoidal signals, both with the same frequency ${\omega }_0$, which goes as follows:

$$\begin{array}{ccc}x\left(t\right)& =& A_1\cos ({\omega }_{0}t+{\varphi }_1)+A_2\cos ({\omega }_{0}t+{\varphi }_2)\\ & =& (\frac{A_1}{2}{e}^{j{\varphi }_1}+\frac{A_2}{2}{e}^{j{\varphi }_2})\cdot e^{j{\omega }_{0}t}+(\frac{A_1}{2}{e}^{-j{\varphi }_1}+\frac{A_2}{2}{e}^{-j{\varphi }_2})\cdot e^{-j{\omega }_{0}t}\end{array}$$

By defining $$A{e}^{j\varphi }=A_1{e}^{j{\varphi }_1}+A_2{e}^{j{\varphi }_2}$$ we obtain: $$\begin{array}{ccc}x\left(t\right)& =& \left(\frac{A}{2}{e}^{j\varphi }\right)\cdot e^{j{\omega }_{0}t}+\left(\frac{A}{2}{e}^{-j\varphi }\right)\cdot e^{-j{\omega }_{0}t}=A\cos ({\omega }_{0}t+\varphi )\end{array}$$

Thus the amplitude $A$ and phase $\varphi$ of the resulting sinusoidal signal $x\left(t\right)$ are found by complex addition of the amplitude and phase of the individual sinusoidal components. \
This example can be extended to the addition of $N$ sinusoidal signals, all with the same radian frequency ${\omega }_0$, resulting in one sinusoidal signal with the same radian frequency ${\omega }_0$ as follows:

$$x\left(t\right)=\sum _{k=1}^N{A}_k\cos ({\omega }_{0}t+{\varphi }_k)=A\cos ({\omega }_{0}t+\varphi )$$

in which the amplitude $A$ and phase $\varphi$ can be calculated by using complex addition rules as follows: $$A{e}^{j\varphi }=\sum _{k=1}^N{A}_k{e}^{j{\varphi }_k}$$

In words this results in the so called phasor addition rule:

The sum of original sinusoidal signals, all with the same radian frequency ${\omega }_0$, results in one sinusoidal signal with the same radian frequency ${\omega }_0$. Amplitude and phase of this resulting signal can be found by adding the complex representation of amplitude and phase of the individual original sinusoidal signals.