Phasors

Sinusoids are easily expressed in terms of phasors which are more convenient to work with than sine and cosine functions.

A phasor is a complex number that represents the amplitude and phase of a sinusoid.

The complex number \(z\) can be written in rectangular form as

\(\displaystyle z = x + jy\)

where \(j = \sqrt{-1}\)

The complex number \(z\) can also be written in polar form as

\(\displaystyle z = r\angle{\phi} = r e^{\displaystyle\,j\phi} = r\left[\cos{(\phi)} + j\sin{(\phi)}\right]\)

where \(j = \sqrt{-1}\)

Relationship between the rectangular form and the polar form.

\(\boxed{z = x + jy = r\angle{\phi} = r e^{\displaystyle\,j\phi} = r\left[\cos{(\phi)} + j\sin{(\phi)}\right]}\)

where

\(\displaystyle r = \sqrt{x^2 + y^2}\)

\(\displaystyle \phi = \tan^{-1}\left(\frac{y}{x}\right)\)

\(\displaystyle x = r\cos{\left(\phi\right)}\)

\(\displaystyle y = r\sin{\left(\phi\right)}\)

Addition and subtraction of complex numbers are better performed in rectangular form; multiplication and division are better done in polar form.

Given the complex numbers

\(\displaystyle z = x + jy = r\angle{\phi}\)

\(\displaystyle z_1 = x_1 + jy_1 = r_1\angle{\phi_1}\)

\(\displaystyle z_2 = x_2 + jy_2 = r_2\angle{\phi_2}\)

Basic operations:

• Addition

  \(\boxed{z_1 + z_2 = \left(x_1 + x_2\right) + j\left(y_1 + y_2\right)}\)

• Subtraction

  \(\boxed{z_1 - z_2 = \left(x_1 - x_2\right) + j\left(y_1 - y_2\right)}\)

• Multiplication

  \(\boxed{z_1 z_2 = r_1 r_2 \angle{\left(\phi_1 + \phi_2\right)}}\)

• Division

  \(\boxed{\frac{z_1}{z_2} = \frac{r_1}{r_2} \angle{\left(\phi_1 - \phi_2\right)}}\)

• Reciprocal

  \(\boxed{\frac{1}{z} = \frac{1}{r} \angle{\left(-\phi\right)}}\)

• Square root

  \(\boxed{\sqrt{z} = \sqrt{r} \angle{\left(\frac{\phi}{2}\right)}}\)

• Complex conjugate

  \(\boxed{z^{\ast} = x - jy = r\angle{\left(-\phi\right)} = re^{\displaystyle -j\phi}}\)

  \(\boxed{\frac{1}{j} = -j}\)

The idea of phasor representation is based on Euler’s identity.

\(\boxed{e^{\displaystyle \pm{j\phi}} = \cos{(\phi)} \pm j\sin{(\phi)}}\)

\(\displaystyle \cos{(\phi)} = \textrm{Re}\left(e^{\displaystyle\,j\phi}\right)\)

\(\displaystyle \sin{(\phi)} = \textrm{Im}\left(e^{\displaystyle\,j\phi}\right)\)

where Re and Im stand for the real part of and the imaginary part of.

Given a sinusoid \(v\left(t\right) = V_m \cos{\left(\omega t + \phi\right)}\)

\(\displaystyle v\left(t\right) = V_m \cos{\left(\omega t + \phi\right)} = \textrm{Re}\left(V_m \left[\cos\left(\omega t + \phi\right) + j\sin\left(\omega t + \phi\right)\right]\right) = \textrm{Re}\left(V_m e^{\displaystyle\,j\left(\omega t + \phi\right)}\right)\)

\(\displaystyle v\left(t\right) = \textrm{Re}\left(V_m e^{\displaystyle\,j\left(\omega t + \phi\right)}\right) = \textrm{Re}\left(V_m e^{\displaystyle\,j\phi} e^{\displaystyle\,j\omega t}\right)\)

\(\boxed{v\left(t\right) = V_m \cos{\left(\omega t + \phi\right)} = \textrm{Re}\left(\displaystyle\mathbf{V}e^{\displaystyle\,j\omega t}\right)}\)

where \(\mathbf{V}\) is the phasor representation of the sinusoid \(v\left(t\right)\).

\(\boxed{\mathbf{V} = V_m\,e^{\displaystyle\,j\phi} = V_m\angle\phi}\)

If we use sine for the phasor instead of cosine for \(v\left(t\right)\).

\(\displaystyle v\left(t\right) = V_m \sin{\left(\omega t + \phi\right)} = \textrm{Im}\left(V_m e^{\displaystyle\,j\left(\omega t + \phi\right)}\right) = \textrm{Im}\left(V_m e^{\displaystyle\,j\phi} e^{\displaystyle\,j\omega t}\right)\)

\(\boxed{v\left(t\right) = V_m \sin{\left(\omega t + \phi\right)} = \textrm{Im}\left(\displaystyle\mathbf{V}e^{\displaystyle\,j\omega t}\right)}\)

where \(\boxed{\mathbf{V} = V_m\,e^{\displaystyle\,j\phi} = V_m\angle\phi}\)

A phasor may be regarded as a mathematical equivalent of a sinusoid with the time dependence dropped.

As a complex quantity, a phasor may be expressed in rectangular form, polar form, or exponential form. Since a phasor has magnitude and phase (“direction”), it behaves as a vector and is printed in boldface.

A phasor diagram showing \(\mathbf{V} = V_m\angle\phi\) and \(\mathbf{I} = I_m\angle\left(-\theta\right)\).

Sinusoid-phasor transformation (based on the real part of Euler’s identity).

Note that the frequency (or time) factor \(e^{\displaystyle\,jwt}\) is suppressed, and the frequency is not explicitly shown in the phasor domain representation because the angular frequency \(\omega\) is constant. However, the response depends on angular frequency \(\omega\). For this reason, the phasor domain is also known as the frequency domain.

Differentiating a sinusoid is equivalent to multiplying its corresponding phasor \(\mathbf{V}\) by \(j\omega\).

\(\boxed{\underset{\textrm{(Time domain)}}{\frac{dv}{dt}} \Leftrightarrow \underset{\textrm{(Phasor domain)}}{j\omega\mathbf{V}}}\)

This allows the replacement of a derivative with respect to time with multiplication of \(j\omega\) in the phasor domain.

Integrating a sinusoid is equivalent to dividing its corresponding phasor \(\mathbf{V}\) by \(j\omega\).

\(\boxed{\underset{\textrm{(Time domain)}}{\int v\,dt} \Leftrightarrow \underset{\textrm{(Phasor domain)}}{\frac{\mathbf{V}}{j\omega}}}\)

This allows the replacement of an integral with respect to time with division by \(j\omega\) in the phasor domain.

Adding sinusoids of the same frequency is equivalent to adding their corresponding phasors.

The differences between \(v\left(t\right)\) and \(\mathbf{V}\):

• \(v\left(t\right)\) is the instantaneous or time domain representation, while \(\mathbf{V}\) is the frequency or phasor domain representation.
• \(v\left(t\right)\) is time dependent, while \(\mathbf{V}\) is not.
• \(v\left(t\right)\) is always real with no complex term, while \(\mathbf{V}\) is generally complex.

Phasor analysis applies only when frequency is constant; it applies in manipulating two or more sinusoidal signals only if they are of the same frequency.