An online calculator to add, subtract, multiply and divide polar impedances is presented. Operations on polar impedances are needed in order to find equivalent impedances in AC circuits.

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In what follows \( j \) is the imaginary unit such that \( j^2 = -1 \) or \( j = \sqrt{-1} \).

Impedances are represented by complex numbers in polar form as follows:

\( Z = \rho \: \; \angle \; \: \theta \) , where \( \rho \) is the magnitude of \( Z \) and \( \theta \) its phase in degrees or radians.

\( Z \) in standard complex form is written as

\( Z = \rho \cos \theta + j \; \rho \sin \theta \)

1) A capacitor of capacitance \( C \) has an impedance \( Z_C \) whose magnitude is \( \dfrac{1}{\omega C} \) , where \( \omega = 2 \pi f \) and \( f \) is the frequency of the signal, and a phase equal to \( - \dfrac {\pi}{2} \). Hence \( Z_C \) is written

in standard complex form as

\( Z_C = - \dfrac{j}{\omega C} \)

and in polar form as

\( Z_C = \dfrac{1}{\omega C} \; \angle \; - \dfrac {\pi}{2} \)

2) An inductor of inductance \( L \) has an impedance \( Z_L \) whose magnitude is \( \omega L \) , where \( \omega = 2 \pi f \) and \( f \) is the frequency of the signal, and a phase equal to \( \dfrac {\pi}{2} \). Hence \( Z_L \) is written

in standard complex form as

\( Z_L = j \; \omega L \)

and in polar form as

\( Z_L = \omega L \; \angle \; \dfrac {\pi}{2} \)

3) A resistor of resistance \( R \) has an impedance \( Z_R \) whose magnitude is \( R \) and a phase equal to \( 0 \). Hence \( Z_R \) is written

in standard complex form as

\( Z_R = R + j \; 0 \)

and in polar form as

\( Z_R = R \; \angle \; 0 \)

Write \( Z_1 \) and \(Z_2 \) in standard complex forms

\( Z_1 = \rho_1 \cos \theta_1 + j \; \rho_1 \sin \theta_1 \)

\(Z_2 = \rho_2 \cos \theta_2 + j \; \rho_2 \sin \theta_2 \)

\( Z_1 + Z_2 = \rho_1 \cos \theta_1 + \rho_2 \cos \theta_2 + j \; ( \rho_1 \sin \theta_1 + \rho_2 \sin \theta_2) \)

in polar form

\[ Z_1 + Z_2 = \rho \; \; \angle \; \theta \]

where

\( \rho = \sqrt {(\rho_1 \cos \theta_1 + \rho_2 \cos \theta_2)^2 + (\rho_1 \sin \theta_1 + \rho_2 \sin \theta_2)^2} \)

and

\( \theta = \arctan (\dfrac{\rho_1 \sin \theta_1 + \rho_2 \sin \theta_2}{\rho_1 \cos \theta_1 + \rho_2 \cos \theta_2}) \)

\( Z_1 - Z_2 = \rho_1 \cos \theta_1 - \rho_2 \cos \theta_2 + j \; ( \rho_1 \sin \theta_1 - \rho_2 \sin \theta_2) \)

in polar form

\[ Z_1 - Z_2 = \rho \; \; \angle \; \theta \]

where

\( \rho = \sqrt {(\rho_1 \cos \theta_1 - \rho_2 \cos \theta_2)^2 + (\rho_1 \sin \theta_1 - \rho_2 \sin \theta_2)^2} \)

and

\( \theta = \arctan (\dfrac{\rho_1 \sin \theta_1 - \rho_2 \sin \theta_2}{\rho_1 \cos \theta_1 - \rho_2 \cos \theta_2}) \)

It is much easier to multiply and divide polar impedances

\( \rho = \rho_1 \times \rho_2 \)

and

\( \theta = \theta_1 + \theta_2 \)

\[ \dfrac{Z_1}{Z_2} = \rho \; \; \angle \; \theta \] where

\( \rho = \dfrac{\rho_1}{\rho_2} \)

and

\( \theta = \theta_1 - \theta_2 \)

The outputs are:

\( Z_1 \) and \( Z_2 \) in complex standard form

and

\( Z_1+Z_2\) , \( Z_1-Z_2\) , \( Z_1 \times Z_2 \) and \( \dfrac{Z_1}{Z_2} \) in polar form with phase in degrees.

Maths Calculators and Solvers.