Dimensions of Impedance

Derivation of Impedance

To understand the dimension formula of impedance, let's first understand the basics of impedance. The first question that you might think is what is impedance. Well! In simple terms,impedance is resistance in the motion of any object. Under Physics terminology, Impedance is the opposition of a circuit offered to the alternating current.

The symbolic representation of Impedance is Z and measured by physical quantity Ohms (Ω).  It means Impedance is calculated in an electric circuit or components. Impedance tells the amount that circuit resists the movement of electrons. There are two different effects that hinder the current and contribute to the impedance. 

Resistance (R) is defined as a hindrance of current due to the material, the shape of the component. The impact of resistance is the largest in resistors. However, all components have at least a small quantity of resistance.

Reactance (X) is then defined as a hindrance to current due to electric and magnetic fields opposing variations in the current or voltage. It is significantly observed in capacitors and inductors.

The dimensional formula of Impedance is

M\[^{1}\]L\[^{2}\]I\[^{-2}\]T\[^{-3}\]

Let’s read the derivation of impedance to calculate the dimensional formula of impedance.

The Dimensional Formula of Impedance

We know the dimensional formula of Impedance is given by,

M\[^{1}\]L\[^{2}\]I\[^{-2}\]T\[^{-3}\]

Where,

M = Mass

 I = Current

 L = Length 

T = Time

Derivation of Impedance             

(Z) = Voltage × [Electric Current]⁻¹ . . . (1)

Since, Voltage = Electric Field × Distance

V = Force × [Electric Charge]⁻¹  × Distance . . . (2)

⇒ The dimensions of force = [M\[^{1}\]L\[^{1}\]T\[^{-2}\]] … (3)

And, the dimensional formula to represent electric charge = [I\[^{1}\]T\[^{1}\]] . . . (4)

On replacing equation (3) and (4) in equation (2) we get,

V = [M\[^{1}\]L\[^{1}\]T\[^{-2}\]] × [I\[^{1}\]T\[^{1}\]]\[^{-1}\] × [M\[^{0}\]L\[^{1}\]T\[^{0}\]]

Therefore, the dimensional formula to represent voltage = [M\[^{1}\]L\[^{2}\]I\[^{-1}\]T\[^{-3}\]] . . .  (5)

And, the dimensions of Electric Current = M\[^{0}\]L\[^{0}\]I\[^{-1}\]T\[^{0}\] . . .  (6)

On replacing equation (5) and (6) in equation (1) we get,

Impedance = Voltage × [Electric Current]⁻¹ 

Or, Z = [M\[^{1}\]L\[^{2}\]I\[^{-1}\]T\[^{-3}\]] × [M\[^{0}\]L\[^{0}\]I\[^{-1}\]T\[^{0}\]\[^{-1}\]] = [M\[^{1}\]L\[^{2}\]I\[^{-2}\]T\[^{-3}\]] 

Therefore, impedance is dimensionally represented as M\[^{1}\]L\[^{2}\]I\[^{-2}\]T\[^{-3}\]

Now, let us enhance our knowledge on wave impedance derivation.

Wave Impedance Derivation

Wave impedance of an electromagnetic wave is defined as the ratio of strength of electric field to strength of magnetic field. These two traverse components are being those at right angles to the direction of propagation. 

Wave impedance represented as

Z = \[\frac{E_{0}^{-}(x)}{H_{0}^{-}(x)}\]

where E\[_{0}\]\[^{-}\](x) is the electric field and H\[_{0}\]\[^{-}\](x) is the magnetic field, in phasor representation. The impedance is, in general, a complex number. 

When the wave impedance is dependent on parameters of an electromagnetic wave and the medium it travels through, it is calculated by the below formula

Z = \[\sqrt{\frac{j \omega \mu }{\sigma + j \omega \epsilon }}\]

Where  magnetic permeability is denoted by μ , (real) electric permittivity is denoted by 𝜖  and the electrical conductivity of the material the wave  σ is travelling through (corresponding to the fictional piece of the permittivity multiplied by omega).

In the formula, j is the fictional unit, and angular frequency ω of the wave. Similar to  electrical impedance, the impedance is a basis of frequency. A perfect dielectric (whose conductivity is zero), the formula decreases to the real number

Z = \[\frac{\mu }{\epsilon }\]

Wave Impedance in Free Space

The wave impedance of plane waves in free space is:

Z = \[\frac{\mu _{0}}{\epsilon _{0}}\]

(where ε0 is the permittivity constant in free space and μ0 is the permeability constant in free space) and: 

c\[_{0}\] = \[\frac{1}{\sqrt{\mu_{0}\epsilon_{0}}}\] = 299,792,458m/s

 (by the SI definition of the metre)

Hence, because the values of c\[_{0}\] and \[\mu\]\[_{0}\] are exact, the value of Z\[_{0}\] in ohms is exactly:

Z\[_{0}\] = \[\mu\]\[_{0}\]c\[_{0}\]  = 4π x 10\[^{-7}\] H/m x 299,792,458 m/s = 376.730313...Ω ≈ 120πΩ

Fun Facts

• Do you know impedance is used to determine the proportion of body fat , fat -free mass, hydration level and other body composition values in the human body.

• Body Impedance, also called Bioelectrical Impedance is calculated when a small electrical signal is carried by water and fluids through the human body.

Impedance is most significant in fat tissue containing 10-20 percent of water compared to fat -free tissues containing 70-75 percent of water level.