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.
FAQs on Dimensions of Impedance
1. Differentiate Between Resistance and Impedance.
Below are the important differences between resistance and Impedance in tabular format
Resistance | Impedance |
It is a concept related to DC circuits | Impedance is completely related to AC circuits |
Resistance is observed in both AC and DC circuits | Impedance is observed only in AC circuits. |
It occurs due to resistive elements. | It occurs due to the resistance and reactance as well. |
Symbolic representation of Resistance is R | Symbolic representation of Impedance is Z |
Resistance remains constant on varying frequencies of DC. | Impedance changes according to the frequency of AC. |
Doesn’t contain magnitude and phase angle | Impedance carries both magnitude and phase angle |
2. What is the Application of Impedance
Impedance applies to our own house. The panel of fuses controls the electricity in the place. When the electricity surges, fuses interrupt, so the electrical accidents are reduced. Fuses are similar to high capacity resistors that face the blow. Additionally, this situation is handled by impedance in capacitors. Alternating current delivers electricity in the form of a fluctuating pulse. Thus, it is required to govern the electricity flowing smoothly in the electric circuit—capacitor. Impedance manages flow of electricity in a circuit board. Without capacitors regulating the electrical discharge, electrical appliances using alternating current will burn.