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The Derivation of Intrinsic Impedance

Key Takeaways

  • The intrinsic impedance of any uniform medium is dependent on the permittivity and permeability of the medium. When the conductivity of the medium varies,  the intrinsic impedance also changes. 

  • For a uniform plane wave traveling in a given medium, E/H is a constant and gives the intrinsic impedance. 

  • The value of intrinsic impedance corresponding to free space is 120π, which is approximately equal to 377Ω.

Electromagnetic waves in communication systems

In wireless communication systems, when an electromagnetic wave propagates through free space, it encounters the characteristic impedance of the free space, called intrinsic impedance

The atmospheric propagation of electromagnetic waves is a part of all wireless communication systems. When an electromagnetic wave propagates through free space, it encounters the characteristic impedance of the free space, called intrinsic impedance (𝜼). 

Intrinsic impedance describes the magnitude of the magnetic and electric fields present in the free space. The derivation of the intrinsic impedance of any uniform medium is dependent on the permittivity and permeability of the medium. When the conductivity of the medium varies,  the intrinsic impedance also changes

Deriving Intrinsic Impedance

The atmospheric air is typically a low-loss medium with little magnetization. It can be modeled as a vacuum or otherwise called free space. Free space is non-conductive, 𝞂=0, where 𝞂 is the conductivity of the medium. The permeability (𝜇) and permittivity (𝜀) of free space is represented by  𝜇0 and 𝜀0, respectively. As there are no physical conductances or resistances in free space, the equation of intrinsic impedance reduces to one with permeability and permittivity. 

In general, the intrinsic impedance or wave intrinsic impedance of an electromagnetic wave traveling through a medium can be given by the ratio of its electric to magnetic field intensities, that is, E/H. For a uniform plane wave traveling in a given medium, E/H is a constant and provides the impedance. The units of E and H are volts per meter and amperes per meter, respectively. Taking the ratio of E/H, the unit is Volts/ ampere which is equal to Ohms.

The electric field is varying in the z-direction and the magnetic field in the x-direction

An electromagnetic wave traveling in the positive y-direction

Consider a uniform plane wave traveling in the positive y-direction. The electric field is varying in the z-direction and the magnetic field in the x-direction. The electric field can be given by the equation:

electric field equation

Using the relationship given by equation (2):

 electrical field relationship given by equation

The magnetic field can be derived as: 

magnetic field

To derive the intrinsic impedance of the free space, take the ratio of equations (1) and (3):

intrinsic impedance of the free space

The intrinsic impedance is complex-valued and magnitude can be given as follows:

 intrinsic impedance is complex-valued and magnitude

Intrinsic Impedance Values

The intrinsic impedance value varies with each medium, as the 𝝈, 𝜇, 𝜖 are different for different mediums. Any medium in which the electromagnetic wave propagates can be compared with that of free space using the relative permeability and permittivity values represented by 𝜇r, and 𝜖r.  

  1. Lossy Medium

In a lossy medium, the intrinsic wave impedance is complex. In such a medium, the electric and magnetic fields exponentially decay in the direction of wave propagation. The electric and magnetic fields are out of phase by an angle equal to the phase angle of the intrinsic impedance. 

electric and magnetic fields are out of phase by an angle equal to the phase angle of the intrinsic impedance

  1. Lossless Medium

In a lossless medium, the intrinsic wave impedance is purely real. As there is no phase angle associated with intrinsic impedance in a lossless medium, the electric and magnetic fields are in phase with each other. 

electric and magnetic fields are in phase with each other

  1. Free Space

Substituting the values of permeability and permittivity of air in the intrinsic impedance equation, the value corresponding to free space is obtained as 120π, which is approximately equal to 377Ω.

Free Space

From the derivation of intrinsic impedance, it is clear that the ratio of E/H of an electromagnetic wave remains constant at any given instant in a medium. The intrinsic impedance is a parameter that represents the characteristics of the medium of propagation in wireless communication systems

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