Intrinsic Impedance

The intrinsic impedance of a medium is the ratio of the electric field amplitude to the magnetic field amplitude for a uniform plane wave propagating in that medium. It plays the same role in electromagnetic wave propagation that characteristic impedance plays in transmission line theory. Its value determines how E and H relate to each other in any medium and governs the reflection and transmission of waves at boundaries between different media.

Core Concept Explanation

When a uniform plane wave propagates through a medium, the electric and magnetic fields are not independent. They are coupled through Maxwell's equations, and their amplitudes are constrained to maintain a fixed ratio at every point and time. This ratio, |E|/|H|, is the intrinsic impedance eta of the medium. It has units of ohms (V/m divided by A/m gives V/A = ohms).

For a lossless medium, eta = sqrt(mu/epsilon) is a real number. This means E and H are in phase. The wave carries real power in the propagation direction. In free space, eta0 = sqrt(mu0/epsilon0) = 377 ohms, often approximated as 120*pi ohms. This value is of great practical importance: it appears in every antenna gain, effective area, and radar equation calculation.

For a lossy medium with conductivity sigma, the permittivity becomes a complex permittivity epsilon_c = epsilon' - j*(sigma/omega), and the intrinsic impedance becomes complex: eta = sqrt(j*omega*mu / (sigma + j*omega*epsilon)). A complex eta means E and H are phase-shifted relative to each other. The real part of the time-averaged Poynting vector, which represents real power flow, is then reduced compared to a lossless medium of the same epsilon.

Mathematical Expression

The intrinsic impedance is derived directly from the wave equations. For a plane wave E = E0 * e^(-gamma*z) * x_hat, applying Faraday's law in phasor form gives H = (gamma / j*omega*mu) * E0 * e^(-gamma*z) * y_hat. The ratio E/H equals j*omega*mu/gamma. Substituting gamma = sqrt(j*omega*mu*(sigma + j*omega*epsilon)) gives eta = sqrt(j*omega*mu / (sigma + j*omega*epsilon)).

For the lossless case (sigma = 0): eta = sqrt(j*omega*mu / j*omega*epsilon) = sqrt(mu/epsilon). The j*omega cancels perfectly, leaving a real, frequency-independent value. This is a key result: the intrinsic impedance of a lossless medium does not depend on frequency. For a lossy medium, eta is complex and frequency-dependent, meaning wave propagation characteristics change with frequency in lossy materials.

Practical Understanding

The intrinsic impedance governs the reflection coefficient at a boundary between two media. If a wave traveling in medium 1 with impedance eta1 hits an interface with medium 2 having impedance eta2, the reflection coefficient is Gamma = (eta2 - eta1) / (eta2 + eta1). For a perfect conductor, eta2 = 0 and Gamma = -1, meaning total reflection with a 180-degree phase reversal. For a matched medium (eta2 = eta1), Gamma = 0 and there is no reflection.

In antenna design, the intrinsic impedance of free space (377 ohms) is used to compute the radiation resistance and power radiated. In antenna matching problems, the antenna input impedance must be matched to the transmission line impedance, and the free-space wave impedance sets the scale for the electromagnetic environment into which the antenna radiates.

Given:
Lossless dielectric with epsilon_r = 4, mu_r = 1
Electric field amplitude E0 = 100 V/m

Why this formula applies:
For lossless medium: eta = eta0 * sqrt(mu_r / epsilon_r)
Magnetic field amplitude: H0 = E0 / eta

Formula:
eta = 377 * sqrt(mu_r / epsilon_r)

Substitution:
eta = 377 * sqrt(1 / 4) = 377 * 0.5 = 188.5 ohms

Calculation:
H0 = E0 / eta = 100 / 188.5 = 0.5305 A/m

Final Answer:
Intrinsic impedance eta = 188.5 ohms
Magnetic field amplitude H0 = 530.5 mA/m
Note: Power density = E0*H0/2 = 100*0.5305/2 = 26.53 W/m^2

Exam Tip: For GATE, remember eta0 = 377 ohms = 120*pi ohms exactly. For a lossless non-magnetic medium (mu_r=1), eta = 377/sqrt(epsilon_r). If epsilon_r = 4, eta = 377/2 = 188.5 ohms. If epsilon_r = 9, eta = 377/3 = 125.7 ohms. These values come up repeatedly in GATE boundary value and power density problems.

• Intrinsic impedance eta = E/H = sqrt(mu/epsilon) for a lossless medium. Units are ohms.
• Free space: eta0 = sqrt(mu0/epsilon0) = 377 ohms = 120*pi ohms. Real and frequency-independent.
• Lossless dielectric: eta = eta0 * sqrt(mu_r/epsilon_r). Still real; E and H remain in phase.
• Lossy medium: eta is complex. E and H are phase-shifted. Partial reactive power flow occurs.
• Reflection coefficient at boundary: Gamma = (eta2 - eta1)/(eta2 + eta1). Total reflection for perfect conductor (eta2=0).

Quick Revision

• eta = sqrt(mu/epsilon) for lossless medium; complex for lossy medium.
• eta0 = 377 ohms = 120*pi ohms in free space.
• For non-magnetic lossless dielectric: eta = 377/sqrt(epsilon_r).
• E and H in phase for real eta (lossless); phase-shifted for complex eta (lossy).
• Reflection coefficient: Gamma = (eta2-eta1)/(eta2+eta1); transmission: tau = 2*eta2/(eta2+eta1).
• Exam trap: eta is NOT impedance of a circuit element. It is a medium property, not a component property. Do not confuse with characteristic impedance of a transmission line (though they are analogous concepts).
• GATE shortcut: Average power density = |E0|^2 / (2*eta). For free space with E0 = 1 V/m, power density = 1/(2*377) = 1.326 mW/m^2.