Characteristic impedance

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In radio communications, characteristic impedance (acoustic impedance or sound impedance) [Z_0 \ ] of a uniform transmission line is the ratio of the voltage amplitude to the current amplitude of a single wave travelling down it. This is sometimes called surge impedance. The SI unit of characteristic impedance is the ohm. For the input impedance of a transmission line, see the article on transmission lines

Description

A uniform line terminated in its characteristic impedance will have no standing waves, no reflections from the end, and a constant ratio of voltage to current at a given frequency at every point on the line. The characteristic impedance of a linear, homogeneous, isotropic, dielectric propagation medium free of electric charge is given by the relation

[Z_0 = \sqrt = } = c \mu ]

where

[Z_0 \ ]is the characteristic impedance

[\epsilon \ ]is the electric permittivity of the medium (in farads per meter)

[\mu \ ]is the magnetic permeability of the medium (in henries per meter)

[c = \frac} \ ] is the speed of propagation in the medium

When the medium is free space, the magnetic permeability [\mu_0 \ ] and electric permittivity [\epsilon_0 \ ] of free space are used and this defines the universal physical constant, the characteristic impedance of free space (also known as: characteristic impedance of vacuum), defined by:
([ Z_0 = \mu_0 c = \sqrt} \ ]
Where:

[ \mu_0 \ ] = magnetic constant

[ \epsilon_0 \ ] = electric constant

[ c \ ] = speed of light

In SI units, the value is exactly expressed by:

[ Z_0 \ ] = 1.199 169 832 · π · 10² Ω = 376.73 Ω)

[Z_0 = \sqrt = } = c \mu_0 = 376.73 \ \Omega]

where

[c = } \ = 2.998 \times 10^8 \ \mbox] is the speed of light in free space,

[\epsilon_0 = 8.854 \times 10^ \ \mbox ] is the permittivity of free space, and

[\mu_0 = 4 \pi \times 10^ \ \mbox ] is the permeability of free space.

Transmission Line Model

Using the notation for the transmission line model, the general expression for the characteristic impedance of a transmission line is:

[Z_0=\sqrt}]

Lossless line

For a lossless line R, and G are assumed to be zero so the equation reduces to the more familiar

[Z_0=\sqrt}]

variation with frequency

The impedance of a real lossy transmission line is not constant, but varies with frequency. At low frequencies, when

[\omega L \ll R] and [\omega C \ll G],

the characteristic impedance of a transmission line is

[Z_0 = \sqrt].

At high frequencies where

[\omega L \gg R] and [\omega C \gg G],

then the characterstic impedance is

[Z_0 = \sqrt].

So there are two distinct characteristic impedances for every line. Usually G is very small so the low-frequency impedance is high, whereas the high-frequency impedance is low. The break points in the impedance frequency graph are at [\omega_1 = G/C] and [\omega_2 = R/L] (where [\omega =2 \pi f]). If [R/G \gg L/C], it is obvious that [\omega_2 \gg \omega_1]. Between these two break frequencies the cable impedance decreases smoothly.

Example

Take the case of a 50Ω coaxial cable with polyethylene dielectric. R is about 100 mΩ/m and G < 20 pS/m (based on measurements of leakage resistance in a 1 m length). Using [L=CZ^2], L can be calculated at about 250 nH/m. So,

ω₂ = R/L = 200 krad/s (f₂ = 30 kHz)

and

ω₁ = G/C = 0.2 rad/s (f₁ = 30 millihertz)

At 100 Hz the 50 ohm coaxial cable will have an impedance of about 900 ohms, only reaching 50 ohms at about 30 or 40 kHz. The phase angle of the impedance between the two break frequencies is leading (the cable looks capacitive).

See also

• Transmission line
• Maxwell's equations

Source

Adapted from

• Federal Standard 1037C.

References

• Fundamentals Of Applied Electromagnetics 2004 media edition., Ulaby, F.T., pub Prentice Hall, ISBN 0-13-185089-x

 

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