Characteristic impedance
The characteristic impedance or surge impedance (usually written Z_{0}) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of reflections in the other direction. Alternatively, and equivalently, it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.
The characteristic impedance of a lossless transmission line is purely real, with no reactive component. Energy supplied by a source at one end of such a line is transmitted through the line without being dissipated in the line itself. A transmission line of finite length (lossless or lossy) that is terminated at one end with an impedance equal to the characteristic impedance appears to the source like an infinitely long transmission line and produces no reflections.
Transmission line model[edit]
The characteristic impedance of an infinite transmission line at a given angular frequency is the ratio of the voltage and current of a pure sinusoidal wave of the same frequency travelling along the line. This relation is also the case for finite transmission lines until the wave reaches the end of the line. Generally, a wave is reflected back along the line in the opposite direction. When the reflected wave reaches the source, it is reflected yet again, adding to the transmitted wave and changing the ratio of the voltage and current at the input, causing the voltage-current ratio to no longer equal the characteristic impedance. This new ratio including the reflected energy is called the input impedance.
The input impedance of an infinite line is equal to the characteristic impedance since the transmitted wave is never reflected back from the end. Equivalently: The characteristic impedance of a line is that impedance which, when terminating an arbitrary length of line at its output, produces an input impedance of equal value. This is so because there is no reflection on a line terminated in its own characteristic impedance.
Applying the transmission line model based on the telegrapher's equations as derived below,^{[1]}^{[2]} the general expression for the characteristic impedance of a transmission line is:
- is the resistance per unit length, considering the two conductors to be in series,
- is the inductance per unit length,
- is the conductance of the dielectric per unit length,
- is the capacitance per unit length,
- is the imaginary unit, and
- is the angular frequency.
This expression extends to DC by letting tend to 0.
A surge of energy on a finite transmission line will see an impedance of prior to any reflections returning; hence surge impedance is an alternative name for characteristic impedance. Although an infinite line is assumed, since all quantities are per unit length, the “per length” parts of all the units cancel, and the characteristic impedance is independent of the length of the transmission line.
The voltage and current phasors on the line are related by the characteristic impedance as:
Derivation[edit]
Using telegrapher's equation[edit]
The differential equations describing the dependence of the voltage and current on time and space are linear, so that a linear combination of solutions is again a solution. This means that we can consider solutions with a time dependence – doing so is functionally equivalent of solving for the Fourier coefficients for voltage and current amplitudes at some fixed angular frequency . Doing so causes the time dependence to factor out, leaving an ordinary differential equation for the coefficients, which will be phasors, dependent on position (space) only. Moreover, the parameters can be generalized to be frequency-dependent.^{[1]}
Let
Take the positive direction for and in the loop to be clockwise.
We find that
These two first-order equations are easily uncoupled by a second differentiation, with the results:
Notice that both and satisfy the same equation.
Since is independent of and , it can be represented by a single constant . (The minus sign is included for later convenience.) That is:
We can write the above equation as
With this definition of , the position- or -dependent part will appear as in the exponential solutions of the equation, similar to the time-dependent part , so the solution reads
Since the equation for is the same form, it has a solution of the same form:
The above equations are the wave solution for and . In order to be compatible, they must still satisfy the original differential equations, one of which is
Substituting the solutions for and into the above equation, we get
Isolating distinct powers of and combining identical powers, we see that in order for the above equation to hold for all possible values of we must have:
- For the co-efficients of :
- For the co-efficients of :
Since
It can be seen that the constant , defined in the above equations has the dimensions of impedance (ratio of voltage to current) and is a function of primary constants of the line and operating frequency. It is called the “characteristic impedance” of the transmission line, and conventionally denoted by .^{[2]}
Alternative approach[edit]
We follow an approach posted by Tim Healy.^{[3]} The line is modeled by a series of differential segments with differential series and shunt elements (as shown in the figure above). The characteristic impedance is defined as the ratio of the input voltage to the input current of a semi-infinite length of line. We call this impedance That is, the impedance looking into the line on the left is . But, of course, if we go down the line one differential length , the impedance into the line is still . Hence we can say that the impedance looking into the line on the far left is equal to in parallel with and , all of which is in series with and . Hence:
The added terms cancel, leaving
The first-power terms are the highest remaining order. Dividing out the common factor of , and dividing through by the factor , we get
In comparison to the factors whose divided out, the last term, which still carries a remaining factor , is infinitesimal relative to the other, now finite terms, so we can drop it. That leads to
Reversing the sign ± applied to the square root has the effect of reversing the direction of the flow of current.
Lossless line[edit]
The analysis of lossless lines provides an accurate approximation for real transmission lines that simplifies the mathematics considered in modeling transmission lines. A lossless line is defined as a transmission line that has no line resistance and no dielectric loss. This would imply that the conductors act like perfect conductors and the dielectric acts like a perfect dielectric. For a lossless line, R and G are both zero, so the equation for characteristic impedance derived above reduces to:
In particular, does not depend any more upon the frequency. The above expression is wholly real, since the imaginary term j has canceled out, implying that is purely resistive. For a lossless line terminated in , there is no loss of current across the line, and so the voltage remains the same along the line. The lossless line model is a useful approximation for many practical cases, such as low-loss transmission lines and transmission lines with high frequency. For both of these cases, R and G are much smaller than ωL and ωC, respectively, and can thus be ignored.
The solutions to the long line transmission equations include incident and reflected portions of the voltage and current:
Surge impedance loading[edit]
In electric power transmission, the characteristic impedance of a transmission line is expressed in terms of the surge impedance loading (SIL), or natural loading, being the power loading at which reactive power is neither produced nor absorbed:
Loaded below its SIL, the voltage at the load will be greater than the system voltage. Above it, the load voltage is depressed. The Ferranti effect describes the voltage gain towards the remote end of a very lightly loaded (or open ended) transmission line. Underground cables normally have a very low characteristic impedance, resulting in an SIL that is typically in excess of the thermal limit of the cable.
Practical examples[edit]
Standard | Impedance (Ω) |
Tolerance |
---|---|---|
Category 5 | 100 | ±5Ω^{[4]} |
USB | 90 | ±15%^{[5]} |
HDMI | 95 | ±15%^{[6]} |
IEEE 1394 | 108 | ^{+3%} _{−2%}^{[7]} |
VGA | 75 | ±5%^{[8]} |
DisplayPort | 100 | ±20%^{[6]} |
DVI | 95 | ±15%^{[6]} |
PCIe | 85 | ±15%^{[6]} |
Overhead power line | 400 | Typical^{[9]} |
Underground power line | 40 | Typical^{[9]} |
The characteristic impedance of coaxial cables (coax) is commonly chosen to be 50 Ω for RF and microwave applications. Coax for video applications is usually 75 Ω for its lower loss.
See also[edit]
- Ampère's circuital law – Concept in classical electromagnetism
- Characteristic acoustic impedance – Opposition that a system presents to an acoustic pressure
- Iterative impedance, characteristic impedance is a limiting case of this
- Maxwell's equations – Equations describing classical electromagnetism
- Wave impedance – constant related to electromagnetic wave propagation in a medium
- Space cloth – Hypothetical plane with resistivity of 376.7 ohms per square.
References[edit]
- ^ ^{a} ^{b} "The Telegrapher's Equation". mysite.du.edu. Retrieved 9 September 2018.
- ^ ^{a} ^{b} "Derivation of Characteristic Impedance of Transmission line". GATE ECE 2018. 16 April 2016. Archived from the original on 9 September 2018. Retrieved 9 September 2018.
- ^ "Characteristic Impedance". www.ee.scu.edu. Archived from the original on 2017-05-19. Retrieved 2018-09-09.
- ^ "SuperCat OUTDOOR CAT 5e U/UTP" (PDF). Archived from the original (PDF) on 2012-03-16.
- ^ "Chapter 2 – Hardware". USB in a NutShell. Beyond Logic.org. Retrieved 2007-08-25.
- ^ ^{a} ^{b} ^{c} ^{d} "AN10798 DisplayPort PCB layout guidelines" (PDF). Archived (PDF) from the original on 2022-10-09. Retrieved 2019-12-29.
- ^ "Evaluation" (PDF). materias.fi.uba.ar. Archived (PDF) from the original on 2022-10-09. Retrieved 2019-12-29.
- ^ "VMM5FL" (PDF). pro video data sheets. Archived from the original (PDF) on 2016-04-02. Retrieved 2016-03-21.
- ^ ^{a} ^{b} Singh 2008, p. 212.
Sources[edit]
- Guile, A.E. (1977). Electrical Power Systems. ISBN 0-08-021729-X.
- Pozar, D.M. (February 2004). Microwave Engineering (3rd ed.). ISBN 0-471-44878-8.
- Ulaby, F.T. (2004). Fundamentals of Applied Electromagnetics (media ed.). Prentice Hall. ISBN 0-13-185089-X.
- Singh, S. N. (23 June 2008). Electric Power Generation: Transmission and Distribution (2 ed.). PHI Learning Pvt. Ltd. p. 212. ISBN 9788120335608. OCLC 1223330325.
External links[edit]
This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.