The characteristic impedance or surge impedance (usually written Z0) 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.
In radio-frequency engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.
A signal travelling along an electrical transmission line will be partly, or wholly, reflected back in the opposite direction when the travelling signal encounters a discontinuity in the characteristic impedance of the line, or if the far end of the line is not terminated in its characteristic impedance. This can happen, for instance, if two lengths of dissimilar transmission lines are joined together.
The input impedance of an electrical network is the measure of the opposition to current (impedance), both static (resistance) and dynamic (reactance), into the load network that is external to the electrical source. The input admittance (1/impedance) is a measure of the load's propensity to draw current. The source network is the portion of the network that transmits power, and the load network is the portion of the network that consumes power.
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.
In electric and electronic systems, reactance is the opposition of a circuit element to a change in current or voltage, due to that element's inductance or capacitance. The notion of reactance is similar to electric resistance, but it differs in several respects.
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 definition extends to DC by letting tend to 0, and subsists for finite transmission lines until the wave reaches the end of the line. In this case, there will be in general a reflected wave which travels back along the line in the opposite direction. When this wave reaches the source, it adds to the transmitted wave and the ratio of the voltage and current at the input to the line will no longer be the characteristic impedance. This new ratio 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. It can be shown that an equivalent definition is: 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,the general expression for the characteristic impedance of a transmission line is:
The telegrapher's equations are a pair of coupled, linear partial differential equations that describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who developed the transmission line model in the 1880s. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can form along the line.
Although an infinite line is assumed, since all quantities are per unit length, 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:
where the superscripts and represent forward- and backward-traveling waves, respectively. A surge of energy on a finite transmission line will see an impedance of Z0 prior to any reflections arriving, hence surge impedance is an alternative name for characteristic impedance.
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 , and the time dependence will factor out, leaving an ordinary differential equation for the coefficients, which will be phasors depending on space only. Moreover, the parameters can be generalized to be frequency-dependent.
Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named volt. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for volt uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by ∆V, but more often simply as V, for instance in the context of Ohm's or Kirchhoff's circuit laws.
An electric current is the rate of flow of electric charge past a point or region. An electric current is said to exist when there is a net flow of electric charge through a region. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).
In physics and engineering, a phasor, is a complex number representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as phasor, complex amplitude, and sinor or even complexor.
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 . That is:
The minus sign is included for later convenience. Because of it, we can write the above equation as
which is correct for all transmission lines. And for typical transmission lines, that are built to make wire resistance loss small and insulation leakage conductance low, the constant is very close to being a real number:
Further, 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
where and are the constants of integration. When we recombine the time-dependent part we obtain the full solution:
Since the equation for is the same form, it has a solution of the same form:
where and are again constants of integration.
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:
hence, for valid solutions require
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 .
for any transmission line, and for well-functioning transmission lines, with and both very small, or very high, or all of the above, we get
hence the characteristic impedance is typically very close to being a real number (see also the Heaviside condition.)
We follow an approach posted by Tim Healy. 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 line is modeled by a series of differential segments with differential series
The terms cancel, leaving
The first-power terms are the highest remaining order. In comparison to , the term with the factor may be discarded, since it is infinitesimal in comparison, leading to:
Reversing the sign on the square root has the effect of changing the direction of the flow of current.
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 Z0 is purely resistive. For a lossless line terminated in Z0, 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:
When the line is terminated with its characteristic impedance, the reflected portions of these equations are reduced to 0 and the solutions to the voltage and current along the transmission line are wholly incident. Without a reflection of the wave, the load that is being supplied by the line effectively blends into the line making it appear to be an infinite line. In a lossless line this implies that the voltage and current remain the same everywhere along the transmission line. Their magnitudes remain constant along the length of the line and are only rotated by a phase angle.
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:
in which is the line-to-line voltage in volts.
Loaded below its SIL, a line supplies reactive power to the system, tending to raise system voltages. Above it, the line absorbs reactive power, tending to depress the voltage. 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. Hence a cable is almost always a source of reactive power.
|Ethernet Cat.5||100||±5 Ω|
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.
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.
The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.
Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.
A resistor–capacitor circuit, or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.
In electronics, a voltage divider is a passive linear circuit that produces an output voltage (Vout) that is a fraction of its input voltage (Vin). Voltage division is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two resistors connected in series, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.
The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity. It is a degenerate form of a voltage-controlled voltage-source (VCVS) filter topology.
The Smith chart, invented by Phillip H. Smith (1905–1987), is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.
Acoustic impedance and specific acoustic impedance are measures of the opposition that a system presents to the acoustic flow resulting from an acoustic pressure applied to the system. The SI unit of acoustic impedance is the pascal second per cubic metre or the rayl per square metre, while that of specific acoustic impedance is the pascal second per metre or the rayl. In this article the symbol rayl denotes the MKS rayl. There is a close analogy with electrical impedance, which measures the opposition that a system presents to the electrical flow resulting from an electrical voltage applied to the system.
An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.
A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many techniques for calculating these values. However, for the most part, the techniques assume linear components. Except where stated, the methods described in this article are applicable only to linear network analysis.
The Heaviside condition, named for Oliver Heaviside (1850–1925), is the condition an electrical transmission line must meet in order for there to be no distortion of a transmitted signal. Also known as the distortionless condition, it can be used to improve the performance of a transmission line by adding loading to the cable.
Ripple in electronics is the residual periodic variation of the DC voltage within a power supply which has been derived from an alternating current (AC) source. This ripple is due to incomplete suppression of the alternating waveform after rectification. Ripple voltage originates as the output of a rectifier or from generation and commutation of DC power.
Zobel networks are a type of filter section based on the image-impedance design principle. They are named after Otto Zobel of Bell Labs, who published a much-referenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the input impedance is fixed in the design independently of the transfer function. This characteristic is achieved at the expense of a much higher component count compared to other types of filter sections. The impedance would normally be specified to be constant and purely resistive. For this reason, Zobel networks are also known as constant resistance networks. However, any impedance achievable with discrete components is possible.
The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.
Characteristic admittance is the mathematical inverse of the characteristic impedance. The general expression for the characteristic admittance of a transmission line is:
An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.
Performance modelling is the abstraction of a real system into a simplified representation to enable the prediction of performance. The creation of a model can provide insight into how a proposed or actual system will or does work. This can, however, point towards different things to people belonging to different fields of work.