### Abacus of smith

Definition
The abacus of Smith is a diagram which allows, inter alia, of: to represent the variation of impedance of a dipole (antenna, filter) according to the frequency to calculate the impedance of a network (coupling unit, filters) formed of reactive elements to dimension a circuit of adaptation of impédancecalculer the impedance of a load seen through a line Any impedance of the form Z = R ± jX can be represented by a point on the diagram. Resistance R can take any value between 0 and the infinite one and reactance X can be included/understood between - the infinite one and + the infinite one.
Description
Usually the abacus of Smith is appeared as a circular graph and a whole of scales (reference mark H). It is not necessary to know all the details of them to be able to include ⁄ understand basic operation of it. With the first glance one distinguishes: A: surface of inductive reactances (higher half of the circle) b: surface of the capacitive reactances (lower half of the circle) C: center null reactances (or pure resistances) D: origin of the axis C, null resistance E: center circle corresponding to the Z = 1+j0F impedance: end of the axis C, resistance (and reactances) infiniesG: scale of the angles of dephasing and lengths of lignesH: together of scales facilitating calculations of losses, ROS the positioning of an impedance Z is done through its two components R and X.
The values of reactances X and resistances R are locatable using two families of circles: circles of the résistancescercles of the reactances Of other circles are used: circles of the ROS constantscercles of stabilitécercles of the factors of noises circles of the constant factors Q the majority of the concepts making it possible to explain the operation of the lines can be included ⁄ understood thanks to the abacus of Smith.
Circles of constant resistances
By observing the reproduction above one notices a whole of tangent circles at the point F: they are the circles of constant resistance. All the points red on the figure correspond to complex impedances whose resistive component is equal to 0,3.Le circle of null resistances is also the axis of the reactances. It constitutes the limit of the abacus, its center is the point of the impedance of reference; for example: Z = 50+j0Le segment DF is the axis of resistances. The point D corresponds to a null impedance: Z = 0+j0 (for example a short-circuit) the point F represents an impedance whose resistive component has an infinite value (for example an open line).
Circles of the constant reactances
The second whole of circles is that of the reactances. The centers of each one of these circles are located on a tangent line out of F at the circle of null resistances (in gray on the figure) which is also the axis of the reactances. All the points located on the same circle correspond to impedances of which the reactive part with the same value.
For example, all the blue points have the same reactance - j0,4.Les circles of the inductive reactances (in red on the figure) is located at the top of the axis of resistances, the circles of the capacitive reactances (in blue on the figure) are located at the lower part of the axis of resistances. The circle of the null reactances (in black on the figure) has an infinite radius, it is also the axis of resistances.
Positioning of a point, standardization
In the problems utilizing a line, the impedance characteristic of the line is used as value of reference for calculations, it is it which is represented by the central point of the abacus. There exist abacuses on paper with the reference 50 ohms but, as it is not possible to print abacuses for all the values of impedance of line (75, 300,400,600) one makes use of a universal diagram and to be able to treat all the cases some is the value of the impedance characteristic Zo of the line, it exists a standardized paper medium whose central reference is Z = 1,0+j0.
This principle has also other advantages: direct display of the value of the ROS, for example. If Zo = 50 ohms, it is enough to divide by 50 each term to the complex impedance. The result is then a complex number without unit. Example 1: That is to say the impedance Z = 15+j20 ohms. While dividing by 50 ohms each term of Z one standardizes Z and one obtains 0,3 + j0,4 still called reduced impedance.This point is with the intersection of the circle of resistances equal to 0,3 and to the circle of the reactances equal to + 0,4Pour to find the impedance in ohms it is enough to multiply each term by the impedance of reference.
Example 2: The point 1,4-j 1,2 represents the impedance Z = 1,4x50-j1,2x50 = 70-j60L’ abacus of Smith allows the very easy use of the admittances, one standardizes of which also the admittances compared to Yo, the admittance corresponding to the Zo impedance.
Circles of the constant ROS
The whole of the circles of constant ROS is centered on the point corresponding to Zo.Le rings infinite ROS corresponding to that of null resistances and constitutes the limit of the abacus. The circle of ROS=1 is the center of the abacus and corresponds to the point of impedance 1+j0, the impedance of reference. All the points located on the same circle of ROS have an identical ROS and reciprocally, two different impedances which cause the same ROS are located on the same circle of ROS.
Circles of ROS
They are not represented on the abacuses paper not to overload because it is easy to then trace them with a compass so necessary. The center of the circles of ROS is that of the abacus, the circle of infinite ROS corresponding to that of null resistances and constitutes the limit of the abacus.
All the circles of ROS are concentric, which is an exception in the abacus of Smith.Le rings ROS = 1 is the center of the abacus and corresponds to the point of impedance 1 + j0, the impedance of reference. In a circuit using a line, it is the case of a load perfectly adapted to the impedance characteristic of this one, when the totality of transmitted energy is absorbed by the load.
All the points located on the same circle of ROS have an identical ROS and reciprocally, two different impedances which cause the same ROS are located on the same circle of ROS.
Tracing of a circle of ROS
With a standardized sheet (impedance of the point central = 1+j0), the radius of the circle of ROS is easy to measure on the sheet because it is also the distance between the central point and the graduation on the axis of resistances. One can also refer on a radial scale placed in bottom of sheet. In a circuit using a line without losses, the circle of ROS is also used for the calculation of the impedances by taking account of the line as a component (see measurement on the lines with the abacus of Smith.
Band-width of an antenna
The figure opposite represents the variation of the impedance measured with the bottom of a coaxial line (75 ohms) supplying a dipole half-wave for the tape 40 meters. Measurement was taken with a impedancemeter of antenna between 6,4 and 7,5 MHz.
The tape amateur of the 40 meters was underlined by a magenta segment of color. After tracing point by point, two circles of ROS were added. They make it possible in a glance to see how the antenna will function on the plan of the ROS. One note: the antenna is too long, its frequency of resonance is 6,92 MHz.
resonance, the impedance of the antenna is equal to the impedance characteristic of the coaxial line. The ROS is of 1 for this frequency. The ROS will vary from 1,2 to 1,6 inside tape 40 m.Pour to know the ROS with 6,4 MHz it is enough to measure with one triple-decimetre the distance between the point 6,4MHz and sheet centers it and to defer this measurement on the scale placed in bottom of the sheet.
The radial scale located in lower part of the circular abacus of Smith, it constitutes an abacus in itself which makes it possible to find without calculation several parameters resulting from/to each other. It on the same scale as the radius of the abacus of Smith what makes it possible to pass from the one to the other using a compass or one triple-decimetre by deferring the lengths on paper.
Definition of the various scales
In a lossless line, the ROS is constant some is the place of measurement. If the line is not perfect, the amplitude of the considered wave decreases as one moves away from the load since part of the energy which it transports is dissipated because of the losses in the lines. The phenomenon is the same one for the direct wave whose amplitude increases when one approaches the generator.
Thus the coefficient of reflection and the ROS are they larger close to the load than towards the generator. The ROS measured on the line between Za (impedance of the load, an antenna, for example) and Ze (impedance of the generator, the transmitter for example) described a spiral and name pluis a circle. Knowing line length and the loss which it causes, one can find by graphic determination the impedance of the load by measuring the impedance at the other end of the line
The ROS and antennas
The report ⁄ ratio of standing waves (ROS) is an indicator of the good performance of the feeding system of an antenna.
It does not express qualities of the antenna itself but the fact that celleci can be connected to a transmitter without risk for this last. A report ⁄ ratio of standing waves high has several disadvantages: overpressure on the level of the transmitter with risk of destruction of the amplifier of power, startings in the CV bad output of the power supply of the antenna, the transmitter which cannot output all its power.
The recent transceivers have moreover for the majority a device which automatically limits the power of emission in the event of raised ROS, a transmitter of 100 Watts can be limited to a few Watts. the radio ham being a concerned experimenter of the optimization of his station, it is tempted to regard as a failure a figure raised for the ROS of its antenna, its goal is to obtain a value close to 1 (1 ⁄ 1 or 1 out of 1 as it is current to say). The measurement of the ROS is thus for him an occasion to detect a problem to be solved and provides him consequently matter with reflection
Direct wave, reflected wave, standing wave
Let us suppose a characteristic line of impedance Zc = 50 ohms feeding a load made up of a pure resistance of 25 ohms. If the line is fed by a transmitter of 100 Watts, part of the direct wave conveying this power will be reflected, the other part being absorbed by the load. The direct wave corresponds to a Ud tension and a Id intensity which depend on the power provided by the transmitter and the impedance of the line.
In the same way for the considered wave: Ur tension and intensity Ir to facilitate the comprehension of the phenomenon it happens that one speaks about direct power Pd and considered power pr. the considered power that would be provided by the load, which is unusual
These fictitious powers can be measured with a directional wattmeter. Direct wave and considered wave combine and form a standing wave whose amplitude varies throughout the line, passing by the maximum ones (bellies) and minima (nodes) of intensity and tension: Umax: maximum of tension (belly of tension) Umin: minimum of tension (node of tension) Imax: maximum of intensity (belly of intensity) Imin: minimum of intensity (node of intensity)
Highlighting of the standing waves
A simple experiment makes it possible to highlight the standing waves present on a line. A line To lick is fed at an end by a generator at frequency variable and charged at its other end by a nonreactive resistance sufficiently powerful and adjustable value. While varying the resistance of the load one can measure the variations of the amplitude mini Umin and maximum Umax of the tension between wire of the line. When impedance Z of the load is equal to the impedance characteristic of the line (here 220 ohms) Umax=Umin and the ROS is equal to 1.
Coefficient of reflection
The coefficient of reflection R is a report ⁄ ratio which is calculated starting from the tensions or of the currents of the direct wave and the considered wave : ρ = Ur ⁄ Ud = Ir ⁄ Id
And if one considers the notions of power of the direct wave and the considered wave : ρ = √ Pr ⁄ Pd
One can calculate it starting from the complex impedance of the load which, in the case of an antenna, is most of the time reactive.
The impedance characteristic of the line being normally resistive, one can write Zc = Rc (50, 75,300 ohms) ρ = √ [(Ra - Rc)² + (Xa)²] ⁄ [(Ra + Rc)² + (Xa)²]
The coefficient of reflection is a vector (generally indicated by the letter G from which one comes to see module R. His argument is the angle of dephasing Q between the considered wave and the direct wave, it varies recurringly throughout the line, it is positive for the points of the line where the reactances are inductive and negative where the reactance is capacitive.
Calculation of the ROS
By definition the ROS is the relationship between the maximum and the minimum of tension raised on the line on the level of a belly of tension (idem for the currents Imax and Imin).ROS = Umax ⁄ Umin = Imax ⁄ Imin
One can also express it starting from the coefficient of reflection : ROS = 1 + ρ ⁄ 1 - ρ
If the line is charged by a pure resistance : ROS = R ⁄ Zc or else ROS = Zc ⁄ R
According to whether the characteristic impedance Zc is larger or smaller than R, so that the ROS is expressed by a number higher than 1.Il exists several types of ROS measures adapted to each use.
Loss of power because of the ROS
The report ⁄ ratio of the power absorptive by the load with the direct power provided by the transmitter can be given using the following formula : Pa ⁄ Pd = 1 - ρ²
Curve Ci against watch that the fall of power is unimportant as long as the ROS does not exceed 2 and remains acceptable for a ROS of 3. The fall of signal in dB compared to the full power is hardly visible in the correspondent: ROS = 1,5: power not absorbed = 4% (0,2dB) ROS = 2: power not absorbed = 11% (0,5dB) ROS = 3: power not absorbed = 25% (1,2dB) On decametric or when the connection is comfortable, to lose 1dB is not a concern.
On the other hand, when one drives out the decibel as in traffic EME (by reflection on the Moon), a ROS of 2 is already unacceptable. This loss of power, which one must distinguish from the loss of energy due to the higher current corresponding to the reflected wave, can be expressed in decibels. One indicates it under abbreviation RL : Rl = 20 * log (ρ)
Losses in the line
The presence of standing waves thus implies stronger currents in the line of the higher losses in the cable than when the forward current is only present. These losses are visible only for big lengths of cable and at raised frequencies, when the ROS exceeds 2.
Improvement of the ROS
It was seen that the use of a line of poor quality improved the measured figure of the ROS. It is obviously not a solution. Simplest is to adapt the impedance of the load and the specific impedance of the line, either by replacing the line, or while acting on the level of the antenna.
The TOS or standing wave ratio
As rate, the TOS is expressed in %. When the ROS is equal to 1, the standing wave ratio is equal to 0%. A TOS of 100% corresponds to an infinite ROS. One obtains the TOS while multiplying by 100 the coefficient of reflection

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