In microwave and radio-frequency engineering, a stub is a length of transmission line or waveguide that is connected at one end only. The free end of the stub is either left open-circuit or (especially in the case of waveguides) short-circuited. Neglecting transmission line losses, the input impedance of the stub is purely reactive; either capacitive or inductive, depending on the electrical length of the stub, and on whether it is open or short circuit. Stubs may thus be considered to be frequency-dependent capacitors and frequency-dependent inductors.
Because stubs take on reactive properties as a function of their electrical length, stubs are most common in UHF or microwave circuits where the line lengths are more manageable. Stubs are commonly used in antenna impedance matching circuits and frequency selective filters. Smith charts can also be used to determine what length line to use to obtain a desired reactance. In their simplest form, stubs that are some multiple of a quarter wavelengths are used to kill harmonics or sub-harmonics of the operating frequency.
One property of a transmission line is that it inverts the impedance every quarter wavelength, so a quarter-wave stub with a short at one end looks like an open circuit at the other end. At the second harmonic, that same stub is two quarter-waves long, so the short is inverted twice, and looks like a short circuit to the harmonic. At the third harmonic, it’s three quarter-waves, so it’s an open circuit again, then at a full wavelength it’s a short circuit again. Likewise, a quarter wave stub that is open at the far end looks like a short circuit, but an open circuit at the second harmonic.
A stub works by placing a short circuit across the line at the frequency of interference. Placing a stub on a line forms a voltage divider between the line impedance and the stub impedance. The higher the line impedance at the point of connection, the greater the attenuation. If the line is well matched at both ends, the position doesn’t matter, because the impedance at every point along the line is the same. But if there’s a mismatch, the impedance will vary along the length of the line, and if the impedance at the location of the stub is low, it won’t be very effective.
Stub matching In a strip line circuit, a stub may be placed just before an output connector to compensate for small mismatches due to the device’s output load or the connector itself. Stubs can be used to match load impedance to the transmission line characteristic impedance. The stub is positioned a distance from the load. This distance is chosen so that at that point the resistive part of the load impedance is made equal to the resistive part of the characteristic impedance by impedance transformer action of the length of the main line.
The length of the stub is chosen so that it exactly cancels the reactive part of the presented impedance. That is, the stub is made capacitive or inductive according to whether the main line is presenting inductive or capacitive impedance respectively. This is not the same as the actual impedance of the load since the reactive part of the load impedance will be subject to impedance transformer action as well as the resistive part. Matching stubs can be made adjustable so that matching can be corrected on test. A single stub will only achieve a perfect match at one specific frequency.
For wideband matching several stubs may be used spaced along the main transmission line. The resulting structure is filter-like and filter design techniques are applied. For instance, the matching network may be designed as a Chebyshev filter but is optimised for impedance matching instead of passband transmission. The resulting transmission function of the network has a passband ripple like the Chebyshev filter, but the ripples never reach 0dB insertion loss at any point in the passband, as they would do for the standard filter. Stub matching is of two types: i) Single stub matching (ii) Double stub matching (iii) Triple stub matching (i) Single stub matching:- (a) Connection of stub in parallel with transmission line It consists of an open or short circuited section of transmission line of length lt connected in parallel with the main line at distance ls from the load ZR Stub has the same characteristic impedance as the main line. Use of a single stub to provide impedance matching. It is shown that a transmission line having characteristic impedance of ‘Z0’ is terminated in a complex load admittance of (gR +jbR) First step:-
Locate a point nearest to the load on the transmission line where the normalized admittance is (1+jbR) Second step: A stub (short or open circuited transmission line) is added in parallel across the transmission line at a point so as to offer a suscetpance of –jbR. Thus the transmission line with a characteristic impedance of ‘Z0’ gets matched to a complex load up to that point. We connect the stub in parallel with the main line as it is easier to deal with the admittance as they can be added up. (b) Connection of stub in series with transmission line
Here also the first step is to locate a point on the transmission line where the normalized impedance looking towards the load end is(1+j X) . At that point a stub is added with the stub offering a normalized reactance of (-j X). The feed line needs to be cut for insertion of series stub. This technique is therefore not commonly used as it is difficult to fabricate in co-axial and strip lines. (ii) Double stub matching:- In single stub matching, the stub is placed on the line at a specified point. Its location varies with ZR and frequency. This creates some difficulties as the specified point may occur at an undesirable location.
In such cases, double stubs are used. Here the distance between the two stubs is fixed such as ?/16,?/8,3?/16,3?/8 or even closer and the lengths of the two stubs are adjusted to match the load. We can use two stubs permanently attached to the line at fixed points of attachment, and tune by altering the stub lengths. Two values have to be matched (r and x) and we have two variables; the length of each stub. As before, the generator-end stub has reactance -jx’ and is attached at a point where the line impedance, including the effect of the other stub at its fixed point of attachment, is 1+jx’.
Transforming the unit r=1 circle towards the load until one reaches the load-end stub attachment, the circle r=1 transforms to another circle, call it “B”, touching the outside of the SMITH chart, and also passing through its center. The load impedance, when transformed towards the generator up to the load-end stub position, will be a generalized impedance ZL’ different from ZL. The effect of the load-end stub is to add reactance x” to ZL’ so that the impedance value ZL’+jx” lies on the circle “B” above. We chose the length of the stub to make x” the required value for this to happen.
If we write ZL’=r’+jx’ then the effect of adding the stub is to move the reactance j(x’+x”) along the constant r’ curve depending on the size of x”. Double stub matching is preferred over single stub matching. 1. Single stub matching is useful for a fixed frequency so as frequency changes The location of single stub will have to be changed. 2. The single stub matching system is based on the measurement of voltage Minimum. Hence for coaxial line it is very difficult to get such voltage minimum, without using slotted line section. iii) Triple stub tuners and E-H tuners:-
It is just possible for the r’ curve not to intersect the circle “B”, in which case a double stub match is not possible for this value of load impedance, and stub placements. Generalized adjustable tuners are therefore designed with three stubs, which are spaced at unequal intervals. Such a device is called a “Triple Stub Tuner”. Sliding shorts are easily arranged in coax or waveguide. In waveguide only, there is a special type of tuner called an E-H tuner. This has shunt and series side arms consisting of sliding shorts, attached at the same point along the guide.
There is no equivalent in 2-conductor transmission line for geometrical reasons. An E-H tuner can always match any load impedance Short Circuit stubs Shorted quarter-wave stubs are used to kill even-order harmonics (2nd, 4th, 6th, and 8th). For example, a shorted quarter-wave 80M stub passes 80M, but kills 40M, 20M, and 10M. A shorted quarter-wave stub for 40M passes 40M but kills 20M and 10M. Shorted stubs are both transmitting and receiving stubs – they kill harmonics of our transmitter, and they also reject signals on harmonically related bands received from nearby transmitters.
Generally short circuited stubs are preferred comparable to open circuited stub as open circuited stub radiates some energy at high frequencies. A short circuited stub is preferred to an open circuited stub because of greater ease in constructions and because of the inability to maintain high enough insulation resistance at the open –circuit point to ensure that the stub is really open circuited . A shorted stub also has a lower loss of energy due to radiation ,since the short –circuit can be definitely established with a large metal plate ,effectively stopping all field propagation.
Open Circuitstubs An open half-wave stub is primarily a receiving stub, used to minimize the received Signal from transmitters on bands below the one where we’re listening. For example, a half-wave open 40M stub will look “open” on 40M, but will be a quarter-wave on 80M, look like a short, and attenuate the 80M fundamental by about 25dB. How stubs length can be varied to reactance values ? An open ended stub of a fixed length will present a certain reactance to the antenna at the feed point. When this reactance cancels the reactance of the patch, a resonant point will appear.
This resonant point occurs for only one value of reactance, therefore by changing the length of the stub, we change the frequency that corresponds to this reactance and hence the location of the resonance point. Furthermore, at the primary resonance point, a new value of reactance is loaded on to the antenna also shifting its location. In general, the two resonances move in the same direction. This is illustrated in Figures What limits power flow through an overhead transmission line? The answer is it depends.
It depends on what is limiting the power flow and how much of an increase is needed to solve the problem. In most circumstances, power flow limits are the result of concerns over electrical phase shift, voltage drop or thermal effects in lines, cables or substation equipment. Surge Impedance Loading Limits: As power flows along a transmission line, there is an electrical phase shift, which increases with distance and with power flow. As this phase shift increases, the system in which the line is embedded can become increasingly unstable during electrical disturbances.
Typically, for very long lines, the power flow must be limited to what is commonly called the Surge Impedance Loading (SIL) of the line. Surge Impedance Loading is equal to the product of the end bus voltages divided by the characteristic impedance of the line. Since the characteristic impedance of various HV and EHV lines is not dissimilar, the SIL depends approximately on the square of system voltage. Typically, stability limits may determine the maximum allowable power flow on lines that are more than 150 miles in length.
For very long lines, the power flow limitation may be less than the SIL Table 0-1 – Power Flow Limits on Lines and Cables SystemkV Transmission Overhead Line Characteristics| XL(?/mi)| XC (M?-mi)| SurgeImpedance (?)| SIL(MW)| ThermalRating (MW)| 230 | 0. 75| 0. 18| 367| 145| 440| 345 | 0. 60| 0. 15| 300| 400| 1500| 500 | 0. 58| 0. 14| 285| 880| 3000| 765 Transmission Cable Characteristics| 0. 56| 0. 14| 280| 2090| 8000| 345 | 0. 25 | 0. 0060 | 39 | 3050 | 2100| Voltage Drop Limits: In addition to electrical phase shift, voltage magnitude decreases with distance.
Generally, for transmission lines, the maximum allowable drop in voltage is limited to between 5% and 10% of the sending end bus voltage. The power flow (in MVA or MW) that corresponds to the maximum allowable decrease in voltage magnitude is called the lines voltage drop limit. As with phase shift, a transmission lines voltage drop limit. Decreases with transmission distance and is generally higher than the lines thermal limit for short lines but less than the lines stability limit for very long lines Voltage drop. Normally limits power flow on HV or EHV lines that are between 50 and 50 miles in length. Voltage drop limits on power flow can be as low as 40% of the lines thermal limit. Voltage drop limits may be increased by the addition of shunt capacitors at the end of the line. Such solutions are typically much cheaper than rebuilding the line. Thermal Limits: Thermal power flow limits on overhead lines are intended to limit the temperature attained by the energized conductors and the resulting sag and loss of tensile strength. In most cases, the maximum conductor temperature applied to modern transmission lines reflect ground clearance concerns rather than annealing of aluminum.
Thermal limits, as typically calculated, are not a function of line length. Thus for a given line design, a line 1 km long and one 500 km long typically have the same thermal limit. Thermal limits usually determine the maximum power flow for lines less than 50 miles in length. There are a number of possible methods by which the MVA thermal capacity of an existing line may be increased. Some of these methods are technically straightforward, such as reinforcing the structures and restringing the line with a larger conductor. These methods come at a price, however.
In addition to the dollar cost involved, there is construction out on the line, and either outage time or special construction methods to allow service while the work is in progress. Other methods of thermal up rating, such as the use of weather dependent dynamic thermal ratings or voltage up rating by reduction from normal phase spacing, may require little or no line outage time and less capital investment than reconductoring and reinforcing the structures. The price here lies in the greater degree of technical sophistication required to ensure safe and reliable operation at higher loadings
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