1423:
108:
3506:
43:
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direction). Thus, at each boundary there are four reflection coefficients (voltage and current on one side, and voltage and current on the other side). All four are the same, except that two are positive and two are negative. The voltage reflection coefficient and current reflection coefficient on the same side have opposite signs. Voltage reflection coefficients on opposite sides of the boundary have opposite signs.
459:). For two impedances to be complex conjugates their resistances must be equal, and their reactances must be equal in magnitude but of opposite signs. In low-frequency or DC systems (or systems with purely resistive sources and loads) the reactances are zero, or small enough to be ignored. In this case, maximum power transfer occurs when the resistance of the load is equal to the resistance of the source (see
703:
circuit to match a broad range of load impedance and thus simplify the circuit design. This issue was addressed by the stepped transmission line, where multiple, serially placed, quarter-wave dielectric slugs are used to vary a transmission line's characteristic impedance. By controlling the position of each element, a broad range of load impedances can be matched without having to reconnect the circuit.
778:
738:
634:). To match the impedances, both cables must be connected to a matching transformer with a turns ratio of 2:1. In this example, the 300-ohm line is connected to the transformer side with more turns; the 75-ohm cable is connected to the transformer side with fewer turns. The formula for calculating the transformer turns ratio for this example is:
4047:
These principles are useful in the application of highly energetic materials (explosives). If an explosive charge is placed on a target, the sudden release of energy causes compression waves to propagate through the target radially from the point-charge contact. When the compression waves reach areas
2894:
is the source impedance. The source of waves incident from the line are the reflections from the load end. If the source impedance matches the line, reflections from the load end will be absorbed at the source end. If the transmission line is not matched at both ends reflections from the load will
1389:
devices are intended to cancel the reactive and nonlinear characteristics of a load at the end of a power line. This causes the load seen by the power line to be purely resistive. For a given true power required by a load this minimizes the true current supplied through the power lines, and minimizes
702:
devices can match a specific range of load impedances. For example, in order to match an inductive load into a real impedance, a capacitor needs to be used. If the load impedance becomes capacitive, the matching element must be replaced by an inductor. In many cases, there is a need to use the same
618:
to the same waveform at another voltage. The power input to the transformer and output from the transformer is the same (except for conversion losses). The side with the lower voltage is at low impedance (because this has the lower number of turns), and the side with the higher voltage is at a higher
3645:
and musical instruments. Because most driver impedances are poorly matched to the impedance of free air at low frequencies, loudspeaker enclosures are designed to both match impedance and minimize destructive phase cancellations between output from the front and rear of a speaker cone. The loudness
1722:
which side of the boundary is medium 1 and which side is medium 2. With a single transmission line it is customary to define the voltage reflection coefficient for a wave incident on the boundary from the transmission line side, regardless of whether a source or load is connected on the other side.
3983:
If we cannot change the masses of bodies, then we can match their impedance with a lever. Imagine a large ball dropping to the ground, and a small ball lying on the ground. The large ball hits the short end of a lever, and the small ball is launched from the long end of the lever. If the lever arm
1717:
There is also a current reflection coefficient, which is the negative of the voltage reflection coefficient. If the wave encounters an open at the load end, positive voltage and negative current pulses are transmitted back toward the source (negative current means the current is going the opposite
1394:
is used to extract the maximum power from a solar panel and efficiently transfer it to batteries, the power grid or other loads. The maximum power theorem applies to its "upstream" connection to the solar panel, so it emulates a load resistance equal to the solar panel source resistance. However,
3492:
is not heard. All devices used in telephone signal paths are generally dependent on matched cable, source and load impedances. In the local loop, the impedance chosen is 600 ohms (nominal). Terminating networks are installed at the exchange to offer the best match to their subscriber lines. Each
1721:
Because they are all the same except for sign it is traditional to interpret the reflection coefficient as the voltage reflection coefficient (unless otherwise indicated). Either end (or both ends) of a transmission line can be a source or a load (or both), so there is no inherent preference for
1414:. It is only necessary for correction to be achieved at one single frequency, the frequency of the supply. Complex networks are only required when a band of frequencies must be matched and this is the reason why simple capacitors are all that is usually required for power factor correction.
396:, will usually achieve a perfect match at only a single frequency. This is because the added element will either be a capacitor or an inductor, whose impedance in both cases is frequency dependent, and will not, in general, follow the frequency dependence of the source impedance. For wide
3528:
amplifiers, impedance-changing transformers are often used to get a low output impedance, and to better match the amplifier's performance to the load impedance. Some tube amplifiers have output transformer taps to adapt the amplifier output to typical loudspeaker impedances.
288:
may be negligible or zero; the impedance can be considered a pure resistance, expressed as a real number. In the following summary we will consider the general case when resistance and reactance are both significant, and the special case in which the reactance is negligible.
544:(complex impedance matching), to avoid echoes. Given a transmission line source with a fixed source impedance, this "reflectionless impedance matching" at the end of the transmission line is the only way to avoid reflecting echoes back to the transmission line.
1735:
In a transmission line, a wave travels from the source along the line. Suppose the wave hits a boundary (an abrupt change in impedance). Some of the wave is reflected back, while some keeps moving onwards. (Assume there is only one boundary, at the load.)
2899:, losing energy on each transit of the transmission line. This can cause a resonance condition and strongly frequency-dependent behavior. In a narrow-band system this can be desirable for matching, but is generally undesirable in a wide-band system.
3627:
helps transfer acoustic energy from the transducer to the body and back again. Without the gel, the impedance mismatch in the transducer-to-air and the air-to-body discontinuity reflects almost all the energy, leaving very little to go into the body.
1441:)—where the length of the line is long compared to the wavelength of the signal (the signal changes rapidly compared to the time it takes to travel from source to load)— the impedances at each end of the line may be matched to the transmission line's
3297:
672:
484:
system), the source and load resistances were matched at 600 ohms. One reason for this was to maximize power transfer, as there were no amplifiers available that could restore lost signal. Another reason was to ensure correct operation of the
493:. Most modern audio circuits, on the other hand, use active amplification and filtering and can use voltage-bridging connections for greatest accuracy. Strictly speaking, impedance matching only applies when both source and load devices are
2884:
4052:. The greater the mismatch, the greater the effect of creasing and spalling will be. A charge initiated against a wall with air behind it will do more damage to the wall than a charge initiated against a wall with soil behind it.
3021:
216:
Impedance is the opposition by a system to the flow of energy from a source. For constant signals, this impedance can also be constant. For varying signals, it usually changes with frequency. The energy involved can be
2753:
1123:
1210:
3967:
376:
1662:
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466:
Impedance matching is not always necessary. For example, if delivering a high voltage (to reduce signal degradation or to reduce power consumption) is more important than maximizing power transfer, then
505:
Adjusting the source impedance or the load impedance, in general, is called "impedance matching". There are three ways to improve an impedance mismatch, all of which are called "impedance matching":
3650:
is directly related to the ratio of the diameter of the speaker to the wavelength of the sound being produced: larger speakers can produce lower frequencies at a higher level than smaller speakers.
1363:, the reactance in parallel, has a negative reactance because it is typically a capacitor. This gives the L-network the additional feature of harmonic suppression since it is a low pass filter too.
1430:
In RF connections, impedance matching is desirable, because otherwise reflections may be created at the end of the mismatched transmission line. The reflection may cause frequency-dependent loss.
1330:
858:
must be an inductor and the other must be a capacitor. One reactance is in parallel with the source (or load), and the other is in series with the load (or source). If a reactance is in parallel
1366:
The inverse connection (impedance step-up) is simply the reverse—for example, reactance in series with the source. The magnitude of the impedance ratio is limited by reactance losses such as the
3575:
The impedance of the loudspeaker on the secondary coil of the transformer will be transformed to a higher impedance on the primary coil in the circuit of the power pentodes by the square of the
1263:
1021:
3768:
2611:
2544:
1709:
3459:
714:
are frequently used to achieve impedance matching in telecommunications and radio engineering. In general, it is not theoretically possible to achieve perfect impedance matching at all
2477:
2419:
547:
Devices intended to present an apparent source resistance as close to zero as possible, or presenting an apparent source voltage as high as possible. This is the only way to maximize
4042:
392:
is purely resistive, then matching can be achieved by adding a reactance of the same magnitude but opposite sign to the load. This simple matching network, consisting of a single
2763:
At the source end of the transmission line, there may be waves incident both from the source and from the line; a reflection coefficient for each direction may be computed with
2359:
2304:
2021:
2072:
1967:
3641:
in loudspeaker systems are used like transformers in electrical circuits to match the impedance of the transducer to the impedance of the air. This principle is used in both
3402:
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3105:
718:
with a network of discrete components. Impedance matching networks are designed with a definite bandwidth, take the form of a filter, and use filter theory in their design.
3149:
2645:
640:
3654:
speakers are a complex case, acting like large speakers lengthwise and small speakers crosswise. Acoustic impedance matching (or the lack of it) affects the operation of a
567:
There are a variety of devices used between a source of energy and a load that perform "impedance matching". To match electrical impedances, engineers use combinations of
72:
3328:
2240:
2212:
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2100:
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1884:
1854:
1826:
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function as a series of levers, which matches mechanical impedance between the eardrum (which is acted upon by vibrations in air) and the fluid-filled inner ear.
3071:
3478:
systems also use matched impedances to minimise echo on long-distance lines. This is related to transmission-line theory. Matching also enables the telephone
2769:
1374:
matching networks can be modeled as infinitely many L-sections wired in cascade. Optimal matching circuits can be designed for a particular system using
2910:
3073:
accounts for everything that happens to the signal in transit (including delay, attenuation and dispersion). If there is a perfect match at the load,
1472:) to prevent reflections of the signal at the ends of the line. In radio-frequency (RF) systems, a common value for source and load impedances is 50
3623:
of the two media are very different most sound energy will be reflected (or absorbed), rather than transferred across the border. The gel used in
3619:
Similar to electrical transmission lines, an impedance matching problem exists when transferring sound energy from one medium to another. If the
2657:
686:
consisting of two resistors. Power loss is an unavoidable consequence of using resistive networks, and they are only (usually) used to transfer
3516:
typically do not match impedances, but provide an output impedance that is lower than the load impedance (such as < 0.1 ohm in typical
4379:
174:
3690:(that depends on the propagation direction) can be calculated for each medium, and may be used in the transmission-line reflection equation
3862:
is the momentum of the system (which remains constant throughout the collision), the energy of the second body after the collision will be
3773:
to calculate reflection and transmission coefficients for the interface. For non-magnetic dielectrics, this equation is equivalent to the
241:. The concept of electrical impedance is perhaps the most commonly known. Electrical impedance, like electrical resistance, is measured in
31:
4044:, then all energy would be transferred to the small ball if collisions are elastic. This is roughly how the middle ear works (see above).
1032:
185:, or properly proportioned transmission lines. Practical impedance-matching devices will generally provide best results over a specified
729:. This would provide a perfect match at one specific frequency only. Wide bandwidth matching requires filters with multiple sections.
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collides elastically with a second body, maximum energy transfer to the second body will occur when the second body has the same mass
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4348:
4292:
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4243:
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192:
The concept of impedance matching is widespread in electrical engineering, but is relevant in other applications in which a form of
94:
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is the one-way transfer function (from either end to the other) when the transmission line is exactly matched at source and load.
307:
1583:
1493:
3797:. In a head-on collision of equal masses, the energy of the first body will be completely transferred to the second body (as in
1422:
1714:
so the reflection coefficient is the same (except for sign), no matter from which direction the wave approaches the boundary.
4310:
3682:. For non-magnetic materials, the refractive index is inversely proportional to the material's characteristic impedance. An
298:
1271:
4425:
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for example). In this case, the masses act as "mechanical impedances", which must be matched to maximize energy transfer.
397:
1217:
952:
560:
3696:
385:. A conjugate match is different from a reflection-less match when either the source or load has a reactive component.
55:
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of the inductor. Multiple L-sections can be wired in cascade to achieve higher impedance ratios or greater bandwidth.
489:
used at central exchange equipment to separate outgoing from incoming speech, so these could be amplified or fed to a
4048:
of high acoustic impedance mismatch (such as the opposite side of the target), tension waves reflect back and create
834:
A simple electrical impedance-matching network requires one capacitor and one inductor. In the figure to the right, R
65:
59:
51:
4420:
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2483:
4090:
Qian, Chunqui; Brey, William W. (July 2009). "Impedance matching with an adjustable segmented transmission line".
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country has its own standard for these networks, but they are all designed to approximate about 600 ohms over the
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3413:
2425:
721:
Applications requiring only a narrow bandwidth, such as radio tuners and transmitters, might use a simple tuned
595:(sometimes called ATUs or roller-coasters, because of their appearance), acoustic horns, matching networks, and
76:
4375:
3556:) in the anode circuit of a vacuum-tube-based power stage. A loudspeaker should not be subjected to DC current.
2370:
1442:
3602:
has a very low output impedance. If they are properly balanced, there is no need for a transformer or a large
587:. These passive (and active) impedance-matching devices are optimized for different applications and include
3624:
1386:
3987:
3509:
Typical push–pull audio tube power amplifier, matched to loudspeaker with an impedance-matching transformer
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2648:
1484:
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430:
163:
123:
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2255:
1972:
3292:{\displaystyle V_{L}=V_{S}{\frac {T(1-\Gamma _{S})(1+\Gamma _{L})}{2(1-T^{2}\Gamma _{S}\Gamma _{L})}}\,}
2026:
1921:
548:
524:
460:
409:
254:
667:{\displaystyle {\text{turns ratio}}={\sqrt {\frac {\text{source resistance}}{\text{load resistance}}}}}
3488:
to the central office (or exchange), cancellation is necessary at the telephone earpiece so excessive
4099:
3373:
3339:
3076:
699:
285:
266:
222:
218:
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2619:
523:* (complex conjugate matching). Given a source with a fixed voltage and fixed source impedance, the
4066:
3545:
1026:
If the imaginary part of the above impedance is canceled by the series reactance, the real part is
611:
4366:
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3484:(2- to 4-wire conversion) to operate correctly. As the signals are sent and received on the same
1396:
726:
584:
552:
469:
452:
393:
230:
226:
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107:
1395:
the maximum power theorem does not apply to its "downstream" connection. That connection is an
497:; however, matching may be obtained between nonlinear devices within certain operating ranges.
138:
of an electrical device for a desired value. Often, the desired value is selected to maximize
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In older audio systems (reliant on transformers and passive filter networks, and based on the
447:
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on a transmission line will be transmitted without reflections if the transmission line is
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2879:{\displaystyle -\Gamma _{ST}=\Gamma _{TS}={Z_{s}-Z_{c} \over Z_{s}+Z_{c}}=\Gamma _{S}\,}
1577:
while the voltage reflection coefficient for a wave moving from medium 2 to medium 1 is
4133:
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connection; it emulates a high-voltage, low-resistance source to maximize efficiency.
4414:
4338:
4282:
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3517:
592:
146:. For example, impedance matching typically is used to improve power transfer from a
946:
is in parallel with the source impedance, the combined impedance can be written as:
682:
Resistive impedance matches are easiest to design and can be achieved with a simple
3587:
3553:
1438:
3678:(or any electromagnetic wave) hits the interface between two media with different
3016:{\displaystyle Z_{in}=Z_{c}{\frac {(1+T^{2}\Gamma _{L})}{(1-T^{2}\Gamma _{L})}}\,}
1798:
be the voltage and current that is incident on the boundary from the source side.
1410:. Consequently, power factor correction is most commonly achieved with banks of
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are sometimes used to match the impedances of circuits. A transformer converts
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563:; it is also used in modern audio amplifiers and signal-processing devices.
30:"Impedance mismatch" redirects here. For the computer science concept, see
27:
Adjusting input/output impedances of an electrical circuit for some purpose
2748:{\displaystyle \Gamma _{TL}={Z_{L}-Z_{c} \over Z_{L}+Z_{c}}=\Gamma _{L}\,}
626:
transformer. This transformer allows interfacing a balanced line (300-ohm
3489:
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576:
572:
3777:. Unwanted reflections can be reduced by the use of an anti-reflection
551:, and so it is used at the beginning of electrical power lines. Such an
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says this is the only way to extract the maximum power from the source.
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be the voltage and current that is reflected back toward the source.
1390:
power wasted in the resistance of those power lines. For example, a
1118:{\displaystyle R_{2}={\frac {R_{1}X_{1}^{2}}{R_{1}^{2}+X_{1}^{2}}}}
800:
may each be either positive (inductor) or negative (capacitor). If
4049:
3974:
3675:
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is the open circuit (or unloaded) output voltage from the source.
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be re-reflected at the source and re-re-reflected at the load end
776:
736:
683:
623:
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419:
106:
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At a boundary, voltage and current must be continuous, therefore
865:
The analysis is as follows. Consider a real source impedance of
3564:
631:
1205:{\displaystyle \left\vert X_{1}\right\vert ={\frac {R_{1}}{Q}}}
3962:{\displaystyle E_{2}={\frac {2P^{2}m_{2}}{(m_{1}+m_{2})^{2}}}}
781:
L networks for narrowband matching a source or load impedance
509:
Devices intended to present an apparent load to the source of
36:
862:, the effective network matches from high to low impedance.
371:{\displaystyle Z_{\mathsf {load}}=Z_{\mathsf {source}}^{*}\,}
1657:{\displaystyle \Gamma _{21}={Z_{1}-Z_{2} \over Z_{1}+Z_{2}}}
1567:{\displaystyle \Gamma _{12}={Z_{2}-Z_{1} \over Z_{2}+Z_{1}}}
1856:
be the voltage and current that is transmitted to the load.
830:
network (a) can be used; otherwise network (b) can be used.
3979:
The middle ear matches mechanical impedance, like a lever.
437:
is delivered to the load when the impedance of the load (
200:, is transferred between a source and a load, such as in
3858:
are the masses of the moving and stationary bodies, and
1487:
for a wave moving from medium 1 to medium 2 is given by
630:) and an unbalanced line (75-ohm coaxial cable such as
400:
applications, a more complex network must be designed.
3548:
component (which contains the audio signals) from the
1426:
Coaxial transmission line with one source and one load
1325:{\displaystyle Q={\sqrt {\frac {R_{1}-R_{2}}{R_{2}}}}}
773:
must be an inductor and the other must be a capacitor.
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to a transmission line with characteristic impedance
643:
310:
3333:Note that if there is a perfect match at both ends
4374:. Radiating Systems Design / Avionics Engineering,
3972:which is analogous to the power-transfer equation.
1476:. A typical RF load is a quarter-wave ground plane
1258:{\displaystyle \left\vert X_{2}\right\vert =QR_{2}}
1016:{\displaystyle {\frac {jR_{1}X_{1}}{R_{1}+jX_{1}}}}
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850:may be the source and the other the load. One of X
765:may be the source and the other the load. One of X
666:
370:
3763:{\displaystyle r={Z_{2}-Z_{1} \over Z_{1}+Z_{2}}}
622:One example of this method involves a television
530:Devices intended to present an apparent load of
388:If the source has a reactive component, but the
64:but its sources remain unclear because it lacks
4368:A Practical Introduction to Impedance Matching
4337:Young, E. C. (1988), "maximum power theorem",
2651:going from the transmission line to the load.
1726:Single-source transmission line driving a load
619:impedance (as it has more turns in its coil).
253:value; this means that loads generally have a
130:is the practice of designing or adjusting the
4305:(1st ed.), Saunders College Publishing,
2606:{\displaystyle I_{t}=(1-\Gamma _{TL})I_{i}\,}
2539:{\displaystyle V_{t}=(1+\Gamma _{TL})V_{i}\,}
451:of the impedance of the source (that is, its
8:
1704:{\displaystyle \Gamma _{21}=-\Gamma _{12}\,}
3540:-based amplifiers has two basic functions:
3454:{\displaystyle V_{L}=V_{S}{\frac {T}{2}}\,}
4319:Stutzman, Warren L.; Thiele, Gary (2012),
2472:{\displaystyle I_{r}=-\Gamma _{TL}I_{i}\,}
280:In simple cases (such as low-frequency or
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2079:
2063:
2057:
2047:
2034:
2028:
2012:
2006:
1996:
1980:
1974:
1958:
1952:
1942:
1929:
1923:
1903:
1897:
1891:
1875:
1869:
1863:
1845:
1839:
1833:
1817:
1811:
1805:
1787:
1781:
1775:
1759:
1753:
1747:
1700:
1694:
1678:
1672:
1645:
1632:
1620:
1607:
1600:
1591:
1585:
1555:
1542:
1530:
1517:
1510:
1501:
1495:
1456:
1450:
1347:
1341:
1313:
1302:
1289:
1281:
1273:
1249:
1229:
1219:
1191:
1185:
1172:
1162:
1139:
1133:
1106:
1101:
1088:
1083:
1071:
1066:
1056:
1049:
1040:
1034:
1004:
988:
976:
966:
956:
954:
930:
924:
903:
897:
876:
870:
652:
644:
642:
367:
361:
340:
339:
316:
315:
309:
169:Techniques of impedance matching include
95:Learn how and when to remove this message
4217:
1421:
297:Complex conjugate matching is used when
4407:Impedance Matching with the Smith Chart
4082:
3559:Reducing the output impedance of power
4274:Transmission Lines and Filter Networks
4193:
4154:Introduction to Radio Frequency Design
2364:All these conditions are satisfied by
1480:(37 ohms with an ideal ground plane).
356:
353:
350:
347:
344:
341:
326:
323:
320:
317:
4340:The Penguin Dictionary of Electronics
4181:
4037:{\displaystyle l_{1}m_{1}=l_{2}m_{2}}
418:such as an electric signal source, a
7:
4205:
3594:semiconductor-based end stages with
381:where a superscript * indicates the
32:object-relational impedance mismatch
2354:{\displaystyle I_{t}=I_{i}+I_{r}\,}
2299:{\displaystyle V_{t}=V_{i}+V_{r}\,}
2016:{\displaystyle V_{r}=-Z_{c}I_{r}\,}
3520:amplifiers), for improved speaker
3378:
3344:
3273:
3263:
3226:
3204:
3081:
2997:
2963:
2866:
2793:
2777:
2735:
2662:
2624:
2577:
2510:
2446:
2388:
2067:{\displaystyle V_{t}=Z_{L}I_{t}\,}
1962:{\displaystyle V_{i}=Z_{c}I_{i}\,}
1691:
1675:
1588:
1498:
25:
1918:On the line side of the boundary
811:is inside the 1+jx circle on the
245:. In general, impedance (symbol:
3646:of sound produced in air from a
3606:to separate AC from DC current.
1483:The general form of the voltage
1433:In electrical systems involving
173:, adjustable networks of lumped
41:
4385:from the original on 2023-07-23
4238:(5th ed.), Prentice Hall,
4236:Principles of Electric Circuits
3397:{\displaystyle \Gamma _{S}=0\,}
3363:{\displaystyle \Gamma _{L}=0\,}
3100:{\displaystyle \Gamma _{L}=0\,}
425:or a mechanical sound (e.g., a
293:Maximum power transfer matching
4301:Sadiku, Matthew N. O. (1989),
3947:
3920:
3282:
3243:
3235:
3216:
3213:
3194:
3144:{\displaystyle Z_{in}=Z_{c}\,}
3006:
2977:
2972:
2943:
2640:{\displaystyle \Gamma _{TL}\,}
2589:
2567:
2522:
2500:
741:Basic schematic for matching R
1:
4287:(3rd ed.), McGraw-Hill,
4258:(5th ed.), McGraw-Hill,
4092:Journal of Magnetic Resonance
3674:A similar effect occurs when
416:with a fixed output impedance
4303:Elements of Electromagnetics
4255:Engineering Electromagnetics
4140:(3rd ed.). p. 223.
1406:the overall load is usually
561:electromagnetic interference
166:with a matching impedance.
3552:component (supplied by the
1392:maximum power point tracker
892:and real load impedance of
463:for a mathematical proof).
414:Whenever a source of power
4447:
4272:Karakash, John J. (1950),
4072:Line isolation transformer
555:connection also minimizes
501:Impedance-matching devices
407:
29:
4323:, John Wiley & Sons,
4321:Antenna Theory and Design
4276:(1st ed.), Macmillan
4112:10.1016/j.jmr.2009.04.005
694:Stepped transmission line
4376:Douglas Aircraft Company
3581:impedance scaling factor
1443:characteristic impedance
284:power transmission) the
150:via the interconnecting
50:This article includes a
4281:Kraus, John D. (1984),
3625:medical ultrasonography
3610:Non-electrical examples
3323:{\displaystyle V_{S}\,}
2235:{\displaystyle I_{t}\,}
2207:{\displaystyle I_{r}\,}
2179:{\displaystyle I_{i}\,}
2151:{\displaystyle V_{t}\,}
2123:{\displaystyle V_{r}\,}
2095:{\displaystyle V_{i}\,}
1907:{\displaystyle I_{r}\,}
1879:{\displaystyle V_{r}\,}
1849:{\displaystyle I_{t}\,}
1821:{\displaystyle V_{t}\,}
1791:{\displaystyle I_{i}\,}
1763:{\displaystyle V_{i}\,}
1387:Power factor correction
1382:Power factor correction
433:, the maximum possible
79:more precise citations.
4252:Hayt, William (1989),
4234:Floyd, Thomas (1997),
4038:
3980:
3963:
3852:
3825:
3764:
3604:electrolytic capacitor
3510:
3501:Loudspeaker amplifiers
3455:
3398:
3364:
3324:
3293:
3145:
3101:
3067:
3046:
3017:
2880:
2749:
2649:reflection coefficient
2641:
2607:
2540:
2473:
2415:
2355:
2300:
2236:
2208:
2180:
2152:
2124:
2096:
2068:
2017:
1963:
1908:
1880:
1850:
1822:
1792:
1764:
1705:
1658:
1568:
1485:reflection coefficient
1466:
1427:
1357:
1326:
1259:
1206:
1149:
1119:
1017:
940:
913:
886:
831:
774:
668:
372:
299:maximum power transfer
124:electrical engineering
119:
4151:Hayward, Wes (1994).
4138:Microwave Engineering
4039:
3978:
3964:
3853:
3851:{\displaystyle m_{2}}
3826:
3824:{\displaystyle m_{1}}
3765:
3508:
3456:
3399:
3365:
3325:
3294:
3146:
3102:
3068:
3047:
3018:
2881:
2759:Source-end conditions
2750:
2642:
2608:
2541:
2474:
2416:
2356:
2301:
2237:
2209:
2181:
2153:
2125:
2097:
2069:
2023:and on the load side
2018:
1964:
1909:
1881:
1851:
1823:
1793:
1765:
1706:
1659:
1569:
1467:
1465:{\displaystyle Z_{c}}
1425:
1358:
1356:{\displaystyle X_{1}}
1327:
1260:
1207:
1150:
1148:{\displaystyle X_{1}}
1120:
1018:
941:
939:{\displaystyle X_{1}}
914:
912:{\displaystyle R_{2}}
887:
885:{\displaystyle R_{1}}
780:
740:
669:
525:maximum power theorem
461:maximum power theorem
410:Maximum power theorem
373:
142:transfer or minimize
110:
4426:Electronics concepts
3988:
3876:
3835:
3808:
3697:
3586:The output stage in
3414:
3374:
3340:
3306:
3162:
3111:
3077:
3056:
3045:{\displaystyle T\ ,}
3030:
2911:
2903:Source-end impedance
2770:
2658:
2620:
2551:
2484:
2426:
2371:
2311:
2256:
2218:
2190:
2162:
2134:
2106:
2078:
2027:
1973:
1922:
1890:
1862:
1832:
1804:
1774:
1746:
1671:
1584:
1494:
1449:
1340:
1272:
1218:
1161:
1132:
1033:
953:
923:
896:
869:
641:
308:
301:is required, namely
4208:, pp. 398–401)
4196:, pp. 505–507)
4104:2009JMagR.199..104Q
4067:Standing wave ratio
3466:Electrical examples
3066:{\displaystyle T\,}
1731:Load-end conditions
1437:(such as radio and
1111:
1093:
1076:
842:, however, either R
757:, however, either R
612:alternating current
487:hybrid transformers
366:
269:component (symbol:
257:component (symbol:
4405:Impedance Matching
4365:Thomas, Robert L.
4034:
3981:
3959:
3848:
3821:
3789:If a body of mass
3760:
3680:refractive indices
3621:acoustic impedance
3579:, which forms the
3544:Separation of the
3511:
3451:
3394:
3360:
3320:
3289:
3141:
3097:
3063:
3042:
3013:
2876:
2745:
2637:
2603:
2536:
2469:
2411:
2351:
2296:
2232:
2204:
2176:
2148:
2120:
2092:
2064:
2013:
1959:
1904:
1876:
1846:
1818:
1788:
1760:
1701:
1654:
1564:
1462:
1435:transmission lines
1428:
1418:Transmission lines
1397:impedance bridging
1353:
1322:
1255:
1202:
1145:
1115:
1097:
1079:
1062:
1013:
936:
909:
882:
832:
775:
664:
585:transmission lines
553:impedance bridging
470:impedance bridging
453:internal impedance
445:) is equal to the
429:) operates into a
368:
335:
273:) which forms the
261:) which forms the
196:, not necessarily
128:impedance matching
120:
52:list of references
4421:Electronic design
4220:, pp. 52–57)
3957:
3775:Fresnel equations
3758:
3643:horn loudspeakers
3631:The bones in the
3600:power transistors
3471:Telephone systems
3448:
3286:
3154:Transfer function
3038:
3010:
2860:
2729:
1652:
1562:
1372:Transmission line
1320:
1319:
1200:
1113:
1011:
919:. If a reactance
749:with an L pad. R
678:Resistive network
662:
661:
660:
657:
656:source resistance
647:
549:energy efficiency
491:four-wire circuit
448:complex conjugate
383:complex conjugate
152:transmission line
148:radio transmitter
144:signal reflection
118:impedance circuit
105:
104:
97:
16:(Redirected from
4438:
4393:
4391:
4390:
4384:
4373:
4353:
4333:
4315:
4297:
4284:Electromagnetics
4277:
4268:
4248:
4221:
4215:
4209:
4203:
4197:
4191:
4185:
4179:
4173:
4172:
4157:. ARRL. p.
4148:
4142:
4141:
4130:
4124:
4123:
4087:
4062:Ringing (signal)
4043:
4041:
4040:
4035:
4033:
4032:
4023:
4022:
4010:
4009:
4000:
3999:
3984:lengths satisfy
3968:
3966:
3965:
3960:
3958:
3956:
3955:
3954:
3945:
3944:
3932:
3931:
3918:
3917:
3916:
3907:
3906:
3893:
3888:
3887:
3857:
3855:
3854:
3849:
3847:
3846:
3830:
3828:
3827:
3822:
3820:
3819:
3769:
3767:
3766:
3761:
3759:
3757:
3756:
3755:
3743:
3742:
3732:
3731:
3730:
3718:
3717:
3707:
3592:common-collector
3514:Audio amplifiers
3486:two-wire circuit
3460:
3458:
3457:
3452:
3449:
3441:
3439:
3438:
3426:
3425:
3403:
3401:
3400:
3395:
3386:
3385:
3369:
3367:
3366:
3361:
3352:
3351:
3329:
3327:
3326:
3321:
3318:
3317:
3298:
3296:
3295:
3290:
3287:
3285:
3281:
3280:
3271:
3270:
3261:
3260:
3238:
3234:
3233:
3212:
3211:
3189:
3187:
3186:
3174:
3173:
3150:
3148:
3147:
3142:
3139:
3138:
3126:
3125:
3106:
3104:
3103:
3098:
3089:
3088:
3072:
3070:
3069:
3064:
3051:
3049:
3048:
3043:
3036:
3022:
3020:
3019:
3014:
3011:
3009:
3005:
3004:
2995:
2994:
2975:
2971:
2970:
2961:
2960:
2941:
2939:
2938:
2926:
2925:
2885:
2883:
2882:
2877:
2874:
2873:
2861:
2859:
2858:
2857:
2845:
2844:
2834:
2833:
2832:
2820:
2819:
2809:
2804:
2803:
2788:
2787:
2754:
2752:
2751:
2746:
2743:
2742:
2730:
2728:
2727:
2726:
2714:
2713:
2703:
2702:
2701:
2689:
2688:
2678:
2673:
2672:
2646:
2644:
2643:
2638:
2635:
2634:
2612:
2610:
2609:
2604:
2601:
2600:
2588:
2587:
2563:
2562:
2545:
2543:
2542:
2537:
2534:
2533:
2521:
2520:
2496:
2495:
2478:
2476:
2475:
2470:
2467:
2466:
2457:
2456:
2438:
2437:
2420:
2418:
2417:
2412:
2409:
2408:
2399:
2398:
2383:
2382:
2360:
2358:
2357:
2352:
2349:
2348:
2336:
2335:
2323:
2322:
2305:
2303:
2302:
2297:
2294:
2293:
2281:
2280:
2268:
2267:
2241:
2239:
2238:
2233:
2230:
2229:
2213:
2211:
2210:
2205:
2202:
2201:
2185:
2183:
2182:
2177:
2174:
2173:
2157:
2155:
2154:
2149:
2146:
2145:
2129:
2127:
2126:
2121:
2118:
2117:
2101:
2099:
2098:
2093:
2090:
2089:
2073:
2071:
2070:
2065:
2062:
2061:
2052:
2051:
2039:
2038:
2022:
2020:
2019:
2014:
2011:
2010:
2001:
2000:
1985:
1984:
1968:
1966:
1965:
1960:
1957:
1956:
1947:
1946:
1934:
1933:
1913:
1911:
1910:
1905:
1902:
1901:
1885:
1883:
1882:
1877:
1874:
1873:
1855:
1853:
1852:
1847:
1844:
1843:
1827:
1825:
1824:
1819:
1816:
1815:
1797:
1795:
1794:
1789:
1786:
1785:
1769:
1767:
1766:
1761:
1758:
1757:
1710:
1708:
1707:
1702:
1699:
1698:
1683:
1682:
1663:
1661:
1660:
1655:
1653:
1651:
1650:
1649:
1637:
1636:
1626:
1625:
1624:
1612:
1611:
1601:
1596:
1595:
1573:
1571:
1570:
1565:
1563:
1561:
1560:
1559:
1547:
1546:
1536:
1535:
1534:
1522:
1521:
1511:
1506:
1505:
1471:
1469:
1468:
1463:
1461:
1460:
1362:
1360:
1359:
1354:
1352:
1351:
1331:
1329:
1328:
1323:
1321:
1318:
1317:
1308:
1307:
1306:
1294:
1293:
1283:
1282:
1264:
1262:
1261:
1256:
1254:
1253:
1238:
1234:
1233:
1211:
1209:
1208:
1203:
1201:
1196:
1195:
1186:
1181:
1177:
1176:
1154:
1152:
1151:
1146:
1144:
1143:
1124:
1122:
1121:
1116:
1114:
1112:
1110:
1105:
1092:
1087:
1077:
1075:
1070:
1061:
1060:
1050:
1045:
1044:
1022:
1020:
1019:
1014:
1012:
1010:
1009:
1008:
993:
992:
982:
981:
980:
971:
970:
957:
945:
943:
942:
937:
935:
934:
918:
916:
915:
910:
908:
907:
891:
889:
888:
883:
881:
880:
829:
673:
671:
670:
665:
663:
658:
655:
654:
653:
648:
645:
475:voltage bridging
457:output impedance
377:
375:
374:
369:
365:
360:
359:
331:
330:
329:
136:output impedance
100:
93:
89:
86:
80:
75:this article by
66:inline citations
45:
44:
37:
21:
18:Matching network
4446:
4445:
4441:
4440:
4439:
4437:
4436:
4435:
4411:
4410:
4401:
4388:
4386:
4382:
4371:
4364:
4361:
4359:Further reading
4356:
4351:
4336:
4331:
4318:
4313:
4300:
4295:
4280:
4271:
4266:
4251:
4246:
4233:
4229:
4224:
4216:
4212:
4204:
4200:
4192:
4188:
4180:
4176:
4169:
4150:
4149:
4145:
4132:
4131:
4127:
4089:
4088:
4084:
4080:
4058:
4024:
4014:
4001:
3991:
3986:
3985:
3946:
3936:
3923:
3919:
3908:
3898:
3894:
3879:
3874:
3873:
3868:
3838:
3833:
3832:
3811:
3806:
3805:
3799:Newton's cradle
3787:
3779:optical coating
3747:
3734:
3733:
3722:
3709:
3708:
3695:
3694:
3672:
3617:
3612:
3503:
3495:voice frequency
3473:
3468:
3430:
3417:
3412:
3411:
3377:
3372:
3371:
3343:
3338:
3337:
3309:
3304:
3303:
3272:
3262:
3252:
3239:
3225:
3203:
3190:
3178:
3165:
3160:
3159:
3156:
3130:
3114:
3109:
3108:
3080:
3075:
3074:
3054:
3053:
3028:
3027:
2996:
2986:
2976:
2962:
2952:
2942:
2930:
2914:
2909:
2908:
2905:
2865:
2849:
2836:
2835:
2824:
2811:
2810:
2792:
2776:
2768:
2767:
2761:
2734:
2718:
2705:
2704:
2693:
2680:
2679:
2661:
2656:
2655:
2623:
2618:
2617:
2592:
2576:
2554:
2549:
2548:
2525:
2509:
2487:
2482:
2481:
2458:
2445:
2429:
2424:
2423:
2400:
2387:
2374:
2369:
2368:
2340:
2327:
2314:
2309:
2308:
2285:
2272:
2259:
2254:
2253:
2221:
2216:
2215:
2193:
2188:
2187:
2165:
2160:
2159:
2137:
2132:
2131:
2109:
2104:
2103:
2081:
2076:
2075:
2053:
2043:
2030:
2025:
2024:
2002:
1992:
1976:
1971:
1970:
1948:
1938:
1925:
1920:
1919:
1893:
1888:
1887:
1865:
1860:
1859:
1835:
1830:
1829:
1807:
1802:
1801:
1777:
1772:
1771:
1749:
1744:
1743:
1733:
1728:
1690:
1674:
1669:
1668:
1641:
1628:
1627:
1616:
1603:
1602:
1587:
1582:
1581:
1551:
1538:
1537:
1526:
1513:
1512:
1497:
1492:
1491:
1452:
1447:
1446:
1420:
1384:
1343:
1338:
1337:
1309:
1298:
1285:
1284:
1270:
1269:
1245:
1225:
1221:
1216:
1215:
1187:
1168:
1164:
1159:
1158:
1135:
1130:
1129:
1078:
1052:
1051:
1036:
1031:
1030:
1000:
984:
983:
972:
962:
958:
951:
950:
926:
921:
920:
899:
894:
893:
872:
867:
866:
860:with the source
857:
853:
849:
845:
841:
837:
826:
816:
809:
790:
772:
768:
764:
760:
756:
752:
748:
744:
735:
709:
696:
680:
659:load resistance
639:
638:
605:
543:
536:
522:
515:
503:
477:is often used.
443:input impedance
412:
406:
311:
306:
305:
295:
235:electromagnetic
214:
132:input impedance
101:
90:
84:
81:
70:
56:related reading
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
4444:
4442:
4434:
4433:
4428:
4423:
4413:
4412:
4409:
4408:
4400:
4399:External links
4397:
4396:
4395:
4360:
4357:
4355:
4354:
4349:
4334:
4330:978-0470576649
4329:
4316:
4311:
4298:
4293:
4278:
4269:
4264:
4249:
4244:
4230:
4228:
4225:
4223:
4222:
4218:Karakash (1950
4210:
4198:
4186:
4184:, p. 407)
4174:
4167:
4143:
4125:
4098:(1): 104–110.
4081:
4079:
4076:
4075:
4074:
4069:
4064:
4057:
4054:
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4027:
4021:
4017:
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3998:
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3750:
3746:
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3729:
3725:
3721:
3716:
3712:
3705:
3702:
3688:wave impedance
3671:
3668:
3616:
3613:
3611:
3608:
3573:
3572:
3571:configuration.
3569:common-cathode
3557:
3502:
3499:
3472:
3469:
3467:
3464:
3463:
3462:
3447:
3444:
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3259:
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3248:
3245:
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3237:
3232:
3228:
3224:
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3218:
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3210:
3206:
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3199:
3196:
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3185:
3181:
3177:
3172:
3168:
3155:
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3133:
3129:
3124:
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3117:
3095:
3092:
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2823:
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2814:
2807:
2802:
2799:
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2757:
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2755:
2741:
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2733:
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2721:
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2712:
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2700:
2696:
2692:
2687:
2683:
2676:
2671:
2668:
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2633:
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2626:
2614:
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2599:
2595:
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2569:
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2557:
2546:
2532:
2528:
2524:
2519:
2516:
2512:
2508:
2505:
2502:
2499:
2494:
2490:
2479:
2465:
2461:
2455:
2452:
2448:
2444:
2441:
2436:
2432:
2421:
2407:
2403:
2397:
2394:
2390:
2386:
2381:
2377:
2362:
2361:
2347:
2343:
2339:
2334:
2330:
2326:
2321:
2317:
2306:
2292:
2288:
2284:
2279:
2275:
2271:
2266:
2262:
2228:
2224:
2200:
2196:
2172:
2168:
2144:
2140:
2116:
2112:
2088:
2084:
2060:
2056:
2050:
2046:
2042:
2037:
2033:
2009:
2005:
1999:
1995:
1991:
1988:
1983:
1979:
1955:
1951:
1945:
1941:
1937:
1932:
1928:
1916:
1915:
1900:
1896:
1872:
1868:
1857:
1842:
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1814:
1810:
1799:
1784:
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1756:
1752:
1732:
1729:
1727:
1724:
1712:
1711:
1697:
1693:
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1686:
1681:
1677:
1665:
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1648:
1644:
1640:
1635:
1631:
1623:
1619:
1615:
1610:
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1575:
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1558:
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1525:
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1516:
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1459:
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1419:
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1383:
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1316:
1312:
1305:
1301:
1297:
1292:
1288:
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1277:
1266:
1252:
1248:
1244:
1241:
1237:
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1228:
1224:
1213:
1199:
1194:
1190:
1184:
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1175:
1171:
1167:
1142:
1138:
1126:
1125:
1109:
1104:
1100:
1096:
1091:
1086:
1082:
1074:
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1065:
1059:
1055:
1048:
1043:
1039:
1024:
1023:
1007:
1003:
999:
996:
991:
987:
979:
975:
969:
965:
961:
933:
929:
906:
902:
879:
875:
855:
851:
847:
843:
839:
835:
824:
807:
788:
770:
766:
762:
758:
754:
750:
746:
742:
734:
731:
708:
705:
700:lumped-element
695:
692:
679:
676:
675:
674:
651:
604:
601:
593:antenna tuners
565:
564:
545:
541:
534:
528:
520:
513:
502:
499:
439:load impedance
408:Main article:
405:
404:Power transfer
402:
379:
378:
364:
358:
355:
352:
349:
346:
343:
338:
334:
328:
325:
322:
319:
314:
294:
291:
282:direct current
213:
210:
187:frequency band
103:
102:
60:external links
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4443:
4432:
4431:Filter theory
4429:
4427:
4424:
4422:
4419:
4418:
4416:
4406:
4403:
4402:
4398:
4381:
4377:
4370:
4369:
4363:
4362:
4358:
4352:
4350:0-14-051187-3
4346:
4342:
4341:
4335:
4332:
4326:
4322:
4317:
4314:
4308:
4304:
4299:
4296:
4294:0-07-035423-5
4290:
4286:
4285:
4279:
4275:
4270:
4267:
4265:0-07-027406-1
4261:
4257:
4256:
4250:
4247:
4245:0-13-232224-2
4241:
4237:
4232:
4231:
4226:
4219:
4214:
4211:
4207:
4202:
4199:
4195:
4190:
4187:
4183:
4178:
4175:
4170:
4168:0-87259-492-0
4164:
4160:
4156:
4155:
4147:
4144:
4139:
4135:
4129:
4126:
4121:
4117:
4113:
4109:
4105:
4101:
4097:
4093:
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3992:
3977:
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3913:
3909:
3903:
3899:
3895:
3889:
3884:
3880:
3872:
3871:
3870:
3865:
3861:
3843:
3839:
3816:
3812:
3802:
3800:
3796:
3792:
3784:
3782:
3780:
3776:
3752:
3748:
3744:
3739:
3735:
3727:
3723:
3719:
3714:
3710:
3703:
3700:
3693:
3692:
3691:
3689:
3685:
3681:
3677:
3669:
3667:
3665:
3664:soundproofing
3661:
3657:
3653:
3649:
3644:
3640:
3636:
3634:
3629:
3626:
3622:
3614:
3609:
3607:
3605:
3601:
3597:
3593:
3589:
3584:
3582:
3578:
3570:
3566:
3563:(such as the
3562:
3558:
3555:
3551:
3547:
3543:
3542:
3541:
3539:
3535:
3530:
3527:
3523:
3519:
3518:semiconductor
3515:
3507:
3500:
3498:
3496:
3491:
3487:
3483:
3482:
3477:
3470:
3465:
3445:
3442:
3435:
3431:
3427:
3422:
3418:
3410:
3409:
3408:
3390:
3387:
3382:
3356:
3353:
3348:
3336:
3335:
3334:
3331:
3314:
3310:
3277:
3267:
3257:
3253:
3249:
3246:
3240:
3230:
3222:
3219:
3208:
3200:
3197:
3191:
3183:
3179:
3175:
3170:
3166:
3158:
3157:
3153:
3151:
3135:
3131:
3127:
3122:
3119:
3115:
3093:
3090:
3085:
3059:
3039:
3033:
3001:
2991:
2987:
2983:
2980:
2967:
2957:
2953:
2949:
2946:
2935:
2931:
2927:
2922:
2919:
2915:
2907:
2906:
2902:
2900:
2898:
2893:
2870:
2862:
2854:
2850:
2846:
2841:
2837:
2829:
2825:
2821:
2816:
2812:
2805:
2800:
2797:
2789:
2784:
2781:
2773:
2766:
2765:
2764:
2758:
2739:
2731:
2723:
2719:
2715:
2710:
2706:
2698:
2694:
2690:
2685:
2681:
2674:
2669:
2666:
2654:
2653:
2652:
2650:
2631:
2628:
2597:
2593:
2584:
2581:
2573:
2570:
2564:
2559:
2555:
2547:
2530:
2526:
2517:
2514:
2506:
2503:
2497:
2492:
2488:
2480:
2463:
2459:
2453:
2450:
2442:
2439:
2434:
2430:
2422:
2405:
2401:
2395:
2392:
2384:
2379:
2375:
2367:
2366:
2365:
2345:
2341:
2337:
2332:
2328:
2324:
2319:
2315:
2307:
2290:
2286:
2282:
2277:
2273:
2269:
2264:
2260:
2252:
2251:
2250:
2247:
2245:
2226:
2222:
2198:
2194:
2170:
2166:
2142:
2138:
2114:
2110:
2086:
2082:
2058:
2054:
2048:
2044:
2040:
2035:
2031:
2007:
2003:
1997:
1993:
1989:
1986:
1981:
1977:
1953:
1949:
1943:
1939:
1935:
1930:
1926:
1898:
1894:
1870:
1866:
1858:
1840:
1836:
1812:
1808:
1800:
1782:
1778:
1754:
1750:
1742:
1741:
1740:
1737:
1730:
1725:
1723:
1719:
1715:
1695:
1687:
1684:
1679:
1667:
1666:
1646:
1642:
1638:
1633:
1629:
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1613:
1608:
1604:
1597:
1592:
1580:
1579:
1578:
1556:
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1548:
1543:
1539:
1531:
1527:
1523:
1518:
1514:
1507:
1502:
1490:
1489:
1488:
1486:
1481:
1479:
1475:
1457:
1453:
1444:
1440:
1436:
1431:
1424:
1417:
1415:
1413:
1409:
1405:
1400:
1398:
1393:
1388:
1381:
1379:
1377:
1373:
1369:
1364:
1348:
1344:
1314:
1310:
1303:
1299:
1295:
1290:
1286:
1278:
1275:
1267:
1250:
1246:
1242:
1239:
1235:
1230:
1226:
1222:
1214:
1197:
1192:
1188:
1182:
1178:
1173:
1169:
1165:
1157:
1156:
1155:
1140:
1136:
1107:
1102:
1098:
1094:
1089:
1084:
1080:
1072:
1067:
1063:
1057:
1053:
1046:
1041:
1037:
1029:
1028:
1027:
1005:
1001:
997:
994:
989:
985:
977:
973:
967:
963:
959:
949:
948:
947:
931:
927:
904:
900:
877:
873:
863:
861:
827:
820:
814:
810:
803:
799:
795:
791:
784:
779:
739:
732:
730:
728:
724:
719:
717:
713:
706:
704:
701:
693:
691:
689:
685:
677:
649:
637:
636:
635:
633:
629:
625:
620:
617:
613:
609:
602:
600:
598:
594:
590:
586:
582:
578:
574:
570:
562:
558:
554:
550:
546:
540:
537: =
533:
529:
526:
519:
516: =
512:
508:
507:
506:
500:
498:
496:
492:
488:
483:
478:
476:
472:
471:
464:
462:
458:
454:
450:
449:
444:
440:
436:
432:
428:
424:
421:
417:
411:
403:
401:
399:
395:
391:
386:
384:
362:
336:
332:
312:
304:
303:
302:
300:
292:
290:
287:
283:
278:
276:
272:
268:
264:
260:
256:
252:
248:
244:
240:
236:
232:
228:
224:
220:
211:
209:
207:
203:
199:
195:
190:
188:
184:
180:
176:
172:
167:
165:
161:
157:
153:
149:
145:
141:
137:
133:
129:
125:
117:
113:
109:
99:
96:
88:
78:
74:
68:
67:
61:
57:
53:
48:
39:
38:
33:
19:
4387:. Retrieved
4367:
4339:
4320:
4302:
4283:
4273:
4254:
4235:
4213:
4201:
4194:Sadiku (1989
4189:
4177:
4153:
4146:
4137:
4134:Pozar, David
4128:
4095:
4091:
4085:
4046:
3982:
3971:
3863:
3859:
3803:
3794:
3790:
3788:
3772:
3687:
3683:
3673:
3637:
3630:
3618:
3588:common-drain
3585:
3580:
3574:
3554:power supply
3531:
3512:
3479:
3474:
3406:
3332:
3301:
3025:
2897:ad infinitum
2896:
2891:
2889:
2762:
2615:
2363:
2248:
1917:
1738:
1734:
1720:
1716:
1713:
1576:
1482:
1439:fiber optics
1432:
1429:
1401:
1385:
1376:Smith charts
1365:
1335:
1128:Solving for
1127:
1025:
864:
859:
833:
822:
818:
805:
801:
797:
793:
786:
782:
720:
710:
697:
681:
621:
608:Transformers
606:
603:Transformers
569:transformers
566:
538:
531:
517:
510:
504:
479:
474:
468:
465:
446:
415:
413:
387:
380:
296:
279:
270:
258:
246:
215:
191:
171:transformers
168:
127:
121:
91:
85:October 2023
82:
71:Please help
63:
4394:(175 pages)
4343:, Penguin,
4182:Kraus (1984
3648:loudspeaker
3577:turns ratio
3538:vacuum-tube
3534:transformer
3532:The output
3526:vacuum tube
3481:hybrid coil
813:Smith chart
716:frequencies
646:turns ratio
597:terminators
427:loudspeaker
423:transmitter
265:part and a
179:capacitance
77:introducing
4415:Categories
4389:2023-07-23
4312:0030134846
4227:References
4206:Hayt (1989
3652:Elliptical
3633:middle ear
3407:and then
1412:capacitors
1404:power grid
725:such as a
688:line level
581:capacitors
557:distortion
255:resistance
223:mechanical
219:electrical
198:electrical
183:inductance
175:resistance
164:terminated
3785:Mechanics
3720:−
3656:megaphone
3615:Acoustics
3476:Telephone
3379:Γ
3345:Γ
3274:Γ
3264:Γ
3250:−
3227:Γ
3205:Γ
3201:−
3082:Γ
2998:Γ
2984:−
2964:Γ
2867:Γ
2822:−
2794:Γ
2778:Γ
2774:−
2736:Γ
2691:−
2663:Γ
2625:Γ
2578:Γ
2574:−
2511:Γ
2447:Γ
2443:−
2389:Γ
1990:−
1692:Γ
1688:−
1676:Γ
1614:−
1589:Γ
1524:−
1499:Γ
1408:inductive
1296:−
815:(i.e. if
733:L-section
690:signals.
628:twin-lead
577:inductors
573:resistors
482:telephone
398:bandwidth
363:∗
286:reactance
275:imaginary
267:reactance
202:acoustics
4380:Archived
4120:19406676
4056:See also
4050:spalling
3561:pentodes
3490:sidetone
828:)>1),
249:) has a
231:magnetic
227:acoustic
4100:Bibcode
3684:optical
3596:MOSFETs
3567:) in a
3524:. For
3522:damping
2647:is the
2244:phasors
1478:antenna
1402:On the
712:Filters
707:Filters
616:voltage
614:at one
394:element
251:complex
239:thermal
160:Signals
156:antenna
154:to the
73:improve
4347:
4327:
4309:
4291:
4262:
4242:
4165:
4118:
3670:Optics
3497:band.
3302:where
3037:
3026:where
2890:where
2616:where
2214:, and
2074:where
1336:Note,
1268:where
838:> R
753:> R
723:filter
589:baluns
521:source
495:linear
277:part.
212:Theory
206:optics
194:energy
112:Source
4383:(PDF)
4372:(PDF)
4078:Notes
3676:light
3658:, an
3639:Horns
698:Most
684:L pad
624:balun
435:power
420:radio
237:, or
140:power
58:, or
4345:ISBN
4325:ISBN
4307:ISBN
4289:ISBN
4260:ISBN
4240:ISBN
4163:ISBN
4116:PMID
3831:and
3662:and
3660:echo
3565:EL34
3370:and
3107:and
2242:are
1969:and
1886:and
1828:and
1770:and
1739:Let
1474:ohms
854:or X
846:or R
796:and
769:or X
761:or R
745:to R
727:stub
632:RG-6
583:and
559:and
542:line
535:load
514:load
431:load
390:load
263:real
243:ohms
181:and
116:load
114:and
4159:138
4108:doi
4096:199
3804:If
3686:or
3598:or
3590:or
3536:in
2158:,
817:Re(
473:or
455:or
441:or
208:.
204:or
189:.
134:or
122:In
4417::
4378:.
4161:.
4136:.
4114:.
4106:.
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3781:.
3666:.
3583:.
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3546:AC
2892:Zs
2246:.
2186:,
2130:,
2102:,
1696:12
1680:21
1593:21
1503:12
1378:.
792:.
599:.
591:,
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575:,
571:,
233:,
229:,
225:,
221:,
177:,
158:.
126:,
62:,
54:,
4392:.
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4122:.
4110::
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3896:2
3890:=
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3461:.
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2855:c
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2041:=
2036:t
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2008:r
2004:I
1998:c
1994:Z
1987:=
1982:r
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1954:i
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1944:c
1940:Z
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1639:+
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1458:c
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1445:(
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1047:=
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1002:X
998:j
995:+
990:1
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978:1
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960:j
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821:/
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808:0
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794:X
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650:=
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327:d
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