Impedance matching
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In
Techniques of impedance matching include
The concept of impedance matching is widespread in electrical engineering, but is relevant in other applications in which a form of energy, not necessarily electrical, is transferred between a source and a load, such as in acoustics or optics.
Theory
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
In simple cases (such as low-frequency or direct current power transmission) the reactance 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.
Maximum power transfer matching
Complex conjugate matching is used when maximum power transfer is required, namely
where a superscript * indicates the complex conjugate. A conjugate match is different from a reflection-less match when either the source or load has a reactive component.
If the source has a reactive component, but the load 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 element, 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 bandwidth applications, a more complex network must be designed.
Power transfer
Whenever a source of power with a fixed output impedance such as an
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 impedance bridging or voltage bridging is often used.
In older audio systems (reliant on transformers and passive filter networks, and based on the
Impedance-matching devices
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":
- Devices intended to present an apparent load to the source of Zload = Zsource* (complex conjugate matching). Given a source with a fixed voltage and fixed source impedance, the maximum power theoremsays this is the only way to extract the maximum power from the source.
- Devices intended to present an apparent load of Zload = Zline (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.
- 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 energy efficiency, and so it is used at the beginning of electrical power lines. Such an impedance bridging connection also minimizes distortion and electromagnetic interference; it is also used in modern audio amplifiers and signal-processing devices.
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
Transformers
Transformers are sometimes used to match the impedances of circuits. A transformer converts alternating current at one voltage 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 impedance (as it has more turns in its coil).
One example of this method involves a television balun transformer. This transformer allows interfacing a balanced line (300-ohm twin-lead) and an unbalanced line (75-ohm coaxial cable such as RG-6). 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:
Resistive network
Resistive impedance matches are easiest to design and can be achieved with a simple L pad consisting of two resistors. Power loss is an unavoidable consequence of using resistive networks, and they are only (usually) used to transfer line level signals.
Stepped transmission line
Most
Filters
Filters 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 frequencies 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.
Applications requiring only a narrow bandwidth, such as radio tuners and transmitters, might use a simple tuned filter such as a stub. This would provide a perfect match at one specific frequency only. Wide bandwidth matching requires filters with multiple sections.
L-section
A simple electrical impedance-matching network requires one capacitor and one inductor. In the figure to the right, R1 > R2, however, either R1 or R2 may be the source and the other the load. One of X1 or X2 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 with the source, the effective network matches from high to low impedance.
The analysis is as follows.[3] Consider a real source impedance of and real load impedance of . If a reactance is in parallel with the source impedance, the combined impedance can be written as:
If the imaginary part of the above impedance is canceled by the series reactance, the real part is
Solving for
- .
- .
- where .
Note, , 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.
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 Q of the inductor. Multiple L-sections can be wired in cascade to achieve higher impedance ratios or greater bandwidth. Transmission line matching networks can be modeled as infinitely many L-sections wired in cascade. Optimal matching circuits can be designed for a particular system using Smith charts.
Power factor correction
On the
Transmission lines
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.
In electrical systems involving
The general form of the voltage reflection coefficient for a wave moving from medium 1 to medium 2 is given by
while the voltage reflection coefficient for a wave moving from medium 2 to medium 1 is
so the reflection coefficient is the same (except for sign), no matter from which direction the wave approaches the boundary.
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 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.
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 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.
Single-source transmission line driving a load
Load-end conditions
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.)
Let
- and be the voltage and current that is incident on the boundary from the source side.
- and be the voltage and current that is transmitted to the load.
- and be the voltage and current that is reflected back toward the source.
On the line side of the boundary and and on the load side where , , , , , and are phasors.
At a boundary, voltage and current must be continuous, therefore
All these conditions are satisfied by
where is the reflection coefficient going from the transmission line to the load.
Source-end conditions
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
- ,
where Zs 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 be re-reflected at the source and re-re-reflected at the load end ad infinitum, 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.
Source-end impedance
where is the one-way transfer function (from either end to the other) when the transmission line is exactly matched at source and load. accounts for everything that happens to the signal in transit (including delay, attenuation and dispersion). If there is a perfect match at the load, and
Transfer function
where is the open circuit (or unloaded) output voltage from the source.
Note that if there is a perfect match at both ends
- and
and then
- .
Electrical examples
Telephone systems
Loudspeaker amplifiers
The output
- Separation of the AC component (which contains the audio signals) from the DC component (supplied by the power supply) in the anode circuit of a vacuum-tube-based power stage. A loudspeaker should not be subjected to DC current.
- Reducing the output impedance of power common-cathodeconfiguration.
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 turns ratio, which forms the impedance scaling factor.
The output stage in
Non-electrical examples
Acoustics
Similar to electrical transmission lines, an impedance matching problem exists when transferring sound energy from one medium to another. If the
The bones in the middle ear provide impedance matching between the eardrum (which is acted upon by vibrations in air) and the fluid-filled inner ear.
Optics
A similar effect occurs when light (or any electromagnetic wave) hits the interface between two media with different refractive indices. For non-magnetic materials, the refractive index is inversely proportional to the material's characteristic impedance. An optical or wave impedance (that depends on the propagation direction) can be calculated for each medium, and may be used in the transmission-line reflection equation
to calculate reflection and transmission coefficients for the interface. For non-magnetic dielectrics, this equation is equivalent to the Fresnel equations. Unwanted reflections can be reduced by the use of an anti-reflection optical coating.
Mechanics
If a body of mass m collides elastically with a second body, maximum energy transfer to the second body will occur when the second body has the same mass m. In a head-on collision of equal masses, the energy of the first body will be completely transferred to the second body (as in Newton's cradle for example). In this case, the masses act as "mechanical impedances",[dubious ] which must be matched. If and are the masses of the moving and stationary bodies, and P is the momentum of the system (which remains constant throughout the collision), the energy of the second body after the collision will be E2:
which is analogous to the power-transfer equation.
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 of high acoustic impedance mismatch (such as the opposite side of the target), tension waves reflect back and create spalling. 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.
See also
Notes
- PMID 19406676.
- ^ Pozar, David. Microwave Engineering (3rd ed.). p. 223.
- ISBN 0-87259-492-0.
- ^ Kraus (1984, p. 407)
- ^ Sadiku (1989, pp. 505–507)
- ^ Hayt (1989, pp. 398–401)
- ^ Karakash (1950, pp. 52–57)
References
- Floyd, Thomas (1997), Principles of Electric Circuits (5th ed.), Prentice Hall, ISBN 0-13-232224-2
- Hayt, William (1989), Engineering Electromagnetics (5th ed.), McGraw-Hill, ISBN 0-07-027406-1
- Karakash, John J. (1950), Transmission Lines and Filter Networks (1st ed.), Macmillan
- Kraus, John D. (1984), Electromagnetics (3rd ed.), McGraw-Hill, ISBN 0-07-035423-5
- Sadiku, Matthew N. O. (1989), Elements of Electromagnetics (1st ed.), Saunders College Publishing, ISBN 0030134846
- Stutzman, Warren L.; Thiele, Gary (2012), Antenna Theory and Design, John Wiley & Sons, ISBN 978-0470576649
- Young, E. C. (1988), "maximum power theorem", The Penguin Dictionary of Electronics, Penguin, ISBN 0-14-051187-3
Further reading
- Thomas, Robert L. A Practical Introduction to Impedance Matching (PDF). Radiating Systems Design / Avionics Engineering, Douglas Aircraft Company. Archived (PDF) from the original on 2023-07-23. Retrieved 2023-07-23. (175 pages)
External links
- Impedance Matching Impedance Matching with the Smith Chart