Attenuators and Impedance Matching, Physics tutorial

Introduction:

An attenuator is the electronic device which decreases the amplitude or power of a signal devoid of appreciably distorting its waveform and they are generally passive devices made up from simple voltage divider networks.

Impedance matching schemes were initially developed for electrical power; however it can as well be applied to any other field where a form of energy (not essentially electrical) is transferred between a source and a load.

An option to impedance matching is impedance bridging, where the load impedance is selected to be much bigger than the source impedance and where the maximizing voltage transfer, instead of maximum power transfer, is the goal.

Attenuators:

Fixed attenuators in circuits are employed to lower voltage, dissipate power and to enhance impedance matching. In measuring signals, attenuator pads or adaptors are employed to lower the amplitude of the signal a known amount to allow measurements, or to protect the measuring device from signal levels which might damage it. Attenuators are as well employed to 'match' impedances through lowering apparent Standing Wave Ratio.

There are mainly four typical attenuator circuits:

1195_Attenuators.jpg

The fundamental circuits employed in attenuators are 'pi' pads (π-type) and 'T' pads. These might be needed to be balanced or unbalanced networks based on whether the line geometry by which they are to be employed is balanced or unbalanced. Attenuators employed by coaxial lines are the unbalanced form whereas attenuators for use having twisted pair are needed to be the balanced form.

Four basic attenuator circuit diagrams are represented in the above figure. As an attenuator circuit comprises solely of passive resistor elements, it is linear and reciprocal. When the circuit is as well made symmetrical that is generally the case as it is often needed that the input and output impedances Z1 and Z2 are equivalent, then the input and output ports are not differentiated, however by convention the left and right sides of the circuits are termed to as input and output.

Attenuator characteristics:

Attenuators Specifications:

1) The given parameters are utilized in commercial attenuator specifications. Attenuation is represented in decibels of relative power and you might roughly state that a 3dB attenuator decreases power to one half whereas a 6dB attenuator to one fourth, 10dB to one tenth, 20dB to one hundredth, 30dB to one thousandth and so forth.

2) Frequency bandwidth is represented in Hertz.

3) The power dissipation is based on mass and surface area of resistance material and also possible additional cooling fins and heat sinks. 

4) The standing Wave Ratio is the standing wave ratio for the input and the output ports.

5) The other specifications comprise Repeatability and Accuracy.

Impedance Matching:

Impedance matching is the practice of designing the input impedance of the electrical load or the output impedance of its equivalent signal source in order to make the most of the power transfer and minimize reflections from the load.

2451_Impedance Matching.jpg

In case of a complex source impedance ZS and load impedance ZL, matching is achieved if

ZS = ZL*

Here, * points out the complex conjugate.

Description:

The word impedance is employed for the resistance of a system to an energy source. For constant signals, this resistance can as well be constant. For varying signals, it generally alters with frequency. The energy comprised can be electrical, mechanical, magnetic or even thermal. The theory of electrical impedance is possibly the most generally recognized. Electrical impedance, such as electrical resistance, is evaluated in ohms. In common, impedance consists of a complex value that signifies that loads usually encompass a resistance to the source which is in phase by a sinusoidal source signal and reactance that is out of phase having a sinusoidal source signal. The total impedance (symbol: Z) is the vector sum of the resistance (symbol: R; a real number) and the reactance (symbol: X; an imaginary number).

In simple cases, like low-frequency or direct-current power transmission, the reactance is negligible or zero and the impedance can be assumed as a pure resistance, represented as a real number. In the given summary, we will assume that the general case whenever the resistance and reactance are both significant, and as well the special case in which the reactance is negligible.

Reflection less matching

Impedance matching to reduce the reflections and maximize power transfer over a comparatively high bandwidth (as well termed as reflection less matching or broadband matching) is the most generally used. To prevent all the reflections of the signal back into the source, the load (that should be totally resistive) should be matched precisely to the source impedance (that again should be entirely resistive). In this case, when a transmission line is employed to join the source and load altogether, it should as well be the similar impedance: 

Zload = Zline = Zsource, here Zline is the characteristic impedance of the transmission line. However source and load should each be totally resistive for this form of matching to work, the more general word 'impedance' is still employed to explain the source and load characteristics. Any and all reactance in reality present in the source or the load will influence the 'match'.

Complex conjugate matching:

This is employed in cases in which the source and load are reactive. This form of impedance matching can merely maximize the power transfer between a reactive source and a reactive load at a single frequency. In this situation:

Zload = Zsource*

Here, * points out the complex conjugate.

When the signals are kept in the narrow frequency range, for which the matching network was designed, reflections (that is, in this narrow frequency band only) are as well minimized. For the case of purely resistive source and load impedances, all the reactance terms are zero and the formula above decreases to:

Zload = Zsource

Power transfer:

If a source of power having fixed output impedance, like an electric signal source, a radio transmitter, or even mechanical sound (example: a loudspeaker) operates into a load, the maximum possible power is delivered to the load if the impedance of the load (that is, load impedance or input impedance) is equivalent to the complex conjugate of the impedance of the source (that is, its internal impedance or output impedance). For two impedances to be complex conjugates, their resistances should be equivalent, and their reactance should be equivalent in magnitude however of opposite sign.

In low-frequency or DC systems or in systems having purely resistive sources and loads, the reactance is zero, or small adequate to be ignored. In this case, the maximum power transfer takes place if the resistance of the load is equivalent to the resistance of the source.

Impedance matching is not for all time desirable. For illustration, if a source having low impedance is joined to a load having high impedance, then the power that can pass via the connection is limited through the higher impedance, however the electrical voltage transfer is higher and less prone to corruption than when the impedances had been matched. This maximum voltage connection is a general configuration termed as impedance bridging or voltage bridging and is broadly employed in signal processing. In such applications, delivering a high voltage (to minimize signal degradation throughout transmission and/or to consume less power by decreasing the currents) is frequently more significant than the maximum power transfer.

In older audio systems, dependent on transformers and passive filter networks, dependent on the telephone system, the source and load resistances were matched at 600 ohms. One of the reasons for this was to maximize the power transfer, as there were no amplifiers available which could restore lost signal. The other reason was to make sure correct operation of the hybrid transformers employed at central exchange equipment to separate outgoing from incoming speech so that these could be augmented or fed to a four-wire circuit. Most of the modern audio circuits, on the other hand, utilize active amplification and filtering, and they can make use of voltage bridging connections for best accuracy.

Strictly speaking, impedance matching merely applies if both source and load devices are linear, though useful matching might be obtained between the nonlinear devices by certain operating ranges.

Impedance matching devices:

Adjusting the source impedance or the load impedance, in common, is termed as impedance matching. There are three possible manners to enhance an impedance mismatch, all of which are termed as impedance matching:

1) Devices proposed to present an apparent load to the source of Rload = Rsource* (complex conjugate matching). Given a source by a fixed voltage and fixed source impedance, the maximum power theorem states this is the only mode to extract the maximum power from the source. 

2) Devices intended to present an apparent load of Rload = Rline (that is, complex impedance matching), to avoid the echoes. Given a transmission line source having fixed source impedance, this 'reflectionless impedance matching at the end of the transmission line is the only mode to avoid reflecting echoes back to the transmission line.

3) Devices intended to present the apparent source resistance as close to zero as possible, or presenting an apparent source voltage as high as possible. This is the only mode to maximize energy efficiency, and therefore it is employed at the starting of electrical power lines. Such an impedance bridging connection as well minimizes distortion and electromagnetic interference, and therefore it is as well used in the modern audio amplifiers and signal processing devices. 

There is a diversity of devices which are used between some sources of energy and a few loads which carry out impedance matching. To match electrical impedances, engineers make use of combinations of transformers, resistors, inductors, capacitors and transmission lines. Such passive and active impedance matching devices are optimized for various applications, and are termed as baluns, antenna tuners (at times termed as ATUs or roller coasters due to the reason of their appearance), acoustic horns, matching networks and terminators.

Transformers:

Transformers are at times employed to match the impedances of circuits by various impedances. A transformer transforms or converts alternating current at one voltage to the similar waveform at another voltage. The power input to the transformer and output from the transformer is similar (apart from for conversion losses). The side having the lower voltage is at low impedance, as this consists of the lower number of turns, and the side having the higher voltage is at higher impedance as it consists of more turns in its coil.

Resistive network:

Resistive impedance matches are simplest to design and can be accomplished by a simple 'L' pad comprising of just two resistors. Power loss is an unavoidable consequence of employing resistive networks and they are as a result only usually used to transfer the line level signals.

Stepped transmission line:

Most of the lumped element devices can match a particular range of load impedance. For illustration, in order to match an inductive load into real impedance, a capacitor requires be employed. And if the load impedance becomes capacitive for several reasons, the matching element should be substituted through an inductor. In most of the practical cases though, there is a requirement to employ the similar circuit to match a broad range of load impedance, therefore simplify the circuit design. This issue was addressed through the stepped transmission line where multiple serially placed quarter wave dielectric slugs are employed to vary transmission line's characteristic impedance. By controlling the position of each and every individual element, a broad range of load impedance can be matched devoid of having to reconnect the circuit.

Several special situations - like radio tuners and transmitters - make use of tuned filters like stubs to match the impedances at particular frequencies. These can distribute diverse frequencies to various places in the circuit.

Moreover, there is closely associated idea of:

- Power factor correction devices meant to cancel out the reactive and nonlinear features of a load at the end of a power line. This causes the load observed by the power line to be purely resistive. For a given true power needed by a load, this minimizes the true current supplied via the power lines, and so minimizes the power wasted in the resistance of such power lines. 

For illustration, a maximum power point tracker is employed to take out the maximum power from a solar panel, and proficiently transfer it to batteries, the power grid, or other loads. The maximum power theorem applies to its upstream connection to the solar panel; therefore it emulates a load resistance equivalent to the solar panel source resistance. Though, the maximum power theorem doesn't apply to its downstream connection, in such a way that connection is an impedance bridging connection - this emulates a high-voltage, low-resistance source, to maximize the efficiency.

L-section:

One of the simple electrical impedance matching networks needs one capacitor and one inductor. One reactance is in parallel by the source (or load) and the other is in series having the load (or source). Whenever a reactance is in parallel by the source, the effective network matches from high impedance to the low impedance. The L-section is inherently a narrowband matching network.

The analysis is as discussed. Assume a real source impedance of R1 and real load impedance of R2. Whenever a reactance X1 is in parallel by the source impedance, the combined impedance can be represented as:

(jR1X1)/(R1 + jX1)

Whenever the imaginary part of the above impedance is fully canceled by the series reactance, then the real part is:

R2 = (R1X12)/(R12 + X12)

Solving for X1:

X1 = √[(R2R12)/(R1 - R2)]

If R1 >> R2 then the above equation can be approximated as:

X1 ≈ √(R1R2)

The inverse connection, impedance step up, is merely the reverse, example: reactance in series having the source. The magnitude of the impedance ratio is limited through reactance losses like the Q of the inductor. Multiple L-sections can be wired in cascade to accomplish higher impedance ratios or greater bandwidth. Transmission line matching networks can be modeled as infinitely many L-sections wired in cascade. You can achieve optimal matching circuits can be designed for a specific system by the utilization of the Smith chart.

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