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Important Filters

Chebyshev filter:

Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (In type I or Chebyshev) or stopband ripple (In type II or Inverse Chebyshev) than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named so because its mathematical characteristics are derived from Chebyshev polynomials.

Butterworth filter:

The Butterworth filter is a type of signal processing filter designed to have a flat(i.e no ripples in the passband and rolls off towards zero in the stopband) frequency response in the passband. It is also referred to as a maximally flat magnitude filter.

Compared with a Chebyshev Type I/Type II filter or an elliptic filter, the Butterworth filter has a slower roll-off, and thus will require a higher order to implement a particular stopband specification, but Butterworth filters have a more linear phase response in the pass-band than Chebyshev Type I/Type II and elliptic filters can achieve.

Elliptic filter/Caur filter:

An elliptic filter (also known as a Cauer filter) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in gain between the passband and the stopband, for the given values of ripple (whether the ripple is equalized or not). Alternatively, one may give up the ability to independently adjust the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations.

As the ripple in the stopband approaches zero, the filter becomes a type I Chebyshev filter. As the ripple in the passband approaches zero, the filter becomes a type II Chebyshev filter and finally, as both ripple values approach zero, the filter becomes a Butterworth filter.

Gaussian filter:

In electronics and signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it). Gaussian filters have the properties of having no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay.Gaussian filter is considered the ideal time domain filter, just as the sinc is the ideal frequency domain filter.These properties are important in areas such as oscilloscopes and digital telecommunication systems.

Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass.

All-pass filter:

An all-pass filter is a signal processing filter that passes all frequencies equally, but changes the phase relationship between various frequencies. It does this by varying its propagation delay with frequency. Generally, the filter is described by the frequency at which the phase shift crosses 90° (i.e., when the input and output signals go into quadrature --when there is a quarter wavelength of delay between them).They are generally used to compensate for other undesired phase shifts that arise in the system, or for mixing with an unshifted version of the original to implement a notch comb filter.

The operational amplifier circuit shown in Figure 1 implements an active all-pass filter with the transfer function

$H(s) = \frac{ sRC - 1 }{ sRC + 1 }, \,$

which has one pole at -1/RC and one zero at 1/RC (i.e., they are reflections of each other across the imaginary axis of the complex plane). Themagnitude and phase of H(iω) for some angular frequency ω are

$|H(i\omega)|=1 \quad \text{and} \quad \angle H(i\omega) = 180^{\circ} - 2 \arctan(\omega RC). \,$

As expected, the filter has unity-gain magnitude for all ω. The filter introduces a different delay at each frequency and reaches input-to-output quadratureat ω=1/RC (i.e., phase shift is 90 degrees).

At high frequencies, the capacitor is a short circuit, thereby creating a unity-gain voltage buffer (i.e., no phase shift).
At low frequencies and DC, the capacitor is an open circuit and the circuit is an inverting amplifier (i.e., 180 degree phase shift) with unity gain.
At the corner frequency ω=1/RC of the high-pass filter (i.e., when input frequency is 1/(2πRC)), the circuit introduces a 90 degree shift (i.e., output is in quadrature with input; it is delayed by a quarter wavelength).

In fact, the phase shift of the all-pass filter is double the phase shift of the high-pass filter at its non-inverting input.

APF Using Latice Topology:

pole zero plot of APF:(Alternate poles and zeros)

Smoothing of Ripple in Power supply unit

(Q)Is it ever practical to use an inductor to smooth the output ripple in power supply unit ????

· we typically use large capacitors to smooth out the output ripple from a rectifying/switching power supply.Until recently, a capacitor was usually chosen over an inductor because of weight. Also, from a storage point of view, an inductor stores it's energy in the form of a magnetic field. When the current stops flowing, the magnetic field collapses. When the magnetic field is fully dissipated, the stored energy is depleted. This happens whether or not anything is utilizing the stored energy or not.

A capacitor stores energy in the form of accumulated electrons on the plates of the capacitor. A capacitor charge will deplete in direct proportion to the amount of current drain between it's terminals. This current drain is in the form of both, the internal resistance of the capacitor and the load resistance externally connected to the capacitor terminals. But the capacitor, unlike the inductor, will hold the accumulated charge if not connected to an external load - only to be discharged by the internal resistance of that capacitor.

· Also inductors resist current change and thus smooth current ripple, capacitors smooth voltage ripple.

A simple power supply has this arrangement:

AC supply ........>> diode or diode bridge.......>>.capacitor ......>>ground. The load goes across the capacitor.

If you wanted to add an Inductor, it should go after the capacitor in line with the load and there should be another capacitor after it to ground. The load goes across the second capacitor.The inductor ideally has no effect on DC current but it will help to limit the AC component on the DC.The input capacitor is important as the inductor can cause voltage spikes which might destroy the diodes if the capacitor wasn't there.
Also, the inductor should be a special one with an air gap in the iron core so that it does not saturate easily with DC passing through it.An inductor is not normally used because voltage regulator ICs are used and these give almost perfect smoothing.

Smoothing using Capacitors in power supply unit

(Q)Why do output capacitors always come in pairs in power supply unit ??? A big electrolytic and a small ceramic??? Why???

The answer to the large electrolytic is little bit tough. The simplest answer is that the caps are based on the design. Sometimes two caps and an inductor are used in a 'pi' filter(two big electrolytic and an inductor). Sometimes it is easier to use two caps to create one big one rather than the expense of buying one big cap. There is nothing wrong with using an inductor, but it is also based on the design.

Second question regarding a big/small pairing .The typical Electrolytic/tantalum or ceramic cap combination is simple, the electrolytic can handle low frequencies, the tantalums/ceramics handle high frequencies, therefore you are improving decoupling. Electrolytic can also be a reserve energy source for current demand 'spikes'.

In some cases the electrolytic is for bulk storage and the ceramic is to get the ESR down. It's all for improved efficiency.

Can we use capacitor instead of battery ??????

I've read that capacitors can be used as batteries – does that mean they can be used interchangeably?
Is there any advantage to using a capacitor as a battery or in place of a battery?

Capacitors can indeed be used to store small amounts of energy. However, compared to a battery they have very low energy densities. As for being interchangeable with batteries, no not really. i.e. if you have a device that uses AA batteries, you will not be able to obtain a AA sized capacitor that you could simply place in that device. Even if you could, the AA batteries will last many hours of use, whereas a capacitor that size may only be seconds (at best minutes).

If however you are thinking about some sort of device that needs a few seconds to shut down gracefully in the event of a power failure, then a capacitor may be a better choice than a battery. The battery has a finite shelf life, whereas the capacitor will last indefinitely and there is no chance of a chemical spill, such as occurs with an aging battery.

Over the useful life of a battery, its voltage will remain relatively constant as it discharges, within a few tenths of a volt (above or below its nominal voltage). For a rechargeable battery, it is a slightly more pronounced at the end of its discharge cycle. Unlike the battery, the voltage of a capacitor is dependent on its level of charge, i.e. from q = CV we have V = q/C. When the charge (q) is high the voltage (V) is high and when the charge is low the voltage is low.

Having said all of that, capacitors serve many useful purposes other than storing energy, uses that cannot be performed by a battery

Log & Anti-log Amplifier

A log amplifier (logarithmic converter) is one for which the output voltage V_out is K times the natural log of the input voltage V_in. This can be expressed as,

$V_\mathrm{out} = K \ln\frac{V_\mathrm{in}}{V_\mathrm{ref}}$

where V_ref is the normalization constant in volts and K is the scale factor.

A necessary condition for successful operation of a log amplifier is that the input voltage, V_in is always positive. This may be ensured by using a rectifier and filter to condition the input signal before applying to the log amp input. As V_in is positive, V_out is obliged to be negative (since the op amp is in the inverting configuration) and is large enough to forward bias the emitter-base junction of the BJT keeping it in the active mode of operation. Now,

$V_\mathrm{BE} = -V_\mathrm{out}\,\!$

$I_\mathrm C = I_\mathrm{SO}(e^{V_\mathrm{BE} / V_\mathrm T} - 1) \approx I_\mathrm {SO} e^{V_\mathrm{BE} /V_\mathrm T}$

$\Rightarrow V_\mathrm{BE} = V_\mathrm T \ln \frac{I_\mathrm C}{I_\mathrm{SO}}$

where $I_\mathrm{SO}\,$ is the saturation current of the emitter-base diode and $V_\mathrm T\,$ is the thermal voltage. Due to the virtual ground at the op amp differential input,

$I_\mathrm C = \frac{V_\mathrm{in}}{R_1}$ , and

$V_\mathrm{out} = -V_\mathrm T \ln \frac{V_\mathrm{in}}{I_\mathrm{SO} R_1}$

The output voltage is expressed as the natural log of the input voltage. Both the saturation current $I_\mathrm{SO}\,$ and the thermal voltage $V_\mathrm T\,$ are temperature dependent, hence, temperature compensating circuits may be required.

The relationship between the input voltage $V_{\text{in}}$ and the output voltage $V_{\text{out}}$ is given by:

$V_{\text{out}} = -V_{\text{T}} \ln \left( \frac{V_{\text{in}}}{I_{\text{S}} \, R} \right)$

where $I_{\text{S}}$ and $V_{\text{T}}$ are the saturation current and the thermal voltage of the diode respectively.

EXPONENTIAL/ANTILOG AMPLIFIER:

The relationship between the input voltage $v_{\text{in}}$ and the output voltage $v_{\text{out}}$ is given by:

$v_{\text{out}} = -R I_{\text{S}} e^{\frac{v_{\text{in}}}{V_{\text{T}}}}$

where $I_{\text{S}}$ is the saturation current and $V_{\text{T}}$ is the thermal voltage.

Considering the operational amplifier ideal, then the negative pin is virtually grounded, so the current through the diode is given by:

$I_{\text{D}} = I_{\text{S}} \left( e^{\frac{V_{\text{D}}}{V_{\text{T}}}} - 1 \right)$

when the voltage is greater than zero, it can be approximated by:

$I_{\text{D}} \simeq I_{\text{S}} e^{\frac{V_{\text{D}}}{V_{\text{T}}}}.$

The output voltage is given by:

$v_{\text{out}} = -R I_{\text{D}}.\,$

Relay-Electrically Operated Switch

A relay is an electrically operated switch. Many relays use an electromagnet to operate a switching mechanism, but other operating principles are used. Relays find applications where it is necessary to control a circuit by a low-power signal, or where several circuits must be controlled by one signal. moving parts, instead using a semiconductor device triggered by light to perform switching. Relays with calibrated operating characteristics and sometimes multiple operating coils are used to protect electrical circuits from overload or faults; in modern electric power systems these functions are performed by digital instruments still called "protection relays".The first relays were used in long distance telegraph circuits, repeating the signal coming in from one circuit and re-transmitting it to another. Relays found extensive use in telephone exchanges and early computers to perform logical operations. A type of relay that can handle the high power required to directly drive an electric motor is called a contactor.

It consists of a coil of wire surrounding a soft iron core, an iron yoke, which provides a low reluctance path for magnetic flux, a movable iron armature, and a set, or sets, of contacts; two in the relay pictured. The armature is hinged to the yoke and mechanically linked to a moving contact or contacts. It is held in place by a spring so that when the relay is de-energised there is an air gap in the magnetic circuit. In this condition, one of the two sets of contacts in the relay pictured is closed, and the other set is open. Other relays may have more or fewer sets of contacts depending on their function. The relay in the picture also has a wire connecting the armature to the yoke. This ensures continuity of the circuit between the moving contacts on the armature, and the circuit track on the Printed Circuit Board (PCB) via the yoke, which is soldered to the PCB.

When an electric current is passed through the coil, the resulting magnetic field attracts the armature, and the consequent movement of the movable contact or contacts either makes or breaks a connection with a fixed contact. If the set of contacts was closed when the relay was de-energised, then the movement opens the contacts and breaks the connection, and vice versa if the contacts were open. When the current to the coil is switched off, the armature is returned by a force, approximately half as strong as the magnetic force, to its relaxed position. Usually this force is provided by a spring, but gravity is also used commonly in industrial motor starters. Most relays are manufactured to operate quickly. In a low voltage application, this is to reduce noise. In a high voltage or high current application, this is to reduce arcing.

If the coil is energized with DC, a diode is frequently installed across the coil, to dissipate the energy from the collapsing magnetic field at deactivation, which would otherwise generate a voltage spike dangerous to circuit components. Some automotive relays already include that diode inside the relay case. Alternatively a contact protection network, consisting of a capacitor and resistor in series, may absorb the surge. If the coil is designed to be energized with AC, a small copper ring can be crimped to the end of the solenoid. This "shading ring" creates a small out-of-phase current, which increases the minimum pull on the armature during the AC cycle.By analogy with the functions of the original electromagnetic device, a solid-state relay is made with a Thrusters or other solid-state switching device. To achieve electrical isolation an opt coupler can be used which is a light-emitting diode (LED) coupled with a photo transistor.

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