Thursday, April 20, 2017

Basic Electronics On The Go - Passive Low Pass RC Filters

From http://www.electronicshub.org/passive-low-pass-rc-filters/

Introduction

Filter is a circuit which is used to filter the signals so that it will pass only required signals and avoid unwanted signals. Generally filters are designed by either passive components or active components.
Passive components are resistors, inductors and capacitors.
Active components are transistors, FETs and Op-amps.
Low Pass Filter is a filter which will pass only low frequency signals and attenuate high frequency signals. It allows signals only from 0Hz to cut off frequency ‘fc’. This cut off frequency value will depend on the value of the components used in the circuit. Generally the cut off frequency of these filters are preferably below 100 kHz. The cut off frequency is also called break frequency or turn over frequency.

Passive Low Pass Filter

The Low Pass Filter circuit which is designed by passive components is a passive low pass filter.
The RC Low Pass Filter is shown below

Simply by connecting resistor ‘R’ in series with a capacitor ‘C’ gives  the RC Low Pass Filter. The  resistor is independent to the variations of the applied frequencies in the circuit but the capacitor is a sensitive component which means that it will  respond to the changes in the circuit.

Since it has only one reactive component this circuit can also termed as ‘one pole filter’ or ‘First order filter’. The input voltage ‘Vin’ is applied to whole series loop and the output voltage is taken only across the capacitor.
Since capacitor is a sensitive component the main concentration to be observed is about “capacitive reactance”.  Capacitive reactance is the opposition response created due to the capacitor in the circuit.
In order to maintain the capacitance of the capacitor, the capacitor will oppose a small amount of current flow in the circuit. This opposition to the current flow in the circuit is called impedance. Thus the capacitive reactance decreases with increase of opposing current.

By this we can say that the capacitive reactance is inversely proportional to the frequency applied to the circuit. The resistive value of the resistor is stable whereas the capacitive reactance value varies. The above circuit can acts as a ‘frequency variable voltage divider’ circuit.

The capacitive reactance can be formulated as follows:



Output voltage calculation


In order to get the potential divider equation we have to consider impedance, capacitive reactance, input voltage and output voltage. By using these terms we can formulate the equation for RC potential divider equation as follows

By using this equation we can calculate the value of the output at any applied frequency.



Frequency Response of Low pass filter


From the introduction to filters we already saw that the magnitude |H(jω)| of the filter is taken as the gain of the circuit. This gain is measured as  20 log (Vout / Vin) and for any RC circuit the angle of the slope ‘roll-off ‘ is at -20 dB/ decade.

The band of frequencies below the cut off region is referred to  as ‘Pass Band’ and the band of frequencies after the cut off frequency are referred to as ‘Stop Band’. From the plot it can be observed that the pass band is the bandwidth of the filter.

From this plot it is clear that until cut off frequency the gain is constant because the output voltage is proportional to the frequency value at the low frequencies. This is due to the capacitive reactance which acts like open circuit at low frequencies and allows maximum current through the circuit for high frequencies. The value of the capacitive reactance is very high at low frequencies thus it has greater ability to block the current flow through the circuit.

Once it reaches the cut off frequency value, the output voltage decreases gradually and reaches  zero. The gain also decreases along with the output voltage. After the cut off frequency, the response of the circuit slope will have a  roll-off point of -20 dB/ decade.

This is mainly due to the increase of the frequency. When frequency increases the capacitive reactance value decreases and thus the ability to block the current through the capacitor decreases.The circuit acts as a short circuit. Thus the output voltage of the filter is zero at high frequencies.

The only way to avoid this problem is to choose the frequency ranges up to which these resistor and capacitor can withstand. The values of the capacitor and the resistor play a major role because the cut off frequency ‘fc’ will depend on these values. If the frequency ranges are within the cut off frequency range then we can overcome the short circuit problem.

This cut off point will occurs when the resistance value and the capacitive reactance value coincides which means the vector sum of the resistance and reactive capacitance are equal. That is when R = Xand at this situation the input signal is attenuated by -3dB/decade.

This attenuation is approximately 70.7 % of input signal.The time taken to charge and discharge  the plates of the capacitor varies according to the sine wave. Due to this, the phase angle (ø) of the output signal lags behind the input signal after cut-off frequency.At cut-off frequency the output signal is -45° out of phase.

If the input frequency of the filter increases the lagging angle of the circuit output signal increases. Simply for the higher frequency value, the circuit is more out of phase.
The capacitor has more time to charge and discharge the plates at low frequencies because the switching time of the sine wave is more. But with the increase of frequency the time taken to switch to the next pulse gradually decreases. Due to this, the time variations occurs which leads to phase shift of the output wave.

Cut-off frequency of a passive low pass filter mainly depends on the resistor and capacitor values used in the filter  circuit.This cut-off frequency is inversely proportional to both resistor and capacitor values. The cut-off frequency of a passive low pass filter is given as
fC = 1/(2πRC)
The phase shift of a passive low pass filter is given as
Phase shift (ø) = – tan-1 (2πfRc)


Time Constant (τ)

As we already seen that the time taken by the capacitor for charging and discharging of the plates with respect to the input sinusoidal wave results in the phase difference. The resistor and the capacitor in series connection will produce this charging and discharging effect.

The time constant of a series RC circuit is defined as the time taken by the capacitor to charge up to 63.2% of the final steady state value and also it is defined as the time taken by the capacitor   to discharge to 36.8% of steady state value. This Time constant is represented with symbol ‘τ ’.

The relationship between the time constant and the cut off frequency is as follows
Time constant τ = RC = 1/ 2πfc and ωc = 1/τ = 1/RC

By this we can say that the output of the filter depends on the frequencies applied at the input and on the time constant.
(to be updated)

Wednesday, April 5, 2017

Basic Electronics On The Go - Introduction to Filters and Capacitive Reactance

http://www.electronicshub.org/introduction-to-filters-and-capacitive-reactance/

Introduction

Electrical filter is a circuit, designed to reject all unwanted frequency components of an electrical signal and allows only desired frequencies. In other words a filter is a circuit which allows only a certain band of frequencies.The main applications of the filters are at audio equalizers and in sensitive electronic devices whose input signals should be conditional.These filters are mainly categorized into 2 types. They are active filters and passive filters.


Passive Filters


Passive filters do not contain any amplifying elements. Examples of them are resistors, capacitors and inductors (passive elements). These filters will not draw any additional power from the external battery supply. The capacitor will allow the high frequency signals and the inductor will allow low frequency signals. Similarly inductor restricts the flow of high frequency signals and capacitor restricts the lower frequency signals. In these filters, output signal amplitude is always less than the amplitude of the applied input signal. The gain of passive filters is always less than unity.This shows that the gain of the signals  cannot be improved by these passive filters. Due to this, the characteristics of the filters are affected by the load impedances. These filters can work at higher frequency ranges nearly at 500 MHz also.



Active Filters


Active filters contain amplifying elements such as Op-Amps, Transistors and FET’s (active components)in addition to the passive elements (Resistors, Capacitors and Inductors). By using these filters we can overcome the drawbacks of Passive filters. Active filters will depend on external power supply because it will amplify the output signals. Without any inductor element these can achieve the resonant frequency, that is the input impedance and output impedances are nullified by each other. Inductors are used less in active filters because inductors dissipate some amount of power and generates stray magnetic fields.  Also, due to inductors, the size of the active filter increases. 



Some of the advantages of Active filters

  • The combination of op-amps, resistors, capacitors, transistors and FETs gives an integrated circuit which in turn reduces the size and weight of the filter.
  • The gain of an Op-amp can be easily controlled in the closed loop form. Due to this reason the input signal is not restricted.
  • These are applicable in Butterworth filters, Chebyshev and Cauer filters.
The main drawback in active filters is the operational frequency range is less. In many applications the operational frequency range of active filters is maximized to 500 kHz only. The active filters must require D.C power supply. When compared with the passive filters these active filters are more sensitive. The outputs can also be disturbed  by  environmental changes.

The filter is a sensitive circuit and in which the output components are only frequency terms. To analyze the filter circuit the frequency domain representation is the best one. This representation is as shown below.




The magnitude of the filter M is called as the gain of the filter. Magnitude is generally represented in dB as 20log (M).

One of the important characteristic of the filters is cut-off frequency. It is defined as the frequency which separates both pass band and stop band in frequency response. Pass band is the range of frequencies that are allowed by the filter without any attenuation. Stop band is defined as the band of frequencies that are not allowed by the filter.

Filters are classified based on the frequency of signals that they allow through them. There are four types of filters -  they are Low Pass Filters, Band Pass Filters, High Pass Filters and Band Stop Filters. Due to the usage of high speed op-amps and approximate values of the components, the characteristics of ideal and practical responses are nearly equal.



Low Pass Filter


Low Pass Filters will pass the frequency signals less than cut-off frequency ‘fc’. Practically a small range of frequencies will pass even after the cut-off frequency range. The gain of the filter will depend  on the frequency. If the input signal frequency increases then, gain of the filter decreases. At the end of the transition band the gain becomes zero. This is as shown below.

Where dotted line indicates the ideal filter characteristics and continuous line indicates practical filter characteristics.
Applications of low pass filters are in sound system that is in various types of loudspeakers. To block the harmonic emissions these low pass filters are used in radio transmitters. These are also used at DSL splitters in telephone subscriber lines.


High Pass Filter

They will pass the frequencies after the cut off frequency ‘fc’. In practical cases, a  small range of frequencies will pass below the cut off range. This is as shown below.


The combination of high pass filter with low pass filter forms Band pass filter. Applications of the high pass filters are at RF circuits and  are also used in DSL splitters.


Band Pass Filter

The name of the filter itself indicates that it allows only a certain band of frequencies and blocks all the remaining frequencies. The upper and lower limits of the band pass filter depend on the filter design. Practical and ideal characteristics of band pass filter are shown below.

Applications of band pass filters are at transmitter and receiver circuits. These are mainly used to calculate the sensitivity of the receiver circuits and to optimize the signal to noise ratio.



Band Stop Filter


These are also called as Band rejection or band elimination filters. These filters stop only a particular band of frequencies and allow all other frequencies.The frequency limits of the filter depend on the filter design.The dotted line indicates the ideal case where as a continuous line indicates the practical case. It has two pass bands and one stop band.

Applications of band stop filters are at instrument amplifiers.

Ideal filters frequency response

Now let us see the ideal response of different filters. Here fL indicates the lower cut-off frequency and fH indicates the higher cut-off frequency.

Ideal characteristics of Low pass filter



This response shows that the low pass filter will allow the signals up to lower cutoff frequency and stops the frequencies higher than the lower cutoff frequency.

Ideal characteristics of High pass filter



This shows the high pass filter will allow the frequencies greater than the higher cut off frequency and stops the frequencies lesser than the high cut off frequency.


Ideal characteristics of Band pass filter



This response shows that the band pass filter will pass the frequencies between the lower cutoff region and higher cut-off region only. It stops the frequencies which are lesser than the lower cutoff frequency and also stops the frequencies greater than higher cut-off frequencies.



Ideal characteristics of Band stop filter


The above figure shows that the frequencies which are greater than the lower cut off frequency and the frequencies which are lower than the higher cut off frequency are not processed.


Capacitive Reactance

When a resistor is connected in series with a capacitor, an RC circuit is formed. In RC circuits,  the capacitor will charge from the D.C supply voltage and when the supply voltage is decreased,  the capacitor  discharges. Not only at the time of the D.C supply, but also in the case of A.C supply - according to the supply voltage level -  the capacitor will charge and discharge continuously.
But due to internal resistance there will be some attenuation in the flow of the current through the capacitor. This internal resistance is called Capacitive Reactance. ‘X_C’ Indicates the Capacitive Reactance and it is measured in Ohms same as that of the resistance.

When the frequency is varied in the capacitive circuit according to the amount of frequency change, this capacitive reactance value also changes. The electrons flow from one plate to the other plate causes the current flow in the circuit.  When the frequency through the capacitor increases, the capacitive reactance value decreases and when frequency through the capacitor decreases, the capacitive reactance value increases. Thus, by this we can say that the capacitive reactance is inversely proportional to the applied frequency level. This shows that the capacitor connected in the circuit is dependent on supply frequency. This phenomenon is called complex impedance.


Capacitive Reactance Formula
Xc = 1/(2πfc)
Where X= Capacitive Reactance
π = 3.142
f = Frequency in Hz
c = Capacitance in Farads (F).


Capacitive Reactance Vs Frequency



From the above frequency verses capacitive reactance plot we can observe that when the frequency is zero, the reactance value reaches to infinity -  this shows the phenomenon of the open circuit. When the value of the frequency increases exponentially, the reactance value decreases. When frequency reaches to infinity, the reactance value is  at nearly zero - this gives us closed circuit behaviour.


Voltage Divider concept

We have already studied the voltage divider concept in resistors topic and we know that the voltage divider circuit is able to produce the output voltage which is a fraction of the input voltage.


By replacing the resistor R2 with a capacitor C in the above circuit the voltage drop across the two components changes with the input frequency because reactance of the capacitor varies with the frequency.Now the output voltage across the capacitor depends on the input frequency. Using this concept we can construct passive low and high pass filters by replacing one of the resistors with a capacitor in a voltage divider circuit.


Capacitor behaviour in Low Pass Filter




For the Low pass filter the resistor R2 is replaced by the capacitor C1. At normal frequency the circuit is as shown in the above figure. When the frequency is zero the reactance value is very high which is nearly equal to infinity. At this condition circuit acts as an open circuit. When frequency is very high the reactance value reaches to zero and the circuit acts as a closed circuit. Both of these behaviours are shown in the above figure.



Capacitor behaviour in High Pass filters


For the High pass filter the resistor R1 is replaced by the capacitor C1. From the above figure it is clear that at normal frequency the circuit acts like a high pass filter circuit. Initially at the zero frequency value the circuit behaves like an open circuit. When frequency increases the reactance will decrease exponentially. At some point the frequency reaches to infinity level thus, it affects the reactance to reach the zero state. These circuit behaviors are shown in the above figures.


Wednesday, March 29, 2017

Basic Electronics On The Go - Applications of Inductor

From http://www.electronicshub.org/applications-of-inductor/

Like resistors and capacitors, inductors are also passive elements which are used to store the electric energy in the form of magnetic field. Simply an inductor is a wire or coil of a good electric conducting material with few turns (twists) in it.

They will produce the magnetic flux (field) around them by the flow of an alternating current through it. An ideal inductor has no inductive reactance, so it acts as short circuit. Practically, every inductor has some resistance internally which we call as ‘Inductive reactance’. It is measured in ohms. When the inductive reactance of a coil is very high then the circuit acts as an open circuit and allows maximum current through it. Inductance is the phenomenon of an inductor that opposes the flow of current in the circuit, by generating the back EMF. This inductance is measured in Henry.

Inductors are of many types like air cored, iron cored, coupled or differential type and many more. Based on the requirement, inductors have many applications in electrical transmission.


Inductors in Tuned Circuits


Inductors are used in tuning circuits which are used to select the desired frequency. In a tuned circuit, we have capacitor connected along with the inductor, either in parallel or series. The frequency of the tuning circuit at which the capacitive reactance is equal to the inductive reactance (XC = XL) is called ‘Resonant Frequency’.




The series resonance circuits are used in many electronic circuits like television , radio tuning circuits and filters to vary the frequency and selecting the various frequency channels.



Inductive sensors


Inductors are used in proximity sensors which work on the principle of inductance. We know that inductance is the phenomenon in which , the magnetic field produced in the coil , will oppose the flow of current in it. Thus the inductance will restrict the current flow and reduces the circuit performance.
For better performances, we need to amplify the current in a circuit. We use proximity sensors to find the level of amplification factor at which we need to amplify the current.


The manufacturers design the sensors by twisting the wire into a tight coil. There are 4 components in the inductive proximity sensor; they are an inductor or coil, an oscillator, a detection circuit and an output circuit.

In the inductive proximity sensor, a fluctuating magnetic field is generated by oscillator around the winding of coil, which locates in the sensing face of the device.



When an object moves in the field of inductive proximity area of detection, eddy currents starts building up in the metal object which will reduce the inductive sensor’s magnetic field.

Strength of the oscillator is monitored by the detection circuit and an output is triggered from the output circuitry , when the oscillations are below the sufficient level.

The inductive proximity sensor is a contactless sensor and is very reliable in operation. The inductive sensors are used at traffic lights to detect the traffic density.

Energy Storage Devices


We can store the energy in passive elements like capacitor and inductors. Inductors can store energy for a limited time. As the inductors store the energy in the form of magnetic field, it will collapse when we remove the power supply.
The inductors functions as energy storage devices in switch mode power supplies (generally we use in our computers). In these type of power supplies, the output voltage ratio depends upon the charging time of the inductor.



Induction Motors


The well known and wide range application of inductors is in  Induction motors. In these induction motors or asynchronous motors, the inductors are in a fixed position and they are not allowed to move in the  nearby magnetic field.

Induction motors converts the electrical energy into mechanical energy. The shaft in the motors will rotate because of the magnetic field produced by the alternating current.

The speed of the motor is fixed as it depends on the frequency of the power supplied from source. So we use inductors in these motors to control the speed by connecting them in series or parallel to the shaft. These induction motors are very reliable and robust.

Transformers

Transformer is another popular application of inductors. By combining the inductors of a shared magnetic field, we can design a transformer. Transformer is the basic and fundamental component of the power transmission system.

These are used to increase or decrease the power in transmission lines to required level, as step up and step down transformers respectively. In the transformers, the inductor (wire) is wounded to the core as primary and secondary windings.

The impedance of the inductor increases with increase in the frequency of supply. The impedance produced in the inductor will limit the effectiveness of transformer. In general, the inductance based transformers are limited to very low operational values.



Inductive Filters


Inductors and capacitors are combined to form filters. The filters are the electronic devices which are used to limit the frequency of the input signal entering to a circuit. There are many types of filters like low pass filter, high pass filter, band pass filter, notch filter etc which are designed by using inductors.

As the frequency increases, the impedance of the inductor also increases. Then the properties of the filter will change according to the impedance value. There are many number of filter topologies we can create by using inductors.



Chokes


Inductors are also used as chokes. We know that the inductors will create an opposite current flow, when the alternating current flows through it. This means the inductors will choke the AC current and allows DC current to pass. This property of inductors is used in power supply circuits, where AC supply need to be converted to DC supply.


Ferrite beds

Generally we see the ferrite beds in computer cables and mobile chargers etc. these ferrite beds use inductors to reduce the radio frequency interference created by the cables.



Relays


A relay is like an electrical switch. It uses an  inductor coil to control the current flow in it. When the AC current flows through the inductor of the relay, it produces a magnetic field which effects the switch contacts.



Saturday, March 18, 2017

Basic Electronics On The Go - Inductors in Parallel

From http://www.electronicshub.org/inductors-in-parallel/
From   http://www.electronics-tutorials.ws/inductor/parallel-inductors.html

Inductors are said to be connected in parallel when two terminals of an inductor respectively connected to each terminal of other inductors or inductor. Similar to the parallel connection of resistors, the total inductance in parallel connection of inductors is somewhat lesser than smallest inductance of an inductor in that connection.





When the inductors are connected in parallel, the current flow through each inductor is not exactly equal to the total current, but the sum of each individual current through the  parallel inductors gives the total current (as it divides among parallel inductors).
If the current flow through the each inductor is less than the total current, the magnetic field generated by each inductor is also less than that of field generated by total current through it.

In case of resistors in parallel, most of the current flows through the smallest resistor as it offer the least opposition to the current flow than larger resistor.
Likewise, if the inductors are connected in parallel, the current chooses the least opposition path of the inductor when current in that circuit is decreased or increased while each inductor individually opposes that change (increase or decrease of current).



Inductors Connected in Parallel (Without Magnetic Coupling)

As we discussed above, one end of inductors is connected to a node and other ends of inductors are collectively connected to another node in parallel connection. The parallel connection of n inductors is shown in below figure.
Consider that there is no magnetic coupling between inductors and hence the total inductance equal to the sum of reciprocals of the individual inductances. Let us discuss how this statement can be obtained.

We know that, in a parallel network the voltage remains constant and the current divides at each parallel inductor. If IL1, IL2, IL3 and so on ILn are the individual currents flowing in the parallel connected inductors L1, L2 and so on Ln, respectively, then the total current in the parallel inductors is given by
ITotal = IL1 + IL2 + IL3 . . . . + In
If the individual voltage drops in the parallel connection are VL1, VL2, VL3 and so on VLn, then the total voltage drop between the two terminals VT is
VTotal = VL1 = VL2 = VL3 . . . . = Vn




The voltage drop in terms of self inductance can be expressed as V = L di/ dt. This implies total voltage drop,
VT = LT di/dt
⇒ L d/dt (IL1 + IL2 + IL3 . . . . + In)
⇒ LT ( (di1)/dt + (di2)/dt + (di3)/dt . . . .)
Substituting V / L in place of di/dt, the above equation becomes

VT = LT (V/L1+ V/L2 + V/L3 . . . .)


As the voltage drop is constant across the circuit, then v = VT. So we can write
1/LT = 1/L1 + 1/L2 + 1/L3 . . . . .
This means that the reciprocal of total inductance of the parallel connection is the sum of reciprocals of individual inductances of all inductors. The above equation is true when there is no mutual inductance between the parallel connected coils.

For avoiding complexity in dealing with fractions, we can use product over sum method to calculate the total inductance of two inductors. If two inductors are connected in parallel, and if there is no mutual inductance between them, then the total inductance is given as
LT = (L1× L2)/(L1+ L2)


Mutually Coupled Inductors in Parallel

When there exist magnetic coupling between the inductors, the above derived formula for total inductance must be modified because total inductance can be more or less depending on the magnetic field directions from each inductor. The magnetic flux produced by the parallel connected inductors will link with each other.
Mutually connected inductors in parallel can be classed as either “aiding” or “opposing” the total inductance with parallel aiding connected coils increasing the total equivalent inductance and parallel opposing coils decreasing the total equivalent inductance compared to coils that have zero mutual inductance.
Consider that two inductors are connected in parallel with self inductances L1 and L2, and which are mutually coupled with mutual inductance M as shown in the  figure.below



Parallel Aiding Inductors

Consider the figure (a), in which inductors L1 and L2 are connected in parallel with their magnetic fields aiding. The total current through the circuit is given as
i = i1 + i2
di/dt = (di1)/dt + (di2)/dt …………. (1)
The voltage across the inductor or parallel branch is given as
V = L1 (di1)/dt + M (di2)/dt or L2 (di2)/dt + M (di1)/dt

L1 (di1)/dt + M (di2)/dt = L2 (di2)/dt + M (di1)/dt

(di1)/dt (L1– M) = (di2)/dt (L2– M)
(di1)/dt = (di2)/dt ((L2 – M))/((L1 – M)) …………. (2)

Substituting equation 2 in equation 1, we get
di/dt = (di2)/dt ((L2– M))/((L1– M)) + (di2)/dt
di/dt = (di2)/dt { (L2– M))/((L1– M)) + 1} …………. (3)

If LT is the total inductance of the parallel inductor circuit , then voltage is given by
V = LT di/dt
LT di/dt = L1 (di1)/dt + M (di2)/dt
di/dt = 1/ LT { L1 (di1)/dt + M (di2)/dt }

Substituting equation 2 in above equation, we get
di/dt = 1/ LT { L1 (di2)/dt (L2– M))/((L1– M)) + M (di2)/dt }
di/dt = 1/ LT { L1 (L2– M))/((L1– M)) + M }(di2)/dt …………. (4)

Equating equations 3 and 4, and simplifying the results, we get
LT = (L1 L2– M2)/(L1+ L1 )-2M)

Here 2M represents the magnetic flux of L1 on L2 or L2 on L1.


Parallel Opposing Inductors

Similarly, if we consider the figure (b), in which inductors L1 & L2 are connected in parallel with their magnetic fields opposing, the total inductance is given as
LT = (L1 L2– M2)/(L1+ L2)+ 2M)

Summary

  • When two terminals of the  inductor are connected respectively to other inductor or inductors terminals, then that connection is referred to as “a parallel connection of inductors”.
  • When the fluxes produced by individual inductors are in the same direction, the mutual inductance will be increased; then these coils are called “Aiding” coils. Total inductance for aiding coils is LT = (L1 L2– M2)/(L1+ L2 )-2M).When the fluxes produced by individual inductors are in the opposite direction of magnetic flux, the mutual inductance will be decreased; then these coils are called “opposing” coils. Total inductance for aiding coils is LT = (L1 L2– M2)/(L1+ L2 )+2M)