Tuesday, October 17, 2017

Basic Electronics on the Go - Kirchhoffs Circuit Law

From http://www.electronics-tutorials.ws/dccircuits/dcp_4.html

Gustav Kirchhoff’s Current Law is one of the fundamental laws used for circuit analysis. His current law states that for a parallel path the total current entering a circuits junction is exactly equal to the total current leaving the same junction.


In other words the algebraic sum of ALL the currents entering and leaving a junction must be equal to zero as: Σ IIN = Σ IOUT.


This idea by Kirchhoff is commonly known as the Conservation of Charge, as the current is conserved around the junction with no loss of current.



Here, the 3 currents entering the node, I1, I2, I3 are all positive in value and the 2 currents leaving the node, I4 and I5 are negative in value. Then this means we can also rewrite the equation as;
I1 + I2 + I3 – I4 – I5 = 0





 The term Node in an electrical circuit generally refers to a connection or junction of two or more current carrying paths or elements such as cables and components. Also for current to flow either in or out of a node a closed circuit path must exist.


Kirchhoffs Second Law – The Voltage Law, (KVL)

Kirchhoffs Voltage Law or KVL, states that “in any closed loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop” which is also equal to zero. In other words the algebraic sum of all voltages within the loop must be equal to zero.


Kirchhoffs Voltage Law

 



  Starting at any point in the loop continue in the same direction noting the direction of all the voltage drops, either positive or negative, and returning back to the same starting point. It is important to maintain the same direction either clockwise or anti-clockwise or the final voltage sum will not be equal to zero. 


Common DC Circuit Theory Terms:

  • • Circuit – a circuit is a closed loop conducting path in which an electrical current flows.
  • • Path – a single line of connecting elements or sources.
  • • Node – a node is a junction, connection or terminal within a circuit were two or more circuit elements are connected or joined together giving a connection point between two or more branches. A node is indicated by a dot.
  • • Branch – a branch is a single or group of components such as resistors or a source which are connected between two nodes.
  • • Loop – a loop is a simple closed path in a circuit in which no circuit element or node is encountered more than once.
  • • Mesh – a mesh is a single open loop that does not have a closed path. There are no components inside a mesh.

A Typical DC Circuit

 

Kirchhoffs Circuit Law Example No1

Find the current flowing in the 40Ω Resistor, R3


 

The circuit has 3 branches, 2 nodes (A and B) and 2 independent loops.
Using Kirchhoffs Current Law, KCL the equations are given as;
At node A :    I1 + I2 = I3
At node B :    I3 = I1 + I2
Using Kirchhoffs Voltage Law, KVL the equations are given as;
Loop 1 is given as :    10 = R1 I1 + R3 I3 = 10I1 + 40I3
Loop 2 is given as :    20 = R2 I2 + R3 I3 = 20I2 + 40I3
Loop 3 is given as :    10 – 20 = 10I1 – 20I2
As I3 is the sum of I1 + I2 we can rewrite the equations as;
Eq. No 1 :    10 = 10I1 + 40(I1 + I2)  =  50I1 + 40I2
Eq. No 2 :    20 = 20I2 + 40(I1 + I2)  =  40I1 + 60I2
We now have two “Simultaneous Equations” that can be reduced to give us the values of I1 and I2 
Substitution of I1 in terms of I2 gives us the value of I1 as -0.143 Amps
Substitution of I2 in terms of I1 gives us the value of I2 as +0.429 Amps
As :    I3 = I1 + I2
The current flowing in resistor R3 is given as :    -0.143 + 0.429 = 0.286 Amps
and the voltage across the resistor R3 is given as :    0.286 x 40 = 11.44 volts
The negative sign for I1 means that the direction of current flow initially chosen was wrong, but never the less still valid. In fact, the 20v battery is charging the 10v battery.

Application of Kirchhoffs Circuit Laws

These two laws enable the Currents and Voltages in a circuit to be found, ie, the circuit is said to be “Analysed”, and the basic procedure for using Kirchhoff’s Circuit Laws is as follows:
  • 1. Assume all voltages and resistances are given. ( If not label them V1, V2,… R1, R2, etc. )
  • 2. Label each branch with a branch current. ( I1, I2, I3 etc. )
  • 3. Find Kirchhoff’s first law equations for each node.
  • 4. Find Kirchhoff’s second law equations for each of the independent loops of the circuit.
  • 5. Use Linear simultaneous equations as required to find the unknown currents.

  We can also use loop analysis to calculate the currents in each independent loop which helps to reduce the amount of mathematics required by using just Kirchhoff's laws. In the next tutorial about DC circuits, we will look at Mesh Current Analysis to do just that.


(to be updated)

Monday, October 16, 2017

Basic Electronics on the Go - Electrical Units of Measure

From http://www.electronics-tutorials.ws/dccircuits/dcp_3.html

 The following table gives a list of some of the standard electrical units of measure used in electrical formulas and component values.


Standard Electrical Units

 

Electrical
Parameter
Measuring
Unit
Symbol Description
Voltage Volt V or E Unit of Electrical Potential
V = I × R
Current Ampere I or i Unit of Electrical Current
I = V ÷ R
Resistance Ohm R or Ω Unit of DC Resistance
R = V ÷ I
Conductance Siemen G or ℧ Reciprocal of Resistance
G = 1 ÷ R
Capacitance Farad C Unit of Capacitance
C = Q ÷ V
Charge Coulomb Q Unit of Electrical Charge
Q = C × V
Inductance Henry L or H Unit of Inductance
VL = -L(di/dt)
Power Watts W Unit of Power
P = V × I  or  I2 × R
Impedance Ohm Z Unit of AC Resistance
Z2 = R2 + X2
Frequency Hertz Hz Unit of Frequency
ƒ = 1 ÷ T

 

Multiples and Sub-multiples

There is a huge range of values encountered in electrical and electronic engineering between a maximum value and a minimum value of a standard electrical unit. For example, resistance can be lower than 0.01Ω’s or higher than 1,000,000Ω’s. By using multiples and submultiple’s of the standard unit we can avoid having to write too many zero’s to define the position of the decimal point. The table below gives their names and abbreviations.


Prefix Symbol Multiplier Power of Ten
Terra T 1,000,000,000,000 1012
Giga G 1,000,000,000 109
Mega M 1,000,000 106
kilo k 1,000 103
none none 1 100
centi c 1/100 10-2
milli m 1/1,000 10-3
micro µ 1/1,000,000 10-6
nano n 1/1,000,000,000 10-9
pico p 1/1,000,000,000,000 10-12

 

So to display the units or multiples of units for either Resistance, Current or Voltage we would use as an example:
  • 1kV = 1 kilo-volt  –  which is equal to 1,000 Volts.
  • 1mA = 1 milli-amp  –  which is equal to one thousandths (1/1000) of an Ampere.

As well as the “Standard” electrical units of measure shown above, other units are also used in electrical engineering to denote other values and quantities such as:
  • •  Wh – The Watt-Hour, The amount of electrical energy consumed by a circuit over a period of time. Eg, a light bulb consumes one hundred watts of electrical power for one hour. It is commonly used in the form of: Wh (watt-hours), kWh (Kilowatt-hour) which is 1,000 watt-hours or MWh (Megawatt-hour) which is 1,000,000 watt-hours.
  • •  dB – The Decibel, The decibel is a one tenth unit of the Bel (symbol B) and is used to represent gain either in voltage, current or power. It is a logarithmic unit expressed in dB and is commonly used to represent the ratio of input to output in amplifier, audio circuits or loudspeaker systems.
    For example, the dB ratio of an input voltage (Vin) to an output voltage (Vout) is expressed as 20log10 (Vout/Vin). The value in dB can be either positive (20dB) representing gain or negative (-20dB) representing loss with unity, ie input = output expressed as 0dB.
  • •  ω – Angular Frequency, Another unit which is mainly used in a.c. circuits to represent the Phasor Relationship between two or more waveforms is called Angular Frequency, symbol ω. This is a rotational unit of angular frequency 2πƒ with units in radians per second, rads/s. The complete revolution of one cycle is 360 degrees or 2π, therefore, half a revolution is given as 180 degrees or π rad.
  • •  τ – Time Constant, The Time Constant of an impedance circuit or linear first-order system is the time it takes for the output to reach 63.7% of its maximum or minimum output value when subjected to a Step Response input. It is a measure of reaction time.






Saturday, October 14, 2017

Basic Electronics on the Go - Ohms Law and Power

 From http://www.electronics-tutorials.ws/dccircuits/dcp_2.html

 Georg Ohm found that, at a constant temperature, the electrical current flowing through a fixed linear resistance is directly proportional to the voltage applied across it, and also inversely proportional to the resistance. This relationship between the Voltage, Current and Resistance forms the basis of Ohms Law and is shown below.


Any Electrical device or component that obeys “Ohms Law” that is, the current flowing through it is proportional to the voltage across it ( I α V ), such as resistors or cables, are said to be “Ohmic” in nature, and devices that do not, such as transistors or diodes, are said to be “Non-ohmic” devices.


Electrical Power in Circuits

Electrical Power, ( P ) in a circuit is the rate at which energy is absorbed or produced within a circuit. A source of energy such as a voltage will produce or deliver power while the connected load absorbs it. Light bulbs and heaters for example, absorb electrical power and convert it into either heat, or light, or both. The higher their value or rating in watts the more electrical power they are likely to consume.
The quantity symbol for power is P and is the product of voltage multiplied by the current with the unit of measurement being the WattW ). Prefixes are used to denote the various multiples or sub-multiples of a watt, such as: milliwatts (mW = 10-3W) or kilowatts (kW = 103W).
Then by using Ohm’s law and substituting for the values of V, I and R the formula for electrical power can be found as:

To find the Power (P)

[ P = V x I ]      P (watts) = V (volts) x I (amps)
Also,
[ P = V2 ÷ R ]      P (watts) = V2 (volts) ÷ R (Ω)
Also,
[ P = I2 x R ]      P (watts) = I2 (amps) x R (Ω)


Electrical Power Rating

Electrical components are given a “power rating” in watts that indicates the maximum rate at which the component converts the electrical power into other forms of energy such as heat, light or motion. For example, a 1/4W resistor, a 100W light bulb etc.
Electrical devices convert one form of power into another. So for example, an electrical motor will covert electrical energy into a mechanical force, while an electrical generator converts mechanical force into electrical energy. A light bulb converts electrical energy into both light and heat.

Electrical Energy in Circuits

 Electrical Energy is the capacity to do work, and the unit of work or energy is the jouleJ ). Electrical energy is the product of power multiplied by the length of time it was consumed. Electrical power can be defined as the rate of doing work or the transferring of energy.

The maths involved when dealing with joules, kilojoules or megajoules to express electrical energy, can end up with some big numbers and lots of zero’s, so it is much more easier to express electrical energy consumed in Kilowatt-hours.

Kilowatt-hours are the standard units of energy used by the electricity meter in our homes to calculate the amount of electrical energy we use and therefore how much we pay.




Monday, October 9, 2017

Basic Electronics on the Go - DC Circuits - DC Circuit Theory

From http://www.electronics-tutorials.ws/dccircuits/dcp_1.html



All materials are made up from atoms, and all atoms consist of protons, neutrons and electrons. Protons, have a positive electrical charge. Neutrons have no electrical charge while Electrons, have a negative electrical charge. Atoms are bound together by powerful forces of attraction existing between the atoms nucleus and the electrons in its outer shell.


When these protons, neutrons and electrons are together within the atom they are stable. But if we separate them from each other they want to reform and start to exert a potential of attraction called a potential difference.
If we create a closed circuit these loose electrons will start to move and drift back to the protons due to their attraction creating a flow of electrons. This flow of electrons is called an electrical current.
  The electrons do not flow freely through the circuit as the material they move through creates a restriction to the electron flow. This restriction is called resistance.

  All basic electrical or electronic circuits consist of three separate but very much related electrical quantities called: Voltage, ( v ), Current, ( i ) and Resistance, ( Ω ).

Electrical Voltage

Voltage, ( V ) is the potential energy of an electrical supply stored in the form of an electrical charge. Voltage can be thought of as the force that pushes electrons through a conductor and the greater the voltage the greater is its ability to “push” the electrons through a given circuit. As energy has the ability to do work, this potential energy can be described as the work required in joules to move electrons in the form of an electrical current around a circuit from one point or node to another.

 The difference in voltage between any two points, connections or junctions (called nodes) in a circuit is known as the Potential Difference, ( p.d. ) commonly called the Voltage Drop. The Potential difference between two points is measured in Volts with the circuit symbol V, or lowercase “v“.  The greater the voltage, the greater is the pressure (or pushing force) and the greater is the capacity to do work.

A constant voltage source is called a DC Voltage while a voltage that varies periodically with time is called an AC voltage. Voltage is measured in volts, with one volt being defined as the electrical pressure required to force an electrical current of one ampere through a resistance of one Ohm. Voltages are generally expressed in Volts with prefixes used to denote sub-multiples of the voltage such as microvolts ( μV = 10-6 V ), millivolts ( mV = 10-3 V ) or kilovolts ( kV = 103 V ). Voltage can be either positive or negative.

 Batteries or power supplies are mostly used to produce a steady D.C. (direct current) voltage source such as 5v, 12v, 24v etc in electronic circuits and systems. A.C. (alternating current) voltage sources are for domestic houses, industrial power and lighting as well as power transmission. The mains voltage supply in the United Kingdom is currently 230 volts a.c. and 110 volts a.c. in the USA.

 General electronic circuits operate on low voltage DC battery supplies of between 1.5V and 24V dc The circuit symbol for a constant voltage source usually given as a battery symbol with a positive, + and negative, sign indicating the direction of the polarity. The circuit symbol for an alternating voltage source is a circle with a sine wave inside.

Electronic circuits operate on  DC battery supplies of between 1.5V and 24V dc. The circuit symbol for a constant voltage source is  usually given as a battery symbol with a positive, + and negative, sign indicating the direction of the polarity. The circuit symbol for an alternating voltage source is a circle with a sine wave inside.


Voltage Symbols

 



 Voltage is always measured as the difference between any two points in a circuit and the voltage between these two points is generally referred to as the “Voltage drop“. Note that voltage can exist across a circuit without current, but current cannot exist without voltage and as such any voltage source whether DC or AC likes an open or semi-open circuit condition but hates any short circuit condition as this can destroy it.


Electrical Current

Electrical Current, ( I ) is the movement or flow of electrical charge and is measured in Amperes, symbol i, for intensity). It is the continuous and uniform flow (called a drift) of electrons (the negative particles of an atom) around a circuit that are being “pushed” by the voltage source. In reality, electrons flow from the negative (-ve) terminal to the positive (+ve) terminal of the supply and for ease of circuit understanding conventional current flow assumes that the current flows from the positive to the negative terminal.

Generally in circuit diagrams the flow of current through the circuit usually has an arrow associated with the symbol, I, or lowercase i to indicate the actual direction of the current flow. However, this arrow usually indicates the direction of conventional current flow and not necessarily the direction of the actual flow.

Conventional Current Flow

 

Conventionally this is the flow of positive charge around a circuit, being positive to negative. The diagram at the left shows the movement of the positive charge (holes) around a closed circuit flowing from the positive terminal of the battery, through the circuit and returns to the negative terminal of the battery. This flow of current from positive to negative is generally known as conventional current flow.

 This was the convention chosen during the discovery of electricity in which the direction of electric current was thought to flow in a circuit. To continue with this line of thought, in all circuit diagrams and schematics, the arrows shown on symbols for components such as diodes and transistors point in the direction of conventional current flow.

In electronic circuits, a current source is a circuit element that provides a specified amount of current for example, 1A, 5A 10 Amps etc, with the circuit symbol for a constant current source given as a circle with an arrow inside indicating its direction.

Current is measured in Amps and an amp or ampere is defined as the number of electrons or charge (Q in Coulombs) passing a certain point in the circuit in one second, (t in Seconds).
Electrical current is generally expressed in Amps with prefixes used to denote micro ampsμA = 10-6A ) or milliampsmA = 10-3A ). Note that electrical current can be either positive in value or negative in value depending upon its direction of flow.

 Current that flows in a single direction is called Direct Current, or D.C. and current that alternates back and forth through the circuit is known as Alternating Current, or A.C..  AC or DC current only flows through a circuit when a voltage source is connected to it with its “flow” being limited to both the resistance of the circuit and the voltage source pushing it.

 Even though alternating currents (and voltages) are periodic and vary with time the “effective” or “RMS”, (Root Mean Squared) value given as Irms produces the same average power loss equivalent to a DC current Iaverage. Current sources are the opposite to voltage sources in that they like short or closed circuit conditions but hate open circuit conditions as no current will flow.

Resistance

Resistance, ( R ) is the capacity of a material to resist or prevent the flow of current or, more specifically, the flow of electric charge within a circuit. The circuit element which does this perfectly is called the “Resistor”.

 Resistance is a circuit element measured in Ohms, Greek symbol ( Ω, Omega ) with prefixes used to denote Kilo-ohmskΩ = 103Ω ) and Mega-ohmsMΩ = 106Ω ). Note that resistance cannot be negative in value only positive.

Resistor Symbols

 

The amount of resistance a resistor has is determined by the relationship of the current through it to the voltage across it which determines whether the circuit element is a “good conductor” – low resistance, or a “bad conductor” – high resistance. Low resistance, for example 1Ω or less implies that the circuit is a good conductor made from materials such as copper, aluminium or carbon while a high resistance, 1MΩ or more implies the circuit is a bad conductor made from insulating materials such as glass, porcelain or plastic.
A “semiconductor” on the other hand such as silicon or germanium, is a material whose resistance is half way between that of a good conductor and a good insulator. Hence the name “semi-conductor”. Semiconductors are used to make Diodes and Transistors etc.
Resistance can be linear or non-linear in nature. Linear resistance obeys Ohm’s Law as the voltage across the resistor is linearly proportional to the current through it. Non-linear resistance, does not obey Ohm’s Law but has a voltage drop across it that is proportional to some power of the current.
Resistance is pure and is not affected by frequency with the AC impedance of a resistance being equal to its DC resistance and as a result can not be negative. Remember that resistance is always positive, and never negative.
A resistor is classed as a passive circuit element and as such cannot deliver power or store energy. Instead resistors absorb power that appears as heat and light. Power in a resistance is always positive regardless of voltage polarity and current direction.
For very low values of resistance, for example milli-ohms, ( mΩ´s ) it is sometimes much easier to use the reciprocal of resistance ( 1/R ) rather than resistance ( R ) itself. The reciprocal of resistance is called Conductance, symbol ( G ) and represents the ability of a conductor or device to conduct electricity.
 High values of conductance implies a good conductor such as copper while low values of conductance implies a bad conductor such as wood. The standard unit of measurement given for conductance is the Siemen, symbol (S).
The unit used for conductance is mho (ohm spelled backward), which is symbolized by an inverted Ohm sign . Power can also be expressed using conductance as: p = i2/G = v2G.

 The relationship between Voltage, ( v ) and Current, ( i ) in a circuit of constant Resistance, ( R ) would produce a straight line i-v relationship with slope equal to the value of the resistance as shown.




Voltage, Current and Resistance Summary

 The relationship between Voltage, Current and Resistance forms the basis of Ohm’s law. In a linear circuit of fixed resistance, if we increase the voltage, the current goes up, and similarly, if we decrease the voltage, the current goes down. This means that if the voltage is high the current is high, and if the voltage is low the current is low.
 
Likewise, if we increase the resistance, the current goes down for a given voltage and if we decrease the resistance the current goes up. Which means that if resistance is high current is low and if resistance is low current is high.

Then we can see that current flow around a circuit is directly proportional (  ) to voltage, ( V↑ causes I↑ ) but inversely proportional ( 1/∝ ) to resistance as, ( R↑ causes I↓ ).
A basic summary of the three units is given below.
  • Voltage or potential difference is the measure of potential energy between two points in a circuit and is commonly referred to as its ” volt drop “.
  • When a voltage source is connected to a closed loop circuit the voltage will produce a current flowing around the circuit.
  • In DC voltage sources the symbols +ve (positive) and -ve (negative) are used to denote the polarity of the voltage supply.
  • Voltage is measured in ” Volts ” and has the symbol ” V ” for voltage or ” E ” for energy.
  • Current flow is a combination of electron flow and hole flow through a circuit.
  • Current is the continuous and uniform flow of charge around the circuit and is measured in ” Amperes ” or ” Amps ” and has the symbol ” I “.
  • Current is Directly Proportional to Voltage ( I ∝ V )
  • The effective (rms) value of an alternating current has the same average power loss equivalent to a direct current flowing through a resistive element.
  • Resistance is the opposition to current flowing around a circuit.
  • Low values of resistance implies a conductor and high values of resistance implies an insulator.
  • Current is Inversely Proportional to Resistance ( I 1/∝ R )
  • Resistance is measured in ” Ohms ” and has the Greek symbol ” Ω ” or the letter ” R “.

Quantity Symbol Unit of Measure Abbreviation
Voltage V or E Volt V
Current I Ampere A
Resistance R Ohms Ω




Tuesday, September 26, 2017

Basic Electronics on the Go - Butterworth Filter Design

From http://www.electronics-tutorials.ws/filter/filter_8.html

"High-order” or “nth-order” filters may be needed in communication or control circuits that require a larger roll off.

  A first-order filter has a roll-off rate of 20dB/decade (6dB/octave), a second-order filter has a roll-off rate of 40dB/decade (12dB/octave), and a fourth-order filter has a roll-off rate of 80dB/decade (24dB/octave),  etc.

 High-order filters  are usually formed by cascading together single first-order and second-order filters. As the order increases so does its size and cost but  its accuracy declines.



Decades and Octaves

On the frequency scale, a Decade is a tenfold increase (multiply by 10) or tenfold decrease (divide by 10).  For example, 2 to 20Hz represents one decade, whereas 50 to 5000Hz represents two decades (50 to 500Hz and then 500 to 5000Hz).

An Octave is a doubling (multiply by 2) or halving (divide by 2) of the frequency scale. For example, 10 to 20Hz represents one octave, while 2 to 16Hz is three octaves (2 to 4, 4 to 8 and finally 8 to 16Hz) . Either way, Logarithmic scales are used extensively in the frequency domain to denote a frequency value when working with amplifiers and filters .





Since the frequency determining resistors  and   capacitors are all equal, the cut-off or corner frequency ( ƒC ) for either a first, second, third or even a fourth-order filter must also be equal and is found by using our now old familiar equation:




Filter Approximations

So far we have looked at a low and high pass first-order filter circuits, their resultant frequency and phase responses. An ideal filter would give us specifications of maximum pass band gain and flatness and a sharp transition from the passband to stopband with steep roll-off.

There are a number of “approximation functions” in linear analogue filter design that use a mathematical approach to best approximate the transfer function we require for the filters design. 

Elliptical, Butterworth, Chebyshev, Bessel, Cauer are some of them but the low pass Butterworth filter design will be considered here as it is the most commonly used function.

Low Pass Butterworth Filter Design

The frequency response of the Butterworth Filter approximation function is referred to as  a “maximally flat” (no ripples) response because the pass band is designed to have a frequency response which is as flat as mathematically possible from 0Hz (DC) until the cut-off frequency at -3dB with no ripples. Higher frequencies beyond the cut-off point rolls-off down to zero in the stop band at 20dB/decade or 6dB/octave due to a “quality factor”, “Q” of just 0.707.

 However, one main disadvantage of the Butterworth filter is that it achieves this pass band flatness at the expense of a wide transition band as the filter changes from the pass band to the stop band. It also has poor phase characteristics as well. 


Ideal Frequency Response for a Butterworth Filter

 

Note that the higher the Butterworth filter order, the higher the number of cascaded stages there are within the filter design, and the closer the filter becomes, to the ideal “brick wall” response.
In practice however, Butterworth’s ideal frequency response is unattainable as it produces excessive passband ripple.

The frequency response for the generalised equation representing a “nth” Order Butterworth filter is given as:


Where: n represents the filter order, Omega ω is equal to 2πƒ and Epsilon ε is the maximum pass band gain, (Amax). If Amax is defined at a frequency equal to the cut-off -3dB corner point (ƒc), ε will then be equal to one and therefore ε2 will also be one.

However, if you now wish to define Amax at a different voltage gain value, for example 1dB, or 1.1220 (1dB = 20logAmax) then the new value of epsilon, ε is found by:

  • Where:
  •   H0 = the Maximum Pass band Gain, Amax.
  •   H1 = the Minimum Pass band Gain

Transpose the equation to give:

  •  
  • Where:
  •   Vout = the output signal voltage.
  •   Vin  = the input signal voltage.
  •      j   = to the square root of -1 (√-1)
  •     ω  = the radian frequency (2πƒ)

Note: ( jω ) can also be written as ( s ) to denote the S-domain. and the resultant transfer function for a second-order low pass filter is given as:


Normalised Low Pass Butterworth Filter Polynomials

To help in the design of his low pass filters, Butterworth produced standard tables of normalised second-order low pass polynomials given the values of coefficient that correspond to a cut-off corner frequency of 1 radian/sec.


n Normalised Denominator Polynomials in Factored Form
1 (1+s)
2 (1+1.414s+s2)
3 (1+s)(1+s+s2)
4 (1+0.765s+s2)(1+1.848s+s2)
5 (1+s)(1+0.618s+s2)(1+1.618s+s2)
6 (1+0.518s+s2)(1+1.414s+s2)(1+1.932s+s2)
7 (1+s)(1+0.445s+s2)(1+1.247s+s2)(1+1.802s+s2)
8 (1+0.390s+s2)(1+1.111s+s2)(1+1.663s+s2)(1+1.962s+s2)
9 (1+s)(1+0.347s+s2)(1+s+s2)(1+1.532s+s2)(1+1.879s+s2)
10 (1+0.313s+s2)(1+0.908s+s2)(1+1.414s+s2)(1+1.782s+s2)(1+1.975s+s2)

Saturday, July 29, 2017

Basic Electronics on the Go - Active Band Pass Filter

From http://www.electronicshub.org/active-band-pass-filter/
From http://www.electronics-tutorials.ws/filter/filter_7.html

Introduction

A Band Pass Filter is a circuit which allows only particular band of frequencies to pass through it. This Pass band is mainly between the cut-off frequencies and they are fand fH.  Where fL is the lower cut-off frequency and fH is higher cut-off frequency. The centre frequency is denoted by ‘fC’ and it is also called as resonant frequency or peak frequency.
The fL value must always be less than the value of fH. The pass band of the filter is nothing but the bandwidth. The gain of the filter is maximum at resonant or centre frequency and this is referred as total pass band gain. This pass band gain is denoted by ‘Amax’.

For low pass filter this pass band starts from 0 Hz and continues until it reaches the resonant frequency value at -3 dB down from a maximum pass band gain.
Where as in the case of high pass filter this pass band begins from the -3 dB resonant frequency and ends at the value of the maximum loop gain for active filter. Combination of low pass and high pass responses gives us band pass response as shown below:

Active Band Pass Filter


Depending on the quality factor the band pass filter is classified into Wide band pass filter and Narrow band pass filter. The quality factor is also referred as ‘figure of merit’. By cascading High Pass Filter and Low Pass Filter with an amplifying component we obtain band pass filter.

The cut-off or corner frequency of the low pass filter (LPF) is higher than the cut-off frequency of the high pass filter (HPF) and the difference between the frequencies at the -3dB point will determine the “bandwidth” of the band pass filter while attenuating any signals outside of these points. Obviously, a reasonable separation is required between the two cut-off points to prevent any interaction between the low pass and high pass stages. The amplifier circuit between these high pass and low pass filter will provide isolation and  voltage gain for the circuit.

The circuit diagram for active band pass filter is shown below:





Wide Band Pass Filter


If the value of quality factor is less than ten, then the pass band is wide, which gives us the larger bandwidth. This band pass filter is called Wide Band Pass Filter. In this filter the high cut-off frequency must be greater than the lower cut-off frequency. It uses two amplifying elements (Op-amps) in design.
First the signal will pass through the high pass filter, the output signal of this high pass filter will tends to infinity and thus the signal which tends to infinity is given to the low pass filter at the end. 

By cascading one first order low pass and high pass gives us the second order band pass filter and by cascading two first order low pass filters with two high pass filters forms a fourth order band pass filter. Due to this cascading the circuit produces a low value quality factor. The capacitor in the first order high pass filter will block any DC biasing from the input signal.

The gain rolls off at both the stop bands is ± 20 dB /decade in the case of second order filter (High + Low). The high Pass and low pass filters must be in first order only. Similarly, when the high pass and low pass filters are at second order, then the gain roll off at both the stop bands is ± 40dB/Decade.

The voltage gain expression for band pass filter is given as:
| Vout / Vin | = [Amax * (f/fL)] / √{[1+(f/fL)²][1+(f/fH)²]}
It is obtained by the individual gains of both high pass and low pass filters, the individual gains of both high pass and low pass filter are given below.
Voltage Gain for High Pass filter:
| Vout / Vin | = [Amax1 * (f/fL)] / √[1+(f/fL)²]
Voltage Gain for Low pass filter:
| Vout / Vin | = Amax2 / √[1+(f/fH)²]

Amax = Amax1 * Amax2







Where Amax1 is the gain of the high pass stage and Amax2 is the gain of the low pass stage.

The response of the wide band filter is shown below

Narrow Band Pass Filter


If the value of quality factor is greater than ten then the pass band is narrow and bandwidth of the pass band is also less. This band pass filter is called as Narrow Band Pass Filter. It uses only one active component (op-amp) rather than two and this op-amp is in inverting configuration. In this filter the gain of the op-amp is maximum at centre frequency fc.



Narrow Band Pass Filter Circuit


The input is applied to the inverting input terminal. This shows that the Op-amp is in inverting configuration. This filter circuit produces narrow band pass filter response.

The voltage gain of the above filter circuit is AV = – R2 / R1 
The cut-off frequencies of the filter circuit are

fC1 = 1 / (2πR1C1) and fC2 = 1 / (2πR2C2)


Multi Feedback Active Band Pass Filter

This filter circuit produces a tuned circuit based on the negative feedback of the filter. The important advantage of this multiple feedback is that without any change in the maximum gain at the centre frequency we can change the value of the cut-off frequency. This change in the cut-off frequency can be done by the resistance ‘R3’.
By considering the  active filter circuit below, let us consider the changed resistor value as R3′and the changed cut-off frequency value as fc′, then we can equate for the new resistor value as follows:

R3′ = R3(fc /fc′)²





t consists of two feedback paths, because of this multiple feedback paths it is also referred as ‘Multiple feedback band pass circuit’. This circuit produce an infinity gain multiple feedback band pass filter. Due to this circuit the quality factor value increases maximum up to 20.
fc = 1/√(R1R2C1C2)
Q = fc/Bandwidth = (½){√[R2/R1]}
Amax = -R2/2R1
R1 = Q/{2πfcCAmax}
R2 = Q/πfcC

R3 = Q/{2πfcC(2Q² – Amax)}

The gain at the centre frequency ‘Amax’ must be less than 2Q². That is,
Amax < 2Q²
Where,

fc = cut off frequency in Hz
C = Capacitance, (C1 = C2 = C)
Q = Quality factor
Amax = Maximum gain



The Frequency Response of Active Band Filter

 

It has two centre frequencies, one is from the  high pass filter and the other is from the  low pass filter. The centre frequency of the high pass filter must be lower than the centre frequency of the low pass filter. The centre frequency of band pass filter is the geometric mean of lower and upper cut-off frequencies fr2 = fH * fL. 

The gain of the filter is 20 log (Vout/Vin) dB/Decade. The amplitude response is similar to the responses of the low pass and high pass filter. Let us consider two cut-off frequencies as 300 Hz and 900Hz, then, the bandwidth of the filter is 300 Hz -900 Hz = 600 Hz.



The Quality Factor

The quality factor depends on the bandwidth of the pass band. Quality factor is inversely proportional to the Bandwidth. That means if band width increases the quality factor decrease and if band width decreases the quality factor increases.
Q = fc/Bandwidth

For wide band pass filter the quality factor is low because the pass band width is high. For the narrow band pass filter the quality factor is high. Selectivity and un-selectivity depends on the width of the pass band. This quality factor is also relates to the damping factor. If damping co-efficient value is more, the flatness of the output response is also more. This is equated as follows:
ε = 2/Q


 For different quality factor values the normalized gain response of a second order band pass filter is given as:


By this graph, it is clear that the selectivity is more for higher quality factor.

Friday, July 7, 2017

Basic Electronics on the Go - Active High Pass Filter

From http://www.electronicshub.org/active-high-pass-filter/
From http://www.electronics-tutorials.ws/filter/filter_6.html



Introduction

A high pass filter will allow the frequencies which are higher than the cut-off frequency and attenuate the frequencies lower than the cut off frequency. In some cases this filter is also termed as ‘Low-Cut’ filter or ‘Base-cut’ filter. The amount of attenuation or the pass band range will depend on the designing parameters of the filter.
The pass band gain of an active filter is more than unity gain. The operation of the active high pass filter is same as passive high pass filter, but the main difference is that the active high pass filter uses an  operational  amplifier which provides amplification of the output signals and controls gain.

The ideal characteristics of the high pass filter are shown below



Technically, there is no such thing as an active high pass filter. Unlike Passive High Pass Filters which have an “infinite” frequency response, the maximum pass band frequency response of an active high pass filter is limited by the open-loop characteristics or bandwidth of the operational amplifier being used, making them appear as if they are band pass filters with a high frequency cut-off determined by the selection of op-amp and gain.


Active High Pass Filter

By connecting a passive RC high pass filter circuit to the inverting or non-inverting terminal of the op-amp gives us first order active high pass filter. The passive RC high pass filter circuit connected to the non-inverting terminal of the unity gain operational amplifier is shown below.



The gain Amax = 1 and cutoff frequency fc = 1/2πRC


Active High Pass Filter With High Voltage Gain

The operation is the same as that of the passive high pass filter, but the input signal is amplified by the amplifier at the output. The amount of amplification depends on the gain of the amplifier. The magnitude of the pass band gain is equal to 1 + (R3/R2). Where R3 is the feedback resistor in Ω (ohms) and R2 is the input resistor. The circuit of  an active high pass filter with amplification is given below

Voltage Gain Of An Active High Pass Filter

Voltage Gain Av = Amax (f/fc) / √{1 + (f/fc)²}
Where f = operating frequency
fc = cut-off frequency
Amax = pass band gain of the filter = 1 + (R3/R2)
At low frequencies means when the operating frequency is less than the cut-off frequency, the voltage gain is less than the pass band gain Amax. At high frequencies means when the operating frequency is greater than the cut-off frequency, the voltage gain of the filter is equal to pass band gain. If operating frequency is equal to the cut-off frequency,then the voltage gain of the filter is equal to 0.707 Amax.


Voltage Gain in (dB)

The magnitude of the  voltage gain is generally taken in decibels (dB):
Av(dB) = 20 log10 (Vout/Vin)
-3 dB = 20 log10 (0.707 * Vout/Vin)
The cut-off frequency which separates both pass band and stop band can be calculated using the  formula
fc = 1 / (2πRC)
The phase shift of the active high pass filter is equal to that of the passive filter. It is equal to +45° at the cut-off frequency fc and this phase shift value is equated as
Ø = tan-1(1/2πfcRC)



Frequency Response Of Active High Pass Filter


The frequency response curve with respect to the amplifiers open loop gain is shown below.

In frequency response of the active high pass filter the maximum pass band frequency is limited by the bandwidth or the open loop characteristics of the operational amplifier. Due to this limitation the active high pass filter response will appears like a wide band filter response. By using this op-amp based active high pass filter we can achieve high accuracy with the use of low tolerance resistors and capacitors.



Active High Pass Filter using Inverting Operational Amplifier


We know that the active high pass filter can be designed by using either inverting terminal or the non-inverting terminal of an operational amplifier. Till now we saw the high pass filter circuit and response curves of the non-inverting active high pass filter. Now let us see the active high pass filter using inverting op-amp.

Gain derivation in Laplace form

Let us consider the inverting amplifier as shown below

The input impedance Z1 = 1/sC1
Where s = Laplace Variable
C1 = Capacitance

The currents flowing in the circuit are I1, I2 and Iin,
Where I1 = I2 and Iin = 0
Vin / Z1 = -Vout / R1
Vout / Vin = – R1 / Z1
Vout / Vin = – R1 / (1/sC1)
Vout / Vin = -sR1C1 = Gain


Bode-plot

To analyse the circuit frequency response this bode plot is used. It is nothing but a graph of the transfer function of linear, time variant versus frequency. This is plotted with the log frequency axis. It consists of mainly two plots; one is magnitude plot and the other is phase plot.
The magnitude plot will express the magnitude of the frequency response i.e., gain and the phase plot is used to express the response of the frequency shift.

The frequency response bode-plot according to the values which are tabulated above is given below:


According to the values calculated, at frequency 10 Hz the gain of the filter obtained in dB is -56.48. If we increase the value of frequency to 100 Hz the obtained gain is -36.48 dB and at frequency 500 Hz the gain of the filter is -22.51 dB, at frequency 1000 Hz gain in dB is -16.52.By this we can say that if frequency increases the gain of the filter increases at the rate of 20dB/decade.

Till the cut-off frequency 10 KHz the gain of the filter increases but after the cut-off frequency the gain reaches maximum value and it is constant.


Second Order High Pass Filter

Second order active high pass filter frequency response is exactly opposite to the second order active low pass filter response because this filter will attenuate the voltages below the cut-off frequency. The transfer function of the second order filter is given below
Vout(s) / Vin(s) = -Ks² / s² + (ω0/Q)s + ω0²
Where K = R1/R2 and ω= 1/CR
This is the general form of the second order high pass filter.


Second Order Active High Pass Filter Circuit

The designing procedure for the second order active filter is same as that of the first order filter because the only variation is in the roll-off. If the roll-off of the first order active high pass filter is 20dB/decade, then roll-off of the second order filter is 40 dB/ decade. It means the twice of the value of the first order filter. The circuit of second order filter is shown below
The gain of the filter is 1+ R1/R2 and the equation of the cut-off frequency is fc = 1/ 2π√R3R4C1C2

Higher Order High Pass Filters

By cascading first order filter with second order filter , we can obtain the third order filter. When we cascade two second order filters we can get the fourth order filter. Like this with the help of first order and second order filters we get the higher order filters.
With the increase in the order of the filter,the difference between actual stop band and theoretical stop band increases. But the overall gain of the higher order filter is equal because we already saw that the resistors and capacitors which determine the frequency response values will be the same.


Applications of active high pass filters

  • These are used in the loud speakers to reduce the low level noise.
  • Eliminates rumble distortions in audio applications so these are also called are treble boost filters.
  • These are used in audio amplifiers to amplify the higher frequency signals.
  • These are also used in equalisers.