Category: electronic Measuring Instruments

Electronic Measuring Instruments – Errors The errors, which occur during measurement are known as measurement errors. In this chapter, let us discuss about the types of measurement errors. Types of Measurement Errors We can classify the measurement errors into the following three types. Gross Errors Random Errors Systematic Errors Now, let us discuss about these three types of measurement errors one by one. Gross Errors The errors, which occur due to the lack of experience of the observer while taking the measurement values are known as gross errors. The values of gross errors will vary from observer to observer. Sometimes, the gross errors may also occur due to improper selection of the instrument. We can minimize the gross errors by following these two steps. Choose the best suitable instrument, based on the range of values to be measured. Note down the readings carefully Systematic Errors If the instrument produces an error, which is of a constant uniform deviation during its operation is known as systematic error. The systematic errors occur due to the characteristics of the materials used in the instrument. Types of Systematic Errors The systematic errors can be classified into the following three types. Instrumental Errors − This type of errors occur due to shortcomings of instruments and loading effects. Environmental Errors − This type of errors occur due to the changes in environment such as change in temperature, pressure & etc. observational Errors − This type of errors occur due to observer while taking the meter readings. Parallax errors belong to this type of errors. Random Errors The errors, which occur due to unknown sources during measurement time are known as random errors. Hence, it is not possible to eliminate or minimize these errors. But, if we want to get the more accurate measurement values without any random error, then it is possible by following these two steps. Step1 − Take more number of readings by different observers. Step2 − Do statistical analysis on the readings obtained in Step1. Following are the parameters that are used in statistical analysis. Mean Median Variance Deviation Standard Deviation Now, let us discuss about these statistical parameters. Mean Let $x_{1},x_{2},x_{3},….,x_{N}$ are the $N$ readings of a particular measurement. The mean or average value of these readings can be calculated by using the following formula. $$m = frac{x_{1}+x_{2}+x_{3}+….+x_{N}}{N}$$ Where, $m$ is the mean or average value. If the number of readings of a particular measurement are more, then the mean or average value will be approximately equal to true value Median If the number of readings of a particular measurement are more, then it is difficult to calculate the mean or average value. Here, calculate the median value and it will be approximately equal to mean value. For calculating median value, first we have to arrange the readings of a particular measurement in an ascending order. We can calculate the median value by using the following formula, when the number of readings is an odd number. $$M=x_{left ( frac{N+1}{2} right )}$$ We can calculate the median value by using the following formula, when the number of readings is an even number. $$M=frac{x_{left ( N/2 right )}+x_left ( left [ N/2 right ]+1 right )}{2}$$ Deviation from Mean The difference between the reading of a particular measurement and the mean value is known as deviation from mean. In short, it is called deviation. Mathematically, it can be represented as $$d_{i}=x_{i}-m$$ Where, $d_{i}$ is the deviation of $i^{th}$ reading from mean. $x_{i}$ is the value of $i^{th}$ reading. $m$ is the mean or average value. Standard Deviation The root mean square of deviation is called standard deviation. Mathematically, it can be represented as $$sigma =sqrt{frac{{d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+….+{d_{N}}^{2}}{N}}$$ The above formula is valid if the number of readings, N is greater than or equal to 20. We can use the following formula for standard deviation, when the number of readings, N is less than 20. $$sigma =sqrt{frac{{d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+….+{d_{N}}^{2}}{N-1}}$$ Where, $sigma$ is the standard deviation $d_{1}, d_{2}, d_{3}, …, d_{N}$ are the deviations of first, second, third, …, $N^{th}$readings from mean respectively. Note − If the value of standard deviation is small, then there will be more accuracy in the reading values of measurement. Variance The square of standard deviation is called variance. Mathematically, it can be represented as $$V=sigma^{2}$$ Where, $V$ is the variance $sigma$ is the standard deviation The mean square of deviation is also called variance. Mathematically, it can be represented as $$V=frac{{d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+….+{d_{N}}^{2}}{N}$$ The above formula is valid if the number of readings, N is greater than or equal to 20. We can use the following formula for variance when the number of readings, N is less than 20. $$V=frac{{d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+….+{d_{N}}^{2}}{N-1}$$ Where, $V$ is the variance $d_{1}, d_{2}, d_{3}, …, d_{N}$ are the deviations of first, second, third, …, $N^{th}$ readings from mean respectively. So, with the help of statistical parameters, we can analyze the readings of a particular measurement. In this way, we will get more accurate measurement values. Learning working make money

Introduction The instruments, which are used to measure any quantity are known as measuring instruments. This tutorial covers mainly the electronic instruments, which are useful for measuring either electrical quantities or parameters. Following are the most commonly used electronic instruments. Voltmeter Ammeter Ohmmeter Multimeter Now, let us discuss about these instruments briefly. Voltmeter As the name suggests, voltmeter is a measuring instrument which measures the voltage across any two points of an electric circuit. There are two types of voltmeters: DC voltmeter, and AC voltmeter. DC voltmeter measures the DC voltage across any two points of an electric circuit, whereas AC voltmeter measures the AC voltage across any two points of an electric circuit. An example of practical DC voltmeter is shown in below figure. The DC voltmeter shown in above figure is a $(0-100)V$ DC voltmeter. Hence, it can be used to measure the DC voltages from zero volts to 10 volts. Ammeter As the name suggests, ammeter is a measuring instrument which measures the current flowing through any two points of an electric circuit. There are two types of ammeters: DC ammeter, and AC ammeter. DC ammeter measures the DC current that flows through any two points of an electric circuit. Whereas, AC ammeter measures the AC current that flows through any two points of an electric circuit. An example of practical AC ammeter is shown in below figure − The AC ammeter shown in above figure is a $(0-100)A :$ AC ammeter. Hence, it can be used to measure the AC currents from zero Amperes to 100 Amperes. Ohmmeter Ohmmeter is used to measure the value of resistance between any two points of an electric circuit. It can also be used for finding the value of an unknown resistor. There are two types of ohmmeters: series ohmmeter, and shunt ohmmeter. In series type ohmmeter, the resistor whose value is unknown and to be measured should be connected in series with the ohmmeter. It is useful for measuring high values of resistances. In shunt type ohmmeter, the resistor whose value is unknown and to be measured should be connected in parallel (shunt) with the ohmmeter. It is useful for measuring low values of resistances. An example of practical shunt ohmmeter is shown in the above figure. The ohmmeter shown in above figure is a $(0-100)Omega$ shunt ohmmeter. Hence, it can be used to measure the resistance values from zero ohms to 100 ohms. Multimeter Multimeter is an electronic instrument used to measure the quantities such as voltage, current & resistance one at a time. It can be used to measure DC & AC voltages, DC & AC currents and resistances of several ranges. A practical multimeter is shown in the following figure − As shown in the figure, this multimeter can be used to measure various high resistances, low resistances, DC voltages, AC voltages, DC currents, & AC currents. Different scales and range of values for each of these quantities are marked in above figure. The instruments which we considered in this chapter are of indicating type instruments, as the pointers of these instruments deflect and point to a particular value. We will discuss about these electronic measuring instruments in detail in later chapters. Learning working make money

Passive Transducers passive transducer is a transducer, which produces the variation in passive element. We will consider the passive elements like resistor, inductor and capacitor. Hence, we will get the following three passive transducers depending on the passive element that we choose. Resistive Transducer Inductive Transducer Capacitive Transducer Now, let us discuss about these three passive transducers one by one. Resistive Transducer A passive transducer is said to be a resistive transducer, when it produces the variation (change) in resistance value. the following formula for resistance, R of a metal conductor. $$R=frac{rho :l}{A}$$ Where, $rho$ is the resistivity of conductor $l$ is the length of conductor $A$ is the cross sectional area of the conductor The resistance value depends on the three parameters $rho, l$ & $A$. So, we can make the resistive transducers based on the variation in one of the three parameters $rho, l$ & $A$. The variation in any one of those three parameters changes the resistance value. Resistance, R is directly proportional to the resistivity of conductor, $rho$. So, as resistivity of conductor, $rho$ increases the value of resistance, R also increases. Similarly, as resistivity of conductor, $rho$ decreases the value of resistance, R also decreases. Resistance, R is directly proportional to the length of conductor, $l$. So, as length of conductor, $l$ increases the value of resistance, R also increases. Similarly, as length of conductor, $l$ decreases the value of resistance, R also decreases. Resistance, R is inversely proportional to the cross sectional area of the conductor, $A$. So, as cross sectional area of the conductor, $A$ increases the value of resistance, R decreases. Similarly, as cross sectional area of the conductor, $A$ decreases the value of resistance, R increases. Inductive Transducer A passive transducer is said to be an inductive transducer, when it produces the variation (change) in inductance value. the following formula for inductance, L of an inductor. $L=frac{N^{2}}{S}$Equation 1 Where, $N$ is the number of turns of coil $S$ is the number of turns of coil the following formula for reluctance, S of coil. $S=frac{l}{mu A}$Equation 2 Where, $l$ is the length of magnetic circuit $mu$ is the permeability of core $A$ is the area of magnetic circuit through which flux flows Substitute, Equation 2 in Equation 1. $$L=frac{N^{2}}{left (frac{l}{mu A} right )}$$ $Rightarrow L=frac{N^{2}mu A}{l}$Equation 3 From Equation 1 & Equation 3, we can conclude that the inductance value depends on the three parameters $N,S$ & $mu$. So, we can make the inductive transducers based on the variation in one of the three parameters $N,S$ & $mu$. Because, the variation in any one of those three parameters changes the inductance value. Inductance, L is directly proportional to square of the number of turns of coil. So, as number of turns of coil, $N$ increases the value of inductance, $L$ also increases. Similarly, as number of turns of coil, $N$ decreases the value of inductance, $L$ also decreases. Inductance, $L$ is inversely proportional to reluctance of coil, $S$. So, as reluctance of coil, $S$ increases the value of inductance, $L$ decreases. Similarly, as reluctance of coil, $S$ decreases the value of inductance, $L$ increases. Inductance, L is directly proportional to permeability of core, $mu$. So, as permeability of core, $mu$ increases the value of inductance, L also increases. Similarly, as permeability of core, $mu$ decreases the value of inductance, L also decreases. Capacitive Transducer A passive transducer is said to be a capacitive transducer, when it produces the variation (change) in capacitance value. the following formula for capacitance, C of a parallel plate capacitor. $$C=frac{varepsilon A}{d}$$ Where, $varepsilon$ is the permittivity or the dielectric constant $A$ is the effective area of two plates $d$ is the effective area of two plates The capacitance value depends on the three parameters $varepsilon, A$ & $d$. So, we can make the capacitive transducers based on the variation in one of the three parameters$varepsilon, A$ & $d$. Because, the variation in any one of those three parameters changes the capacitance value. Capacitance, C is directly proportional to permittivity, $varepsilon$. So, as permittivity, $varepsilon$ increases the value of capacitance, C also increases. Similarly, as permittivity, $varepsilon$ decreases the value of capacitance, C also decreases. Capacitance, C is directly proportional to the effective area of two plates, $A$. So, as effective area of two plates, $A$ increases the value of capacitance, C also increases. Similarly, as effective area of two plates, $A$ decreases the value of capacitance, C also decreases. Capacitance, C is inversely proportional to the distance between two plates, $d$. So, as distance between two plates, $d$ increases the value of capacitance, C decreases. Similarly, as distance between two plates, $d$ decreases the value of capacitance, C increases. In this chapter, we discussed about three passive transducers. In next chapter, let us discuss about an example for each passive transducer. Learning working make money

Electronic Measuring Instruments – Bridges If the electrical components are arranged in the form a bridge or ring structure, then that electrical circuit is called a bridge. In general, bridge forms a loop with a set of four arms or branches. Each branch may contain one or two electrical components. Types of Bridges We can classify the bridge circuits or bridges into the following two categories based on the voltage signal with which those can be operated. DC Bridges AC Bridges Now, let us discuss about these two bridges briefly. DC Bridges If the bridge circuit can be operated with only DC voltage signal, then it is a DC bridge circuit or simply DC bridge. DC bridges are used to measure the value of unknown resistance. The circuit diagram of DC bridge looks like as shown in below figure. The above DC bridge has four arms and each arm consists of a resistor. Among which, two resistors have fixed resistance values, one resistor is a variable resistor and the other one has an unknown resistance value. The above DC bridge circuit can be excited with a DC voltage source by placing it in one diagonal. The galvanometer is placed in other diagonal of DC bridge. It shows some deflection as long as the bridge is unbalanced. Vary the resistance value of variable resistor until the galvanometer shows null (zero) deflection. Now, the above DC bridge is said to be a balanced one. So, we can find the value of unknown resistance by using nodal equations. AC Bridges If the bridge circuit can be operated with only AC voltage signal, then it is said to be AC bridge circuit or simply AC bridge. AC bridges are used to measure the value of unknown inductance, capacitance and frequency. The circuit diagram of AC bridge looks like as shown in below figure. The circuit diagram of AC bridge is similar to that of DC bridge. The above AC bridge has four arms and each arm consists of some impedance. That means, each arm will be having either single or combination of passive elements such as resistor, inductor and capacitor. Among the four impedances, two impedances have fixed values, one impedance is variable and the other one is an unknown impedance. The above AC bridge circuit can be excited with an AC voltage source by placing it in one diagonal. A detector is placed in other diagonal of AC bridge. It shows some deflection as long as the bridge is unbalanced. The above AC bridge circuit can be excited with an AC voltage source by placing it in one diagonal. A detector is placed in other diagonal of AC bridge. It shows some deflection as long as the bridge is unbalanced. Vary the impedance value of variable impedance until the detector shows null (zero) deflection. Now, the above AC bridge is said to be a balanced one. So, we can find the value of unknown impedance by using balanced condition. Learning working make money

Data Acquisition Systems The systems, used for data acquisition are known as data acquisition systems. These data acquisition systems will perform the tasks such as conversion of data, storage of data, transmission of data and processing of data. Data acquisition systems consider the following analog signals. Analog signals, which are obtained from the direct measurement of electrical quantities such as DC & AC voltages, DC & AC currents, resistance and etc. Analog signals, which are obtained from transducers such as LVDT, Thermocouple & etc. Types of Data Acquisition Systems Data acquisition systems can be classified into the following two types. Analog Data Acquisition Systems Digital Data Acquisition Systems Now, let us discuss about these two types of data acquisition systems one by one. Analog Data Acquisition Systems The data acquisition systems, which can be operated with analog signals are known as analog data acquisition systems. Following are the blocks of analog data acquisition systems. Transducer − It converts physical quantities into electrical signals. Signal conditioner − It performs the functions like amplification and selection of desired portion of the signal. Display device − It displays the input signals for monitoring purpose. Graphic recording instruments − These can be used to make the record of input data permanently. Magnetic tape instrumentation − It is used for acquiring, storing & reproducing of input data. Digital Data Acquisition Systems The data acquisition systems, which can be operated with digital signals are known as digital data acquisition systems. So, they use digital components for storing or displaying the information. Mainly, the following operations take place in digital data acquisition. Acquisition of analog signals Conversion of analog signals into digital signals or digital data Processing of digital signals or digital data Following are the blocks of Digital data acquisition systems. Transducer − It converts physical quantities into electrical signals. Signal conditioner − It performs the functions like amplification and selection of desired portion of the signal. Multiplexer − connects one of the multiple inputs to output. So, it acts as parallel to serial converter. Analog to Digital Converter − It converts the analog input into its equivalent digital output. Display device − It displays the data in digital format. Digital Recorder − It is used to record the data in digital format. Data acquisition systems are being used in various applications such as biomedical and aerospace. So, we can choose either analog data acquisition systems or digital data acquisition systems based on the requirement. Learning working make money

AC Bridges In this chapter, let us discuss about the AC bridges, which can be used to measure inductance. AC bridges operate with only AC voltage signal. The circuit diagram of AC bridge is shown in below figure. As shown in above figure, AC bridge mainly consists of four arms, which are connected in rhombus or square shape. All these arms consist of some impedance. The detector and AC voltage source are also required in order to find the value of unknown impedance. Hence, one of these two are placed in one diagonal of AC bridge and the other one is placed in other diagonal of AC bridge. The balancing condition of Wheatstone’s bridge as − $$R_{4}=frac{R_{2}R_{3}}{R_{1}}$$ We will get the balancing condition of AC bridge, just by replacing R with Z in above equation. $$Z_{4}=frac{Z_{2}Z_{3}}{Z_{1}}$$ $Rightarrow Z_{1}Z_{4}=Z_{2}Z_{3}$ Here, $Z_{1}$ and $Z_{2}$ are fixed impedances. Whereas, $Z_{3}$ is a standard variable impedance and $Z_{4}$ is an unknown impedance. Note − We can choose any two of those four impedances as fixed impedances, one impedance as standard variable impedance & the other impedance as an unknown impedance based on the application. Following are the two AC bridges, which can be used to measure inductance. Maxwell’s Bridge Hay’s Bridge Now, let us discuss about these two AC bridges one by one. Maxwell”s Bridge Maxwell’s bridge is an AC bridge having four arms, which are connected in the form of a rhombus or square shape. Two arms of this bridge consist of a single resistor, one arm consists of a series combination of resistor and inductor & the other arm consists of a parallel combination of resistor and capacitor. An AC detector and AC voltage source are used to find the value of unknown impedance. Hence, one of these two are placed in one diagonal of Maxwell’s bridge and the other one is placed in other diagonal of Maxwell’s bridge. Maxwell’s bridge is used to measure the value of medium inductance. The circuit diagram of Maxwell’s bridge is shown in the below figure. In above circuit, the arms AB, BC, CD and DA together form a rhombus or square shape. The arms AB and CD consist of resistors, $R_{2}$ and $R_{3}$ respectively. The arm, BC consists of a series combination of resistor, $R_{4}$ and inductor, $L_{4}$. The arm, DA consists of a parallel combination of resistor, $R_{1}$ and capacitor, $C_{1}$. Let, $Z_{1}, Z_{2}, Z_{3}$ and $Z_{4}$ are the impedances of arms DA, AB, CD and BC respectively. The values of these impedances will be $$Z_{1}=frac{R_{1}left ( frac{1}{jomega C_{1}} right )}{R_{1}+frac{1}{jomega C_{1}}}$$ $$Rightarrow Z_{1}=frac{R_{1}}{1+j omega R_{1}C_{1}}$$ $Z_{2}=R_{2}$ $Z_{3}=R_{3}$ $Z_{4}=R_{4}+j omega L_{4}$ Substitute these impedance values in the following balancing condition of AC bridge. $$Z_{4}=frac{Z_{2}Z_{3}}{Z_{1}}$$ $$R_{4}+jomega L_{4}=frac{R_{2}R_{3}}{left ( {frac{R_{1}}{1+j omega R_{1}C_{1}}} right )}$$ $Rightarrow R_{4}+jomega L_{4}=frac{R_{2}R_{3}left (1+j omega R_{1}C_{1} right )}{R_{1}}$ $Rightarrow R_{4}+jomega L_{4}=frac{R_{2}R_{3}}{R_{1}}+frac{j omega R_{1}C_{1}R_{2}R_{3}}{R_{1}}$ $Rightarrow R_{4}+jomega L_{4}=frac{R_{2}R_{3}}{R_{1}}+j omega C_{1}R_{2}R_{3}$ By comparing the respective real and imaginary terms of above equation, we will get $R_{4}=frac{R_{2}R_{3}}{R_{1}}$Equation 1 $L_{4}=C_{1}R_{2}R_{3}$Equation 2 By substituting the values of resistors $R_{1}$, $R_{2}$ and $R_{3}$ in Equation 1, we will get the value of resistor, $R_{4}$. Similarly, by substituting the value of capacitor, $C_{1}$ and the values of resistors, $R_{2}$ and $R_{3}$ in Equation 2, we will get the value of inductor, $L_{4}$. The advantage of Maxwell’s bridge is that both the values of resistor, $R_{4}$ and an inductor, $L_{4}$ are independent of the value of frequency. Hay’s Bridge Hay’s bridge is a modified version of Maxwell’s bridge, which we get by modifying the arm, which consists of a parallel combination of resistor and capacitor into the arm, which consists of a series combination of resistor and capacitor in Maxwell’s bridge. Hay’s bridge is used to measure the value of high inductance. The circuit diagram of Hay’s bridge is shown in the below figure. In above circuit, the arms AB, BC, CD and DA together form a rhombus or square shape. The arms, AB and CD consist of resistors, $R_{2}$ and $R_{3}$ respectively. The arm, BC consists of a series combination of resistor, $R_{4}$ and inductor, $L_{4}$. The arm, DA consists of a series combination of resistor, $R_{1}$ and capacitor, $C_{1}$. Let, $Z_{1}, Z_{2}, Z_{3}$ and $Z_{4}$ are the impedances of arms DA, AB, CD and BC respectively. The values of these impedances will be $$Z_{1}=R_{1}+frac{1}{j omega C_{1}}$$ $Rightarrow Z_{1}=frac{1+j omega R_{1}C_{1}}{j omega C_{1}}$ $Z_{2}=R_{2}$ $Z_{3}=R_{3}$ $Z_{4}=R_{4}+j omega L_{4}$ Substitute these impedance values in the following balancing condition of AC bridge. $$Z_{4}=frac{Z_{2}Z_{3}}{Z_{1}}$$ $R_{4}+j omega L_{4}=frac{R_{2}R_{3}}{left ( frac{1+j omega R_{1}C_{1}}{j omega C_{1}}right )}$ $R_{4}+j omega L_{4}=frac{R_{2}R_{3}j omega C_{1}}{left ( 1+j omega R_{1}C_{1}right )}$ Multiply the numerator and denominator of right hand side term of above equation with $1 – j omega R_{1}C_{1}$. $Rightarrow R_{4}+j omega L_{4}=frac{R_{2}R_{3}j omega C_{1}}{left ( 1+j omega R_{1}C_{1}right )}times frac{left (1 – j omega R_{1}C_{1} right )}{left (1 – j omega R_{1}C_{1} right )}$ $Rightarrow R_{4}+j omega L_{4}=frac{omega^{2}{C_{1}}^{2}R_{1}R_{2}R_{3}+j omega R_{2}R_{3}C_{1}}{left ( 1+omega^{2}{R_{1}}^{2}{C_{1}}^{2}right )}$ By comparing the respective real and imaginary terms of above equation, we will get $R_{4}= frac{omega^{2}{C_{1}}^{2}R_{1}R_{2}R_{3}}{left ( 1+omega^{2}{R_{1}}^{2}{C_{1}}^{2}right )}$Equation 3 $L_{4}= frac{R_{2}R_{3}C_{1}}{left ( 1+omega^{2}{R_{1}}^{2}{C_{1}}^{2}right )}$Equation 4 By substituting the values of $R_{1}, R_{2}, R_{3}, C_{1}$ and $omega$ in Equation 3 and Equation 4, we will get the values of resistor, $R_{4}$ and inductor, $L_{4}$. 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Spectrum Analyzers The electronic instrument, used for analyzing waves in frequency domain is called spectrum analyzer. Basically, it displays the energy distribution of a signal on its CRT screen. Here, x-axis represents frequency and y-axis represents the amplitude. Types of Spectrum Analyzers We can classify the spectrum analyzers into the following two types. Filter Bank Spectrum Analyzer Superheterodyne Spectrum Analyzer Now, let us discuss about these two spectrum analyzers one by one. Filter Bank Spectrum Analyzer The spectrum analyzer, used for analyzing the signals are of AF range is called filter bank spectrum analyzer, or real time spectrum analyzer because it shows (displays) any variations in all input frequencies. The following figure shows the block diagram of filter bank spectrum analyzer. The working of filter bank spectrum analyzer is mentioned below. It has a set of band pass filters and each one is designed for allowing a specific band of frequencies. The output of each band pass filter is given to a corresponding detector. All the detector outputs are connected to Electronic switch. This switch allows the detector outputs sequentially to the vertical deflection plate of CRO. So, CRO displays the frequency spectrum of AF signal on its CRT screen. Superheterodyne Spectrum Analyzer The spectrum analyzer, used for analyzing the signals are of RF range is called superheterodyne spectrum analyzer. Its block diagram is shown in below figure. The working of superheterodyne spectrum analyzer is mentioned below. The RF signal, which is to be analyzed is applied to input attenuator. If the signal amplitude is too large, then it can be attenuated by an input attenuator. Low Pass Filter (LPF) allows only the frequency components that are less than the cut-off frequency. Mixer gets the inputs from Low pass filter and voltage tuned oscillator. It produces an output, which is the difference of frequencies of the two signals that are applied to it. IF amplifier amplifies the Intermediate Frequency (IF) signal, i.e. the output of mixer. The amplified IF signal is applied to detector. The output of detector is given to vertical deflection plate of CRO. So, CRO displays the frequency spectrum of RF signal on its CRT screen. So, we can choose a particular spectrum analyzer based on the frequency range of the signal that is to be analyzed. Learning working make money

Other AC Bridges In previous chapter, we discussed about two AC bridges which can be used to measure inductance. In this chapter, let us discuss about the following two AC bridges. Schering Bridge Wien’s Bridge These two bridges can be used to measure capacitance and frequency respectively. Schering Bridge Schering bridge is an AC bridge having four arms, which are connected in the form of a rhombus or square shape, whose one arm consists of a single resistor, one arm consists of a series combination of resistor and capacitor, one arm consists of a single capacitor & the other arm consists of a parallel combination of resistor and capacitor. The AC detector and AC voltage source are also used to find the value of unknown impedance, hence one of them is placed in one diagonal of Schering bridge and the other one is placed in other diagonal of Schering bridge. Schering bridge is used to measure the value of capacitance. The circuit diagram of Schering bridge is shown in the below figure. In above circuit, the arms AB, BC, CD and DA together form a rhombus or square shape. The arm AB consists of a resistor, $R_{2}$. The arm BC consists of a series combination of resistor, $R_{4}$ and capacitor, $C_{4}$. The arm CD consists of a capacitor, $C_{3}$. The arm DA consists of a parallel combination of resistor, $R_{1}$ and capacitor, $C_{1}$. Let, $Z_{1}$, $Z_{2}$,$Z_{3}$ and $Z_{4}$ are the impedances of arms DA, AB, CD and BC respectively. The values of these impedances will be $Z_{1}=frac{R_{1}left ( frac{1}{j omega C_{1}} right )}{R_{1}+frac{1}{j omega C_{1}}}$ $Rightarrow Z_{1}=frac{R_{1}}{1+j omega R_{1}C_{1}}$ $Z_{2}=R_{2}$ $Z_{3}=frac{1}{j omega C_{3}}$ $Z_{4}=R_{4}+frac{1}{j omega C_{4}}$ $Rightarrow Z_{4}=frac{1+j omega R_{4}C_{4}}{j omega C_{4}}$ Substitute these impedance values in the following balancing condition of AC bridge. $$Z_{4}=frac{Z_{2}Z_{3}}{Z_{1}}$$ $$frac{1+j omega R_{4}C_{4}}{j omega C_{4}}=frac{R_{2}left (frac{1}{j omega C_{3}} right )}{frac{R_{1}}{1+j omega R_{1}C_{1}}}$$ $Rightarrow frac{1+j omega R_{4}C_{4}}{j omega C_{4}}=frac{R_{2}left ( 1+j omega R_{1}C_{1} right )}{j omega R_{1}C_{3}}$ $Rightarrow frac{1+j omega R_{4}C_{4}}{C_{4}}=frac{R_{2}left ( 1+j omega R_{1}C_{1} right )}{R_{1}C_{3}}$ $Rightarrow frac{1}{C_{4}}+j omega R_{4}=frac{R_{2}}{R_{1}C_{3}}+frac{jomega C_{1}R_{2}}{C_{3}}$ By comparing the respective real and imaginary terms of above equation, we will get $C_{4}=frac{R_{1}C_{3}}{R_{2}}$Equation 1 $R_{4}=frac{C_{1}R_{2}}{C_{3}}$Equation 2 By substituting the values of $R_{1}, R_{2}$ and $C_{3}$ in Equation 1, we will get the value of capacitor, $C_{4}$. Similarly, by substituting the values of $R_{2}, C_{1}$ and $C_{3}$ in Equation 2, we will get the value of resistor, $R_{4}$. The advantage of Schering bridge is that both the values of resistor, $R_{4}$ and capacitor, $C_{4}$ are independent of the value of frequency. Wien’s Bridge Wien’s bridge is an AC bridge having four arms, which are connected in the form of a rhombus or square shape. Amongtwo arms consist of a single resistor, one arm consists of a parallel combination of resistor and capacitor & the other arm consists of a series combination of resistor and capacitor. The AC detector and AC voltage source are also required in order to find the value of frequency. Hence, one of these two are placed in one diagonal of Wien’s bridge and the other one is placed in other diagonal of Wien’s bridge. The circuit diagram of Wien’s bridge is shown in the below figure. In above circuit, the arms AB, BC, CD and DA together form a rhombus or square shape. The arms, AB and BC consist of resistors, $R_{2}$ and $R_{4}$ respectively. The arm, CD consists of a parallel combination of resistor, $R_{3}$ and capacitor, $C_{3}$. The arm, DA consists of a series combination of resistor, $R_{1}$ and capacitor, $C_{1}$. Let, $Z_{1}, Z_{2}, Z_{3}$ and $Z_{4}$ are the impedances of arms DA, AB, CD and BC respectively. The values of these impedances will be $$Z_{1}=R_{1}+frac{1}{j omega C_{1}}$$ $$Rightarrow Z_{1}=frac{1+j omega R_{1}C_{1}}{j omega C_{1}}$$ $Z_{2}=R_{2}$ $$Z_{3}=frac{R_{3}left (frac{1}{j omega C_{3}} right )}{R_{3}+frac{1}{j omega C_{3}}}$$ $$Rightarrow Z_{3}= frac{R_{3}}{1+j omega R_{3}C_{3}}$$ $Z_{4}=R_{4}$ Substitute these impedance values in the following balancing condition of AC bridge. $$Z_{1}Z_{4}=Z_{2}Z_{3}$$ $$left (frac{1+j omega R_{1}C_{1}}{j omega C_{1}} right )R_{4}=R_{2}left (frac{R_{3}}{1+j omega R_{3}C_{3}} right )$$ $Rightarrow left (1+j omega R_{1}C_{1}right )left (1+j omega R_{3}C_{3}right )R_{4}=j omega C_{1}R_{2}R_{3}$ $Rightarrow left (1+j omega R_{3}C_{3}+j omega R_{1}C_{1}-omega^{2}R_{1}R_{3}C_{1}C_{3}right )R_{4}=j omega C_{1}R_{2}R_{3}$ $Rightarrow R_{4}left ( omega^{2}R_{1}R_{3}C_{1}C_{3} right )+j omega R_{4}left (R_{3}C_{3}+R_{1}C_{1} right )=j omega C_{1}R_{2}R_{3}$ Equate the respective real terms of above equation. $$R_{4}left (1- omega^{2}R_{1}R_{3}C_{1}C_{3} right )=0$$ $Rightarrow 1- omega^{2}R_{1}R_{3}C_{1}C_{3} =0$ $Rightarrow 1= omega^{2}R_{1}R_{3}C_{1}C_{3}$ $omega = frac{1}{sqrt{R_{1}R_{3}C_{1}C_{3}}}$ Substitute, $omega = 2 pi f$ in above equation. $$Rightarrow 2 pi f= frac{1}{sqrt{R_{1}R_{3}C_{1}C_{3}}}$$ $Rightarrow f= frac{1}{2 pisqrt{R_{1}R_{3}C_{1}C_{3}}}$ We can find the value of frequency, $f$ of AC voltage source by substituting the values of $R_{1}, R_{3}, C_{1}$ and $C_{3}$ in above equation. If $R_{1}=R_{3}=R$ and $C_{1}=C_{3}=C$, then we can find the value of frequency, $f$ of AC voltage source by using the following formula. $$f=frac{1}{2pi RC}$$ The Wein’s bridge is mainly used for finding the frequency value of AF range. Learning working make money

Transducers Basically, Transducer converts one form of energy into another form of energy. The transducer, which converts non-electrical form of energy into electrical form of energy is known as electrical transducer. The block diagram of electrical transducer is shown in below figure. As shown in the figure, electrical transducer will produce an output, which has electrical energy. The output of electrical transducer is equivalent to the input, which has non-electrical energy. Types of Electrical Transducers Mainly, the electrical transducers can be classified into the following two types. Active Transducers Passive Transducers Now, let us discuss about these two types of transducers briefly. Active Transducers The transducer, which can produce one of the electrical quantities such as voltage and current is known as active transducer. It is also called self-generating transducer, since it doesn’t require any external power supply. The block diagram of active transducer is shown in below figure. As shown in the figure, active transducer will produce an electrical quantity (or signal), which is equivalent to the non-electrical input quantity (or signal). Examples Following are the examples of active transducers. Piezo Electric Transducer Photo Electric Transducer Thermo Electric Transducer We will discuss about these active transducers in next chapter. Passive Transducers The transducer, which can’t produce the electrical quantities such as voltage and current is known as passive transducer. But, it produces the variation in one of passive elements like resistor (R), inductor (L) and capacitor (C). Passive transducer requires external power supply. The block diagram of passive transducer is shown in below figure. As shown in the figure, passive transducer will produce variation in the passive element in accordance with the variation in the non-electrical input quantity (or signal). Examples Following are the examples of passive transducers. Resistive Transducer Inductive Transducer Capacitive Transducer We will discuss about these passive transducers in later chapters. Learning working make money

Measurement Of Displacement The physical quantities such as displacement, velocity, force, temperature & etc. are all non-electrical quantities. active transducer converts the physical quantity into an electrical signal. Whereas, passive transducer converts the physical quantity into the variation in passive element. So, based on the requirement we can choose either active transducer or passive transducer. In this chapter, let us discuss how to measure displacement by using a passive transducer. If a body that moves from one point to another point in a straight line, then the length between those two points is called displacement. We have the following three passive transducers Resistive Transducer Inductive Transducer Capacitive Transducer Now, let us discuss about the measurement of displacement with these three passive transducers one by one. Measurement of Displacement using Resistive Transducer The circuit diagram of resistive transducer, which is used to measure displacement is shown in below figure. The above circuit consists of a potentiometer and a voltage source, $V_{S}$. We can say that these two are connected in parallel with respect to the points A & B. Potentiometer has a sliding contact, which can be varied. So, the point C is a variable one. In above circuit, the output voltage, $V_{0}$ is measured across the points A & C. Mathematically, the relation between the voltages and distances can be represented as $$frac{V_{0}}{V_{S}}=frac{AC}{AB}$$ Therefore, we should connect the body whose displacement is to be measured to the sliding contact. So, whenever the body moves in a straight line, the point C also varies. Due to this, the output voltage, $V_{0}$ also changes accordingly. In this case, we can find the displacement by measuring the output voltage, $V_{0}$. Measurement of Displacement using Inductive Transducer The circuit diagram of inductive transducer, which is used to measure displacement is shown in below figure. The transformer present in above circuit has a primary winding and two secondary windings. Here, the ending points of two secondary windings are joined together. So, we can say that these two secondary windings are connected in series opposition. The voltage, $V_{P}$ is applied across the primary winding of transformer. Let, the voltage developed across each secondary winding is 𝑉𝑆1 and 𝑉𝑆2. The output voltage, $V_{0}$ is taken across the starting points of two secondary windings. Mathematically, the output voltage, 𝑉0 can be written as $$V_{0}= V_{S1}-V_{S2}$$ The transformer present in above circuit is called differential transformer, since it produces an output voltage, which is the difference between $V_{S1}$ and $V_{S2}$. If the core is at central position, then the output voltage, $V_{0}$ will be equal to zero. Because, the respective magnitudes & phases of $V_{S1}$ and $V_{S2}$ are same. If the core is not at central position, then the output voltage, $V_{0}$ will be having some magnitude & phase. Because, the respective magnitudes & phases of $V_{S1}$ and $V_{S2}$ are not equal. Therefore, we should connect the body whose displacement is to be measured to the central core. So, whenever the body moves in a straight line, the central position of the core varies. Due to this, the output voltage, $V_{0}$ also changes accordingly. In this case, we can find the displacement by measuring the output voltage, $V_{0}$. The magnitude & phase of output voltage, $V_{0}$ represents the displacement of the body & its direction respectively. Measurement of Displacement using Capacitive Transducer The circuit diagram of capacitive transducer, which is used to measure displacement is shown in below figure. The capacitor, which is present in above circuit has two parallel plates. Among which, one plate is fixed and the other plate is a movable one. Due to this, the spacing between these two plates will also vary. the value of capacitance changes as the spacing between two plates of capacitor changes. Therefore, we should connect the body whose displacement is to be measured to the movable plate of a capacitor. So, whenever the body moves in a straight line, the spacing between the two plates of capacitor varies. Due to this, the capacitance value changes. Learning working make money