Alternating Current

Alternating current signal
AC means Alternating Current and DC means Direct Current. AC and DC are also used when referring to voltages and electrical signals which are not currents! For example: a 12V AC power supply has an alternating voltage (which will make an alternating current flow).

1. UNIT 3 Alternating Current: rms and average value of ac – definition and expressions, Representation of sinusoids by complex numbers (brief explanation), response of LR, CR and LCR series circuit to sinusoidal voltage, series and parallel resonant (LR parallel C) circuits (mention condition for resonance with expressions for impedance and current), expression for Q factor, band width, AC bridge - Maxwell bridge (derivation of condition for balance, determination of self-inductance of a coil). Thermoelectricity: Seebeck effect (brief explanation, experiment and temperature dependence), Thermoelectric series, Neutral temperature, Laws of thermoelectricity (qualitative), Peltier effect, Peltier coefficient (qualitative analysis), Thomson effect, Thomson coefficient (qualitative analysis), Theory of thermoelectric circuits using thermodynamics (Application of thermodynamics to a thermocouple and connected relations with derivation), Applications of thermoelectricity - Boys' Radio-micrometer, thermopile and thermoelectric pyrometer (brief explanation with experimental setup). By: Dr. Mayank Pandey

2. Alternating Current: rms and average value of ac - definition and expressions AC voltage alternates in polarity and AC current alternates in direction. We also know that AC can alternate in a variety of different ways, and by tracing the alternation over time we can plot it as a “waveform.” Ways of Expressing the Magnitude of an AC Waveform One way to express the intensity, or magnitude (also called the  amplitude ), of an AC quantity is to measure its peak height on a waveform graph. This is known as the  peak  or  crest  value of an  AC waveform : Figure below

3. One way of expressing the amplitude of different wave shapes in a more equivalent fashion is to mathematically average the values of all the points on a waveform’s graph to a single, aggregate number. This amplitude measurement is known simply as the  average  value of the waveform. If we average all the points on the waveform algebraically (that is, to consider their  sign , either positive or negative), the average value for most waveforms is technically zero, because all the positive points cancel out all the negative points over a full cycle: Figure below

4. Peak Value Definition:  The maximum value attained by an alternating quantity during one cycle is called its P eak value . It is also known as the maximum value or amplitude or crest value. The sinusoidal alternating quantity obtains its peak value at 90 degrees as shown in the figure below. The peak values of alternating voltage and current is represented by E m  and I m  respectively.

5. Average Value Definition:  The average of all the instantaneous values of an alternating voltage and currents over one complete cycle is called  Average Value. If we consider symmetrical waves like sinusoidal current or voltage waveform, the positive half cycle will be exactly equal to the negative half cycle. Therefore, the average value over a complete cycle will be  zero . The work is done by both, positive and negative cycle and hence the average value is determined without considering the signs. So, the only positive half cycle is considered to determine the average value of alternating quantities of sinusoidal waves.

6. R.M.S Value Definition:  That steady current which, when flows through a resistor of known resistance for a given period of time than as a result the same quantity of heat is produced by the alternating current when flows through the same resistor for the same period of time is called  R.M.S  or effective value of the alternating current. In other words, the R.M.S value is defined as the square root of means of squares of instantaneous values. Let I be the alternating current flowing through a resistor R for time t seconds, which produces the same amount of heat as produced by the direct current ( I eff ). The base of one alteration is divided into n equal parts so that each interval is of t/n seconds as shown in the figure below. Root Mean Square is the actual value of an alternating quantity which tells us an energy transfer capability of an AC source. The ammeter records the RMS value of alternating current and voltmeter record’s the root mean square (R.M.S) value of alternating voltage. The domestic single-phase AC supply is 230 V, 50 hertz, where 230 V is the R.M.S value of alternating voltage.

7. Representation of sinusoids by complex numbers

8. Response of LR, CR and LCR series circuit to sinusoidal voltage When a constant voltage  source  or battery is connected across a resistor,  current  is developed in it. This current has a unique  direction  and flows from the negative terminal of the battery to its positive terminal. The magnitude of current remains constant as well. If the direction of current through this resistor changes periodically or alternately, then the current is called  alternating current . An alternating current or  AC generator  or AC dynamo can be used as AC voltage source.

9. Resonance in Series and Parallel Circuits Series circuit The circuit, with resistance R, inductance L, and a capacitor, C in series (Fig. 17.1a) is connected to a single phase variable frequency ( f ) supply. The total impedance of the circuit is

10. Q Factor and Bandwidth The bandwidth of a parallel resonance circuit is defined in exactly the same way as for the series resonance circuit. The upper and lower cut-off frequencies given as:  ƒ upper  and  ƒ lower  respectively denote the half-power frequencies where the power dissipated in the circuit is half of the full power dissipated at the resonant frequency  0.5 (I 2  R)  which gives us the same -3dB points at a current value that is equal to 70.7% of its maximum resonant value,  (0.707 x I) 2  R As with the series circuit, if the resonant frequency remains constant, an increase in the quality factor,  Q  will cause a decrease in the bandwidth and likewise, a decrease in the quality factor will cause an increase in the bandwidth as defined by:    BW = ƒ r  /Q  or   BW = ƒ upper  - ƒ lower

11. The selectivity or  Q-factor  for a parallel resonance circuit is generally defined as the ratio of the circulating branch currents to the supply current and is given as: Note that the Q-factor of a parallel resonance circuit is the inverse of the expression for the Q-factor of the series circuit. Also in series resonance circuits the Q-factor gives the voltage magnification of the circuit, whereas in a parallel circuit it gives the current magnification.

12. Parallel Resonance Example No 1 A parallel resonance network consisting of a resistor of 60Ω, a capacitor of 120uF and an inductor of 200mH is connected across a sinusoidal supply voltage which has a constant output of 100 volts at all frequencies. Calculate, the resonant frequency, the quality factor and the bandwidth of the circuit, the circuit current at resonance and current magnification. Electric Circuit Studio

13. AC bridge-Maxwell Bridge Definition:  The  bridge  used for the  measurement  of  self-inductance  of the  circuit  is known as the Maxwell bridge. It is the advanced form of the  Wheatstone bridge . The Maxwell bridge  works  on the  principle  of the  comparison , i.e., the value of  unknown   inductance  is determined by  comparing  it with the known value or standard value. Types of Maxwell’s Bridge Two methods are used for determining the self-inductance of the circuit. They are 1. Maxwell’s Inductance Bridge 2. Maxwell’s inductance Capacitance Bridge

14. Maxwell’s Inductance Bridge In such type of bridges, the value of unknown  resistance  is determined by comparing it with the known value of the standard self-inductance. The connection diagram for the balance Maxwell bridge is shown in the figure below. Let, L 1  – unknown inductance of resistance R 1 . L 2  – Variable inductance of fixed resistance r 1 . R 2  – variable resistance connected in series with inductor L 2 . R 3 , R 4  – known non-inductance resistance At balance, 

15. The value of the R 3  and the R 4  resistance varies from 10 to 1000 ohms with the help of the resistance box. Sometimes for balancing the bridge, the additional resistance is also inserted into the circuit. The phasor diagram of Maxwell’s inductance bridge is shown in the figure below.

16. Maxwell’s Inductance Capacitance Bridge In this type of bridges, the unknown resistance is measured with the help of the standard variable  capacitance . The connection diagram of the Maxwell Bridge is shown in the figure below. Let, L 1  – unknown inductance of resistance R 1 . R 1  – Variable inductance of fixed resistance r 1 . R 2 , R 3 , R 4  – variable resistance connected in series with inductor L 2 . C 4  – known non-inductance resistance For balance condition ,

17. By separating the real and imaginary equation we get, The above equation shows that the bridges have two variables R 4  and C 4  which appear in one of the two equations and hence both the equations are independent. The circuit quality factor is expressed as Phasor diagram of Maxwell’s inductance capacitance bridge is shown in the figure below.

18. Advantages of the Maxwell’s Bridges The following are the advantages of the Maxwell bridges The balance equation of the circuit is free from frequency. Both the balance equations are independent of each other. The Maxwell’s inductor capacitance bridge is used for the measurement of the high range  inductance. Disadvantages of the Maxwell’s Bridge The main disadvantages of the bridges are The Maxwell inductor capacitance bridge requires a variable capacitor which is very expensive. Thus, sometimes the standard variable capacitor is used in the bridges. The bridge is only used for the measurement of medium quality coils. Because of the following disadvantages, the Hays bridge is used for the measurement of circuit inductance which is the advanced form of the Maxwell’s Bridge.

19. Fundamentals of Thermoelectrics Thermoelectric phenomena in materials can be described through three thermoelectric effects, the Seebeck effect, the Peltier effect, and the Thomson effect. The Seebeck effect, discovered in 1821 by Thomas Johann Seebeck , represents the generation of an electromotive force by a temperature gradient. When a temperature gradient is applied along a conductive material, charge carriers move from the hot to the cold side. In the case of open-circuit, charge accumulation results in an electric potential difference, as shown in Fig.  1.2 . The Seebeck coefficient  S  of a material shows the magnitude of the induced voltage in response to the temperature difference:

20. where  T C  and T H  are the temperatures of the cold and the hot side, respectively. The Seebeck coefficient of a material is generally a function of temperature. However, if the temperature difference  ΔT=  T H - T C is small enough, and the Seebeck coefficient is nearly constant in the range of applied temperatures, one has: A single thermoelectric conductor, however, is not able to play the role of a battery. This is due to the fact that the net loop voltage would be zero if circuit wires of the same conductor were connected. A non-zero loop voltage can be obtained when two dissimilar conductors are connected in the configuration shown in Fig.  1.3 . The circuit configuration of a thermoelectric generator is shown in Fig.  1.4 -a. Here the generator is composed of two materials, an  n  -type and a  p  -type material. The voltage obtained from this configuration is:

21. where  S p  and S n  are the Seebeck coefficients of the  p  -type and  n  -type materials, respectively, and  S pn  is the Seebeck coefficient of the device (junction). The Seebeck coefficient of a  p  -type material is positive, that of an  n  -type material negative. Therefore, using this configuration one can achieve a high net voltage The Peltier effect, on the other hand, discovered by Athanaseal Jean Charles Peltier in 1834, describes how an electrical current can create a heat flow. This effect has enabled the second application of thermoelectric devices, the thermoelectric cooler, as shown in Fig.  1.4 -b. Here, by applying an external power source, both the electrons of the  n  -type conductor and the holes of the  p  -type conductor move and carry heat from one side to the other. Therefore, one side cools down, whereas the other side heats up. The heat flow absorbed at the hot side is proportional to the current through the junction:

22. where, Π p , Π n and Π pn  are the Peltier coefficients of the  p  -type material, the  n  -type material, and the device, respectively. The Thomson effect, observed by William Thomson in 1851, expresses a relation for heat production in a current-carrying conductor in the presence of temperature gradient: where  qH  is the heat production per volume, ρ  the electrical resistivity,  J  the current density, and  µ  the Thomson coefficient. Considering the second term in Eq.  1.5 , the conductor can either release or absorb the heat, depending on the direction of current with respect to the temperature gradient.

23. According to the Thomson relations, these three thermoelectric coefficients are not independent. The Peltier coefficient and the Seebeck coefficient are linearly related to each other  whereas the Thomson coefficient can be expressed as: Therefore, the Seebeck coefficient as a function of temperature is enough for describing all three thermoelectric phenomena. The computational method for obtaining the Seebeck coefficient, as well as the calculation of the other transport coefficients, is presented. It is, however, worth mentioning that one can only measure the Seebeck coefficient for a pair (junction) of dissimilar materials, but not of an individual conductor, whereas measurement of the Thomson coefficient is possible even for individual materials.

24. so that the thermoelectric power is given by The relationship between V  and  T  is a parabola. The temperature T n = - a /2 b  at which the thermoelectric power is maximum is called the  neutral temperature . The temperatures T i = T  and T i = T – a / b  at which a small change in the difference of the junction temperatures leads to a change in the sign of emf is called the  inversion temperature . A complete understanding of Seebeck effect requires knowledge of behaviour of electron in a metal which is rather complicated. The Seebeck coefficient depends on factors like work functions of the two metals, electron densities of the two components, scattering mechanism within each solid etc. However, a simple minded picture is as follows.

25. Example 2: Compute the thermo-emf of a copper-constantan thermocouple with its junctions at  o C and  100 o C

26. Law of Intermediate Metals: According to this law, inserting an wire of arbitrary material into a thermocouple circuit has no effect on the thermal emf of the original circuit, if the additional junctions introduced in the circuit are at the same temperature. The law is useful in using thermocouples with elements which are both different from the material of the lead wires of a voltmeter. The most convenient way to analyze a thermocouple circuit is to use Kirchhoff's voltage law, which states that the algebraic sum of voltages around a closed loop is zero. Thus when the junction of two metals a  and b  are maintained at a temperature T  and the open circuit voltage  V  is measured under isothermal condition at a temperature T , we may write