8.1 Alternating Voltage and Alternating Current ( A. C. )

Size: px

Start display at page:

Download "8.1 Alternating Voltage and Alternating Current ( A. C. )"

Imogene Hubbard
6 years ago
Views:

1 8 - ALTENATING UENT Page 8. Alternating Voltage and Alternating urrent ( A.. ) The following figure shows N turns of a coil of conducting wire PQS rotating with a uniform angular speed ω with respect to the X-axis in a uniform magnetic field along the Y-axis. Let the angle between the direction of the area vector of the coil and the magnetic field direction be zero at time, t 0 and θ ωt at time t t. The magnetic flux, Φ 0, associated with the coil at time t 0 and Φ t at t me t t are given by Φ 0 N A B N A B cos 0 N A B and Φ t N A B cos ωt The emf induced in t e coil according to Faraday s law is V - d Φ t where V m N A B ω N A B ω sin ωt V m sin ωt ( ) is the maximum value of the induced emf. Equation ( ) shows that the induced emf versus time is a sine curve. This emf is obtained between the brushes B and B which are in contact with the slip rings A and A as shown in the above figure. The voltage is zero at time t 0 and varies as per the function sin ωt reaching maximum value V m at time t π / ω and again zero at time t π / ω. B being at greater potential than B acts like a positive end of the voltage source during this time interval. After time t π / ω, the potential of B starts to rise with respect to B till time ω, reaches maximum in the reverse direction and again becomes zero at time t 3 π / ω,, t π / ω. This cycle keeps repeating in every time interval of T π / ω. The voltage so developed is known as alternating voltage and its graph versus time is shown by a continuous line in the following figure. The arrangement to produce such a voltage is

2 8 - ALTENATING UENT Page known as A.. ( alternating current ) dynamo or generator. In the adjoining figure, an alternating voltage source is connected in a circuit with resistor,. The voltage between a and b is zero at time t 0. Zero current will flow at this time. The voltage in the circuit varies according as V V m sin ωt. The current as per Ohm s law V sin t will be Ι m ω. The current changes sinusoidally in the same way as the v ltage as shown by the broken line in the above graph Here, voltage was considered changing with time as per sin ωt. Both the voltage and the current can be considered to be ch nging as per cos ωt. It is not necessary that the voltage and current should change as above. There are also other ways in which both can change periodically with time. 8. A.. ircuit with Series ombination of esistor, Inductor and apacitor The following figure s ows a series combination of resistor having resistance, inductor of inductance L and a capacitor of capacitance with an alternating voltage source of voltage changing with time as V V m cos ωt. It is assumed that the resistor has zero inductance and the inductor h s zero resistance. Let Ι current in the circuit, Q ch rge on the capacitor and dι ate of change of current at t me t. Hence, by Kirchhoff s law, V V L V V dι Ι L Putting Ι dq Q and Vm cos ωt dι d Q in the above equation,

3 dq d Q Q L Vm cos ωt d Q dq Q V m cos ωt L L L 8 - ALTENATING UENT Page 3 This equation is known as the differential equation of Q for the series L - A.. ircuit. It is similar to the differential equation of forced harmonic oscillations given as under. d y b m dy k y m F 0 m sin ωt The mechanical and electrical quantities in the above equations are com ared in the following table. No. Mechanical quantities Symbol Elec rical quantities Symbol. Displacement y El ctri al charge Q dy dq. Velocity Electric current Ι 3. o-efficient of resistance b esistance 4. Mass m Inductance L 5. Force constant k Inverse of capacitance 6. Angular frequency k Angular frequency m L 7. Periodic force F 0 Periodic voltage V m The function of charge Q versus time t which satisfies the above differential equation of charge Q is lled its solution. omplex function is used to find the solution of the above differential equ tion 8.3 omplex Numbers ( For Information Only ) A ompl x number z x jy, where j. x is the real part and y is the imaginary p rt of the complex number. ( ) The complex number can be represented by a point in a complex plane with x-axis representing real numbers and y-axis representing imaginary numbers. Point P in the figure ( next page ) represents the complex number z x jy. The x-coordinate of P gives the real part of z and y-coordinate gives its imaginary part. The magnitude of complex number is equal to r, i.e., l z l r x y.

4 Now, x r cos θ and y r sin θ z r cos θ j r sin θ r ( cos θ j sin θ ) l z l e j θ ( e j θ cos θ j sin θ ) ( ) z* x - jy is the complex conjugate of the complex number z. ( 3 ) For complex number z, z * z * z z z * l z l x x y The real part of ALTENATING UENT Page 4 x x y j x - jy y y x z x y and the imaginary part of y - j. z x y The real part of the complex number z is denoted by e ( z ) and the imaginary part by Ιm ( z ). 8.4 Solution of the differential equation dι Q The equation for series L-- circ it is Ι L Vm cos ωt dι Ι Ι L V m cos ωt where Q Ι d Ι V Ι Ι m cos t L L L ω eplacing the current Ι by complex current i and cos ωt by e j ω t, and solving the differential equation of complex quantities, the real part of complex current i will give the equation of rea current Ι as a function of time, t. Thus, d i e differential equation of complex quantities is L i L i Vm e j ω t L Let i i m e j ω t be a trial solution. d i i jω e j ω t m Putting the values of i, i j ω e j ω t m d i i m e j ω t L and Ι ( ), where, L, and t are real quantities. i e j ω t m j ω and Ι in equation ( ) above, we have L i e j ω t m j ω Vm e j ω t L

5 i m ( j ω i m ( j ω L i m ( j ω L - i m 8 - ALTENATING UENT Page 5 V ) m L j ω L L j ω ) V m j ω ) V m Vm j ω L - ω Putting this value of i m in the trial solution, we have i j ω t Vm e j ω L - ω j j - j j This equation shows that the resistance off ed by an inductor and a capacitor are j ω L and - j / ω which are known as inductive eactance and capacitive reactance respectively. Their magnitudes are ω L and / ω respec ively. The inductive and capacitive reactance are represented by symbols Z L and Z while their magnitudes are equal to X L and X. Z L j ω L, Z - j / ω, X L ω L, X / ω. The summation of Z L, Z and is called the impedance ( Z ) of the series L-- circuit. The unit of impedance is ohm Z Z L Z j ( ω L - / ω ) i j t Vm e ω Z V Z The above equation represents Ohm s law for complex current, complex voltage and impedan e. Impedance is also complex which can be expressed as Z l Z l e j δ. i j t Vm e ω Z e jδ where, l Z l V Z ω L j ( ω t e - δ ) - ω V [ cos ( ωt - δ ) j sin ( ωt - δ ) ] Z Now, Ι e ( i ) Vm cos ( ω t ω L - - δ ) ω V m cos ( ω t - δ ) Z

6 8 - ALTENATING UENT Page 6 The equation shows that the current in the circuit changes according to cos ( ωt - δ ) and lags the voltage by phase δ as shown in the following graph. In the second graph above, point H shows the compl x impedance, Z j ( ωl - / ω ) in the complex plane where x-coordinate of the poin is which is the real part of the complex impedance and y-coordinate of the poin is ωl - / ω which is the imaginary part of the complex impedance. ωl - HD From the figure, tan δ ω OD OD DH and l Z l OH ( ) ( ) 8.5 Different ases ω L - ω The rules regarding the quivalent resistances for series and parallel connection of are also applicable to jωl and - / ω. Using the above geometrical construction, the relationship between voltage and c rren for various circuits can be derived. ( ) A.. ircuit ontaining only Inductor: The impedan e Z j ω L j X L is represented by the point A in the complex plane as shown in he figure.

7 OA forms an angle 8 - ALTENATING UENT Page 7 π / with the real axis indicating that the current lags the voltage by π / as shown in the above figures. Further OA ω L l Z l. So the equation for current can be written as Ι V m cos ω t - ω L π V m cos ω t XL π - ( ) A.. ircuit containing only a capacitor: The impedance Z - j / ω - j X is represented by the point F in the complex plane as shown in the figure. OF forms an angle - π / with the eal axis indicating that the current leads the voltage by π / as shown in the above figures. Further OF / ω l Z l. So the equation for current can be written as Ι V m cos ω t ω π π V m cos ω t X ( 3 ) A.. ircui containing and L in series: The impedance Z j ω L is represented by the po nt H n the complex plane as shown in the figure. OH l Z l From the figure, ω L δ tan [ ω L V cos ( ω Ι X L m t - δ ) ] tan [ X L ] ω L Here the current lags the voltage by a phase δ, as given by the above equation for δ.

8 ( 4 ) A.. circuit with and connected in series: The impedance represented by the point H in the complex plane as shown in the figure. OH l Z l 8 - ALTENATING UENT Page 8 Z - j / ω ω - j X is ω Here, δ is negative and is given by δ Ι tan ω Vm cos ( ω t δ ) ω tan X and X Vm cos ( ω t δ ) X Here the current leads the voltage by a phase δ, s giv n by the above equation for δ. ( 5 ) An A.. circuit having L and connected in series: The impedance Z j ω L - j / ω j X L - j X is represented by the point G in the compl x plane as shown in the figure. l Z l ω L - / ω Here ω L > / ω phase π /. δ π / and Ι X L - X / ω, and current lags voltage by a π V m cos ( ω t ) ω L ω If ω L < /, current leads voltage by a phase π / and δ - π / and Ι π V m cos ( ω t ) ωl ω ( 6 ) A.. circuit containing parallel combination of L and, connected in series with : If Z resultant impedance of the parallel combination of the inductor and the capacitor, Z Z Z L j ω j ωl j ω ω L

9 Z j ω ω L 8 - ALTENATING UENT Page 9 - ω equivalent impedance of the circuit, j ω L j Z Z - ω ω L If ω >, the impedance Z can be ω L represented in the complex plane as shown in the graph. Here, δ tan δ l Z l is negative and given by HD OD ω ω ω L ω L and Ι V m cos ( ω t δ ) ω ωl 8.6 r.m.s. Values of Alternating Voltage and urrent Special ammeters and v ltmeters are used to measure alternating voltage and current. These meters measure the r. s. ( root mean square ) value of the voltage and current. oot mean square of a quantity varying periodically with time means the square root of the mean of the squar s of the quantity, taken over a time equal to the periodic time. For V V m cos ωt, the av r ge value of V < V > < V m cos ωt > V m cos ωt V m cos ωt The average value of / is / and that of cos ωt is zero for one periodic time. < V > Vm V r.m.s. < V > V m Similarly, Ι r.m.s. Ι m

10 8 - ALTENATING UENT Page Series esonance The current in L series circuit is given by Ι Vm cos ( ω t ω L - - δ ) Ι Ι m cos ( ωt - δ ), where, Ι m Now, Ι r.m.s. Ι m ω Vm ω L - ω Vm ω L V ω L -.m s. ω - ω V r.m.s. Z Thus, value of Ι r.m.s. varies with ω. If ω ω 0 is taken such that ω 0 L / ω 0, then Ι r.m.s. V r.m.s. which is the maximum va ue of Ι r.m.s. The L series circuit is aid to e in resonance when the r.m.s. value of the current becomes maximum for a particular frequency, Now, ω 0 L of the given L series circuit. esonance is achieved when the imaginary part f the impedance is zero. The figure shows the Ι r.m.s. versus ω c rves for two different values of < ). The resonance curve becomes sharper as the value of reduces. Q - Factor: The sharpness of the resonance curve of the series L circuit is measured in terms of a quality known as the Q factor. The maximum power Ι r.m.s.. is obtained during resonance. The power ω 0, of the voltage source. is known as the natural angular frequency or resonant angular frequency

11 8 - ALTENATING UENT Page reduces to half of the maximum power when the Ι r.m.s. is equal to Ι r.m.s. ( max ). This situation is shown in the graph where it is found that there are two values of angular frequency, ω and ω for which power is reduced to half the maximum power. ( ω - ω ) ω is known as the half power bandwih. The sharpness of the r sonance increases with the decrease in the value of the half power bandwih, The Q factor is defined as Q ω0. ω Larger the value of Q, sharper will be the resonance curve. It can be proved that the value of ω L. Also we know hat ω0 Q ω0 ω L Thus, the Q factor depends on the compon nt o the circuit. It gives the information about the tuning of the circuit as well as the selectivity of the circuit. To tune a known frequency source, like a T V. r a radio, one has to select the right value of the inductor or the capacitor. esona ce is obtained only when both L and are present because at the time of resonance hey cancel reactance of each other. Hence, resonance never occurs in case of -L or cir uit. 8.8 Phasor Method onsider a harmonic function Ι Ι m cos ( ωt δ ). A vector of magnitude Ι m is drawn from the origin of a coordinate system a an angle of ωt δ with the X-axis as shown in the figure. The phase ( ωt δ ) changes with time, i.e., the angle between he vector shown in the figure and the X-axis keep on changing with time. The vector rotates with the angular frequency ω in the XY plane. Such a rotat ng vector is known as phasor. The X-component of the vector at any instant t gives the value of. L Ι m cos ( ωt δ ) which is the instantaneous value of the current. To add several functions like Ι cos ( ωt δ ), Ι cos ( ωt δ ),... etc., one has to take the algebraic summation of the X-components of the respective phasors. One can deal with the sine function by taking the Y-component of the vector. Now, consider the example of addition of two harmonic functions, Ι Ι m cos ( ωt δ ) and Ι Ι m cos ( ωt δ ).

12 The figure shows the vectors representing Ι and Ι and their resultant vector Ι. The amplitude of the resultant vector OP and its phase angle is φ using the law of triangle for addition of vectors. The magnitudes of Ι, Ι and Ι are given by Ι m, Ι m and Ι m respectively. The angle between Ι and Ι is δ - δ as can be seen from the figure. Now, Ι m Ι m Ι m Ι m Ι m cos ( δ - δ ) Ι m Ι m where, δ δ - δ 8 - ALTENATING UENT Page Ι m Ι m cos δ 8.9 Use of Phasor in an A.. ircuit An A.. ircuit containing only resisto : In an A.. circuit containing only a resistor, the phase difference between the voltag an current is zero ( δ 0 ). Here, current Ι is d wn along the X-direction and as δ 0, voltage is also drawn in the same direction as shown in the figure. An A.. ircuit Here, current ontaining only inductor: lags the voltage V by π / radian, or in other words, th voltage leads the current by π / radian. Hence, if Ι is rep esented along the X-direction, V will be along the Y-direction as shown in the figure. A A.. circuit containing only capacitor: Here, current Ι leads the voltage V by π / radian, or in other words, the voltage lags the current by π / radian. Hence, if Ι is represented along the X-direction, V will be along the negative Y-direction as shown in the figure.

13 L series A.. circuit: The phasor diagram of each of the component is shown in the figure. Here, since L, and are in series, the same current flows through all the three components. If V is the applied voltage, then V V L V V where, V L, V and V are the voltage drop across the inductor, the capacitor and the resistor respectively. If the current, Ι, flowing through the above circuit is represented along the X-direction, then the phasor diagram of the series circuit will be as shown in the next figure. It can be seen from the figure that V ( V L - V ) V If Ι m is the maximum current, then V Ι m, V L Ι m X L and V Ι m X V Ι m ( X L - X ) Ι m V Ι m ( X X ) L - For maximum current, V V m Ι m ( XL Vm - X ) 8 - ALTENATING UENT Page 3 V m l Z l If δ phase angle between the applied voltage and current phasors and V L > V, then the cu ren lags the voltage by a phase δ. If V L < V, then current would have led the voltage. From the above graph, tan δ - V L V V 8.0 L Oscillations Ιm XL - Ιm X Ιm XL If the two ends of a charged capacitor are connected by a conducting wire or a resistor, then it gets discharged and the energy stored in the form of electric field between its two plates gets dissipated in the form of joule heat. - X

14 8 - ALTENATING UENT Page 4 Now, consider an inductor having negligible resistance connected to a charged capacitor as shown in the figure. Such a circuit is called an L- circuit. Let Q 0 be the charge on the capacitor on the initially charged capacitor to which an inductor is connected. Let Q charge on the capacitor at time t t, and Ι circuit current during discharge of capacitor. applying Kirchhoff s law to the closed circuit, we have dι Q - L 0 But Ι dq - ( negative sign indicates that he charge on the capacitor decreases with time. ) d Q Q L 0 d Q - L Q omparing the above equation with the differential equation of the simple harmonic motion, d y - ω 0 y Q is analogous to the variab e y and ω 0 is analogous to. The solution to the above L equation would be, Q Q m sin ( ω 0 φ ) ( ) Here, Q m and φ e the constants which can be determined from the initial conditions. Putting the value of Q Q 0 at time t 0 in the above equation, Q 0 Q sin φ ( ) Differentiating equation ( ) with respect to time t, we have, Ι dq Qm ω 0 cos ( ω 0 t φ ) But at time t 0, Ι 0. 0 Q m ω 0 cos φ φ π / ( Q m and ω 0 are not zero ) Putting the value of φ π / in equation ( ) gives Q m Q 0

15 8 - ALTENATING UENT Page 5 Putting the value of φ π / and Q m Q 0 in equation ( ), we have, Q Q 0 sin ( ω 0 t π / ) Q 0 cos ω 0 t ( 3 ) This equation shows that the charge on the capacitor changes in a periodic manner. Differentiating the above equation with respect to time t, we have, dq Ι - Q0 ω 0 sin ω 0 t ( 4 ) This equation shows that the current through the inductor also changes n a periodic manner. At time t 0, the charge on the capacitor is maximum and the c rrent through the inductor is zero. The electric field intensity and hence the energy associated with the capacitor ( Q / ) is maximum. The energy associated with the magneti field of the inductor is zero. With the passage of time, the charge and hence the ene gy associated with the capacitor decrease as per the equation ( 3 ). At the same time the current through the inductor and hence the magnetic field and energy associated with it ( LΙ / ) increase as per the equation ( 4 ). It can thus be concluded that the energy of the electric field of the capacitor gets converted into the energy of the magnetic field of the nductor. At time t π / ( ω 0 ), Q 0, and Ι be mes maximum and the entire energy stored in the electric field gets converted into the ene gy s ored in the magnetic field. At time t π / ω 0, the charge on the capacitor again becomes maximum but with reverse polarity and the current in the indu tor between the capacitor and inductor in ecomes zero. This phenomenon of charge oscillating periodic manner is known as L- oscillations. These oscillations of charge esults in the emission of electromagnetic radiations which result in the decrease of ene gy associated with the L- circuit. Such a circuit is known as the tank circuit of the os illator 8. Power and Energy associated with L, and in an A.. ircuit In an A.. c rcuit voltage and current continuously change with time. In a series circuit, the nstantaneous power VΙ V m cos ωt. Ι m cos ( ωt - δ ) V m Ι m cos ωt. V cos ( ωt - δ ) m Ι m [ cos δ cos ( ωt - δ ) eal power, P Average value of instantaneous power for the entire cycle T V mιm T cos δ ωt δ ) T cos ( - T 0 0 V m Ι m. T T cos δ ω δ 0 T cos ( t - ) T 0 V m Ι m cos δ V r.m.s. Ι r.m.s. cos δ ( ) L--

16 8 - ALTENATING UENT Page 6 Special cases: ( i ) ircuit containing only resistor: δ 0 P V r.m.s. Ι r.m.s. ( ii ) ircuit containing only inductor: δ π / cos π / 0 P 0 When the current through the inductor increases, the energy from the voltage source gets stored in the magnetic field linked with the inductor and is given back to he circuit when the current through the inductor decreases. Hence power consumed by the circuit is zero. Thus, there is current in an A.. circuit containing inductor without consuming any power. ( iii ) ircuit containing only capacitor: Here, δ - π / cos ( - π / ) 0 P 0. The energy consumed in charging the capacitor is sto ed in the electric field between the plates of the capacitor and is given back to the circui when current in the circuit decreases. ( iv ) Series L-- circuit: Here, cos δ ω L - ω Putting this value of cos δ i equation ( ), we get the value of the power in the series L-- circuit. It is less than whe only resistance is present in he circuit. Its value is maximum when ω L. Thus, power in A.. circuit containing only inductor or capacitor is zero. The current flowing in such a circuit is called wattless current. 8. Transformer When powe P V Ι has to be transmitted over a long distance from the power station to th city through the cable having resistance, Ι amount of power gets lost in the form of hea which is very large. If this power is transmitted at a very high voltage, then the current Ι wil be less thereby reducing the power loss. Moreover, before supplying this power to the household, its voltage has to be reduced to a proper value. Both the increase as well as decrease of voltage can be done using a transformer. The transformers, which increase the voltage are called step-up transformers, while those which decrease the voltage are called step-down transformers. There is no appreciable loss of power in the transformers. Principle: Transformers work on the principle of electromagnetic induction.

17 8 - ALTENATING UENT Page 7 onstruction: The figure shows the construction of a step-up transformer and its circuit symbol. Two coils of wire are wound very close to each other on the rectangular slab of iron. The rectangular slab of iron is made up of several layers of iron plates to minimize the eddy currents and power loss due to them. One of the coils called primary coil, P, other coil is called the secondary coil, S. is connected to the A.. source. The In the step-up transformer shown in the figure, the numb of turns of the primary coil is less than that of the secondary coil and is made up of thicker copper wire. In the step-down transformer, the arrangement is opposite to that of a s ep-up transformer. The permeability of the material of the slab is h gh A a result, all the flux generated by the primary coil remains confined to the core and gets inked to the secondary coil. Therefore, the fluxes, Φ and Φ linked with the prima y and the secondary coils are proportional to the respective number of turns N and N o the coils. flux Φ linked with secondary co l flux Φ linked with primary co l According to Faraday s law, d Φ the emf induced in the p imary coil, ε and d Φ the emf induced in the primary coil, ε N From equation ( ), we have, Φ Φ N ε N ε ε or N r N ε N No. of secondary turns N No. of primary turns N d Φ is known as the transformer ratio. For a step-up transformer, r > ( ) N d Φ N A the power loss in the transformer is negligible, power input power output ε Ι ε Ι ε ε Ι Ι N N r The above result is valid for an ideal transformer having no power loss. Actually, some power is lost in the primary coil in the form of heat, some in the magnetization and demagnetization of the iron core and some in the form of eddy currents formed on the surface of the iron core. As a result, the output power is less than the input power.

Physics 142 AC Circuits Page 1. AC Circuits. I ve had a perfectly lovely evening but this wasn t it. Groucho Marx

Physics 142 AC Circuits Page 1. AC Circuits. I ve had a perfectly lovely evening but this wasn t it. Groucho Marx Physics 142 A ircuits Page 1 A ircuits I ve had a perfectly lovely evening but this wasn t it. Groucho Marx Alternating current: generators and values It is relatively easy to devise a source (a generator