AC Notes

 2015 Yeow Kok Han Page 1 Alternating Current 11 BBaassiiccss ooff AACC An alternating current is a current whose magnitude and direction varies periodically. We focus on sinusoidal a.c. because any given non-sinusoidal current can be synthesised from a selection of sinusoidal alternating currents of appropriate weightings and frequencies. Period T the time taken for one cycle of variation. Frequency f the number of cycles per unit time. Peak value the maximum magnitude of current I0 or voltage V0. Peak to peak value the magnitude of the interval between the positive peak and negative peak. 2I0 or 2V0. 22 PPoowweerr && RRMMSS vvaalluuee TThhee rreepprreesseennttaattiivvee mmeeaann ppoowweerr P = IV, hence if I and V both vary with time as shown (in phase for resistive load), the resultant power is given by the product of I and V moment by moment, resulting in a sin2 variation. A typical a.c. has frequency 50 Hz, so the instantaneous power fluctuates at 100 Hz. For many applications such as lighting or heating, the fluctuation is not noticeable. What we observe is the mean effect, hence it is more useful to measure the mean power and regard it as representative or effective power of an a.c.. An alternating current is a current whose magnitude and direction varies periodically. In contrast, a d.c. doesn’t change direction. For a fluctuating power of an a.c., the mean power is more useful & representative. I t I0 -I0 2T T 0 t V0 V -V0 2T T 0 t I0V0 P 2T T 0 P0 I0V0 I = I0 sint V = V0 sint P = I0V0 sin2t time I0, V0 Current or voltage -I0, - V0 2T T 0

 2015 Yeow Kok Han Page 2 TThhee rr..mm..ss.. vvaalluuee ffoorr ccuurrrreenntt aanndd vvoollttaaggee For sinusoidal a.c. current or voltage, is the mean value a use ful representative value? No, because the mean is zero for any amplitude so it cannot t ell us its effectiveness in producing useful work . What value then should we look at? Consider the rate of heating in a resistor, P = I2R (note P and I2 are time varying). The representative mean power <P> is = < I2>R, where <I2> is the mean square current. For the same resistance R, i f a constant current Ic gives a power Pc which is = <P>, then we can take Ic to be a good representative or effective current of the a.c. since Ic essentially produces the same outcome. As Pc = <P>, Ic 2 = <I2> and so Ic =  2I . This constant Ic that is representative of the effectiveness of the a.c. is formally known as root mean square or r.m.s current. A similar consideration using P = V2/R leads to the r.m.s. voltage  2V . Hence, the r.m.s. value of an alternating current is defined as How is the r.m.s. value calculated? Looking at the origin of r .m.s. value, it is not difficult to see that it is calculated by firstly squaring the a.c. current or voltage , then finding the mean value over time and finally taking square root of that mean. For sinusoidal a.c., a graphical way to find the r.m.s. value is as follows: The second graph shows the result of squaring. The third graph shows the exploitation of the symmetry to get a mean value of ½ I0 2. Finally, taking the square root of that yields r.m.s. value of I0 / 2 . Hence, for sinusoidal a.c. and resistive load: Instantaneous Power Peak Power Mean Power P = IV = ( I0 sint )( V0 sint ) = I0V0 sin2t where P varies with t P0 = I0V0 = I0 2R = V0 2/R <P> = IrmsVrms = I0V0/2 = Irms 2R = I0 2R/2 = Vrms 2/R = V0 2/2R <P> = P0/2 R.M.S. values are effective or representative constant values of the a.c. current or voltage. The r.m.s. value of an alternating current/voltage is defined as a constant current/voltage which produces the same effective rate of heating (mean power) as the alternating current/voltage for the same resistance. Steps to calculate rms value for fluctuating a.c.: 1 square the a.c. 2 mean 3 square root For sinusoidal a.c. only: Irms = 2 0I Vrms = 2 0V Power for a.c., resistive load: - varies with time - peak value is obtained from using peak current and voltage - mean value is obtained using r.m.s. values <P> = P0/2 A constant current which produces the same effective rate of heating (mean power) as the alternating current for the same resistance. I0 2 I2 2T T I2 = I0 sin2t t I0 2 I2 I2 = I0 sin2t ½ I0 2 2T T t I I0 -I0 2T T I = I0 sint t

 2015 Yeow Kok Han Page 3 33 AACC iinn TTrraannssffoorrmmeerrss A t ransformer is a device that convert s an alternating voltage to another alternating voltage of different magnitude using electromagnetic induction (Faraday’s law). It consists of 2 coils, a primary coil and a secondary coil, wound on an iron core. The main function of the iron core is to drastically increase the flux linkage between the primary and secondary coils and thereby improve the efficiency of conversion. It is useful here to visualise flux as the total number of field lines given by BA where B is the number of lines per unit area. For ideal transformer: Also, Power in = Power out IpVp = IsVs Therefore, Real transformers have efficiency of about 94% to 99% and some r easons for the lost power are: 1 Heating (I2R) in the windings. 2 Induced currents (eddy currents) in the core which also lead to heating. 3 Flux leakage (not all the flux from the primary coil are channelled to the secondary coil). One important use of transformers is to step up the a.c. voltage before transmitting electrical power over long distance transmission lines a nd then step down to smaller voltages near to users. This is to exploit the fact that when stepping up the voltage, the current is correspondingly reduced, thereby reducing the loss due to heating in the transmission lines. A transformer converts an a.c. voltage to another a.c. voltage of different magnitude using electromagnetic induction. The iron core channels and greatly magnifies the magnetic flux linking the coils. The turns ratio Np/Ns is = Vp/Vs. Having a turns ratio > 1 steps up the voltage and steps down the current by the same factor Np/Ns. For ideal transformer, IpVp = IsVs. If say 90% efficient, then 0.9 IpVp = IsVs Primary coil Secondary coil Iron core laminated to reduce eddy currents A different kind of core where the primary and secondary coils are wound one over the other around the central column. This has less flux leakage and better efficiency than the above core. © Mtodorov 69/ Wikimedia Commons / CC-BY-SA-3.0 / GFDL Np Vp Vs Ns Np - number of turns in primary coil Ns - number of turns in secondary coil s p s p V V N N  p s s p I I V V 

 2015 Yeow Kok Han Page 4 44 RReeccttiiffiiccaattiioonn -- AACC ttoo DDCC Some devices (e.g. computer) need d.c. to work. The process of conv erting a.c. to d.c. is called rectification and the circuit that does it is called a rectifier. Diodes are typically used in rectifiers. They are devices which behave like an infinite resistance in forward bias and zero resistance in reverse bias, thus allow current to flow during forward bias but not reverse bias. Picture to the right shows the symbol for a diode as well as how semiconductor diodes may look like. The I -V characteristic s of a n ideal and semiconductor diode are as shown below: A half wave re ctifier circuit, its input and corresponding output voltages are shown below. Extra beyond syllabus material: Below is a full -wave rectifier circuit which produces a full wave rectified d.c. with twice the power of the hal