Asist. Prof. Dr. Aytaç Gören Asist. Prof. Dr. Levent Çetin 30.10.2012
Contents Alternating Voltage Phase Phasor Representation of AC Behaviors of Basic Circuit Components under AC Resistance, Reactance and Impedance Power in AC Circuits 2
Alternating Voltage If the direction of current and voltage value of a source change due to time, then it is called an AC voltage. The grid uses AC, since the generation and converting to mechanical energy of it is easy and efficient, moreover the loss in tranfer is less than DC. 3
Alternating Voltage 4
Alternating Voltage The change of the voltage in grid is defined with a sine function. V AV ( t) Vmax sin( 2 ft) The parameters in this equation are; a) f is oscillation (frequency) b) V max is the maximum value of the voltage (amplitude). Frequency or oscillation of a signal is the value of repetition observed in a changing signal in unit time. In other words, frequency refers how often an evvalue of the fequency of the grid in Turkey is ent occurs. The 50 [Hertz]. [1/s ] is also used instead of [Hertz]. The time needed to complete one cycle is a period. Period is 1/f, so the period of the grid is 0.02 [s] in Turkey. Voltage of the grid. 5
Alternating Voltage 6
Alternating Voltage The maximum value or the amplitude of the alternating voltage is the maximum value of the sine wave during one period. This value is approximately 311 [V] for the grid. But, this value is not used as nominal value. Instead, the RMS value of this sine wave is used. The RMS value may be said as the equivalent value of an alternating voltage to direct voltage. The effective value of a signal is: f RMS ( t) 1 T T 0 f 2 ( t) dt V V max RMS. 707 V max 2 0 For the electrical grid, the effective value is app. 0.707 times of the maximum voltage value of the grid and it is 220 [v]. 7
Alternatif Gerilim 07.10.2011 8
Phase An important point in operations of two time dependent signals is whether they are synchronized or not. For electrical definitions, two voltage signals or two corrent signals or one voltage with one current signals can be either synchronized or with phase diference. If two signals are synchronized, they both pass the zero points and the maximum value points at the same time. 9
Phase If two signals pass the zero points and maximum value points in different moments then a phase shift occurs. Phase or phase shift is the time difference between two signals. The phase shift or phase is denoted with degree in sine functions. If one of the two sine functions is accepted as reference signal; the value of angle of the signal that is not the reference signal when the reference signal reaches zero is the phase value. Accordingly, +45 +90 +180 and 0 degrees of phase shifts are shown in the figure below. 10
Phase 11
Phasor Representaion of AC In order to define the affects of the alternating current to a circuit, frequency, amplitude and phase need to be known. Frequency depends on the electrical grid, so the country or region. So, the voltage/current functions can be defined depending on two parameters. One of the choice in modeling alternating current / voltage is to represent them using rotating vectors (i.e. phasors). 12
Phasor Representaion of AC The projection of a rotating vector around origin in cartesian coordinate system is a sine function as can be seen in figure above. The length (or the radius) of the rotating vector is the amplitude of alternating voltage in this representaion. Similarly, the angle between the vector and the horizantal axis is the phase value (θ). The angular velocity of this rotating vector is the frequency of alternating voltage. 13
Phasor Representaion of AC 14
Phasor Representaion of AC A complex number is a mathematical quantity representing two dimensions of magnitude and direction. Representation of alternating voltage as a rotating vector indicates a complex number in means of mathematics. As known, complex numbers has two parts called real and imaginary which are represented in complex plane. The common representation of a complex number in cartesian form is equation (1) whereas phasor representation that is representation of a complex number as a rotating vector and more used in electrical circuit analysis is the equation (2) below. z a bi (1) z r (2) 15
Phasor Representaion of AC z a bi z r b ( ) a 1 tg 2 2 r a b 16
Phasor Representaion of AC Four basic operations in complex numbers can be seen below. z z a a ) ( b b ) i 1 2 ( 1 2 1 2 z z a a ) ( b b ) i 1 2 ( 1 2 1 2 z z 1 z2 r1 r2 1 2 1 z2 r1 r2 1 2 Implementation of complex number arithmetics to voltage/current signals is examination of total effects of voltage/current sources which have different phases. Before calculating this effect, the state of the two rotating vectors, which have different phases, according to each other should be understood. This relation might be described as vectors which have the same starting point but have different angles respect to the horizontal line. 17
Phasor Representaion of AC 18
Phasor Representaion of AC Created by a combination of current/voltage sources connected to the same circuit is determined by the complex numbers, addition and subtraction operations. 19
Behaviors of Basic Circuit Components under AC 1 2 3 Resistance (R) Coil (L) (Inductance) 20 Capacitance (C) (Capacitor)
Behaviors of Basic Circuit Components under AC Resistance (R) Ohm s Law can be used for resistance under the influence of alternating voltage. V V max sin( wt) V R I V I R max sin( wt) According to equations above, there is no phase shift between current and voltage on a resistor. Nevertheles, the amplitude changes due to Ohm s Law. 21
Behaviors of Basic Circuit Components under AC Coil (L) (Inductor) In contrast with resistors, coils under alternating voltage resists against alternating current. The voltage on a coil (the voltage measured between two terminals) can be calculated using Lenz Law. di( t) V( t) L dt If this equation is studied considering the alternating current, the ralationship between the current and voltage might be predicted. 22
Behaviors of Basic Circuit Components under AC Coil (L) (Inductance) In contrast with resistors, coils under alternating voltage resists against alternating current. The voltage on a coil (the voltage measured between two terminals) can be calculated using Lenz Law. di( t) V( t) L dt If this equation is studied considering the alternating current, the ralationship between the current and voltage might be predicted. I( t) Imax sin( wt) di( t) V( t) L Lcos( wt) dt 23
Behaviors of Basic Circuit Components under AC 24
Behaviors of Basic Circuit Components under AC This result shows us that there is a 90 degrees of phase shift between voltage and current on a coil under AC. The voltage leads current by phase angle of 90 degree. The phase shift results with negative electrical power. Negative power denotes that the coil transfer power to the circuit. The resistance of coils changes due to time or frequency. This is called as reactance (inductive reactance X L ) for this reason. X L wl X L 2 fl 25
Behaviors of Basic Circuit Components under AC Ohm s Law might be implemented easily to alternating current circuits using quantity, the reactance. In that case, the calculations should be made using complex numbers instead of scalars. X V I XL 2 60 10 2 3.7699 I V X 10 3. 7699 2. 6526 A Now, let us calculate the influence of total resistance of a resistor and a coil adding a 5 [Ohm] resistor to this circuit. 26
Behaviors of Basic Circuit Components under AC Resistor value: R 5 0j Inductive reactance of the coil The total effect is called as impedance. 27 X L 0 3. 7699 j Z R X 5 3.7699 j 6.262 37. 016 L Commonly, impedance in alternating voltage circuits is the corresponding definition of resistance. Besides, it can be used as resistor in Ohm s Law as mentioned above.
Behaviors of Basic Circuit Components under AC V Z I Z R X 5 3.7699 j 6.262 37. 016 L 10 0 I 1.597 37. 016A 6.262 37.016 28
Behaviors of Basic Circuit Components under AC 29
Behaviors of Basic Circuit Components under AC Parallel circuit 30
Behaviors of Basic Circuit Components under AC First state: 31
Behaviors of Basic Circuit Components under AC Implementing the Ohm s Law; 32
Behaviors of Basic Circuit Components under AC Implementing the Ohm s Law; 33
Behaviors of Basic Circuit Components under AC The impedance equation of parallel circuits: 34