alternating current - deukisi.deu.edu.tr/aytac.goren/elk2015/w2c.pdf · alternating current 3...

Alternating Current

Alternating Current Asist. Prof. Dr. Aytaç Gören

Asist. Prof. Dr. Levent Çetin

30.10.2012

Alternating Current

Contents

Alternating Voltage

Phase

Phasor Representation of AC

Behaviors of Basic Circuit Components under AC

Resistance, Reactance and Impedance

2

Power in AC Circuits

Alternating Current3


Capacitor

Capacitors react different due to the voltage level applied to them underalternating current. If the voltage level applied is greater than the voltageon a capacitor, the source charges the capacitor; in opposite case,capacitor behaves like a source.The current equation for a capacitor is:

dt

tdvCti

)()(



wCXc

1

fCXC

2

1

The phase difference between voltage and current is 90 degrees on a capacitor or in other words, current leads voltage 90 degrees. This caseresults with negative electrical power which means that capacitor transfers power to the circuit (i.e. Capacitor discharges its electrical charge). The ‘resistance’ of the capacitors change due to the frequency of the alternating voltage. The higher frequency of the AC signal, the more easily that signal will pass through the capacitor. Thus, this is called as capacitive reactance, Xc.



Let us examine the behavior of the capacitor in the circuit mathematically.

The current can be found using the capacitive reactance equation and the implementation of Ohm’s Law to AC easily.



The current of the AC sourceleads the voltage of the source90 degrees. The resistance effectof the capacitor to AC source iscalculated considering this.



jR 05 . jXc 5258260

32579993265258265 .. . jXRZ c

Adding a 5 [Ohm] resistor to this circuit, let us calculate the total effect of the resistor and the capacitor.

The total resistance in this circuit is found as:



Implementing the Ohm’s Law to the circuit, the current is calculated. In this analysis, the phase of the source is always zero (0).

As it can be considered easily, the phase shift is 79.325 degrees in this circuit whereas in the circuit that has only one capacitor it was 90 degrees.


Behaviors of Basic Circuit Components under ACUsing phasor diagrams,

Let us examine the last table carefully. The current and the voltage on the resistor is on the same phase as it is

mentioned. However, the current on a capacitor leads voltage by 90 degrees.



Parallel Resistor – Capacitor Circuit



Resistor, Reactance and Impedance

The resistance against the current can be in three types:

1. Resistor: It is the friction of electrons during motion. Its symbol is “R” andunit is [] (i.e. [Ohm]). It does not form any phase shift.

2. Reactance: It is the inertia of electrons. It occurs if there is a change involtage or current values (if an electric or magnetic field occurs). Thecapacitor and inductor are the main circuit components which this influenceis highly distinct. If there is a reactance effect in a circuit, there is also phaseshift. If the component is a capacitor, the current leads voltage by 90degrees whereas if it is an inductance, the current lags voltage by 90degrees.



3. The impedance, is the strain against the current in an electrical circuit. In aother words, it is the resistance against the motion of electrons. Impedance is the total resistance and reactance effects of all components. The resistance in DC circuits is the impedance in AC’s. The AC implemented Ohm’s Law can be seen as below. Please consider that all the quantities are in complex number form in the equation below :

I

VZ

Like Ohm’s Law, other laws (Kirrschoff’s, grid theorems, etc.) used incircuit analysis can be also implemented in AC in condition of usingcomplex numbers.It is the power calculations that the AC and DC calculations differes.The next subject will be about this case.



Resistor (R)Coil (L)

(Inductor)Capacitor(C)(Condenser)



Serial R-L-C Circuits



It should be considered that the amplitude of the voltage on the capacitor is greater than the voltage supplied to the circuit. The reason is that the influence of the reactance of the coil effects just the opposite according to the influence of the reactance of the coil to the circuit. The influence of the impedance in the whole circuit is smaller than the influence of impedance of any single component. This case causes higher voltages on single components. However, this does not change the Kirchhoff’s Law about the the total voltages in a circuit(0+0j).



Parallel R-L-C Circuit



Serial and Parallel Circuits



The calculation of impedance in this circuit should be completed stepby step. First, serial connection branch of C2 and L, afterwards theparallel branch of resistor and last the serial capacitor effects shouldbe calculated.



Since it was mentioned, there is a phase shift between current and voltage in AC circuits. The reason is the impedance as it was stated. There are three definitions in AC circuits which are related with power. These are:

True power (active power),Reactive power, Apparent power.



The reactive power that is referring to the influence of reactance is:

The apparent power is the power related with the impedance:



The power quantities are scalar quantities.

It is a prependicular triangle that we get if we draw the calculated values,

considering the 90 degrees of direction angle between the resistor

and the reactance and phase shift in the circuit. This triangle is called as

‘Power Triangle’.

As it is seen in this triangle, a part of the power is lost in an AC circuit. The generated effective power is just as the true power.

Power factor is the cosine of the angle between the true and apparent powers (cos). This value is equal to 1 in only circuits

those have just resistors. But if there is a reactance, then the value is between 0 and 1.



To examine the effect of the reactance, let us calculate the

P, Q and S powers in this circuit.

The power factor of this circuit maybe found using the definition below.



The power factor value shows that the 70.5 % of the power used fromthe grid is served for the purpose. This situation is not wanted. So, incircuit design stage, it must be noted that the power factor isapproximately equal to 1. For this reason, the capacitive andinductive ractance values should be equal to each other. If this is notpossible, a capacitor or an inductor should be externally added to thecircuit. This improvement is called as compensation.

Now, let us calculate the reactance of the same circuit.



The ractance in this circuit is inductive. In othe words, the coil is the reason of ractance.To compensate this, let us add a capacitor to the circuit. Using the calculations below, the capacitor value that can form a capacitive ractance to generate the same inductive ractance value but in opposite direction might be found.



The capacitor value found is not a standart value for

capacitors, so the closest standart value shold be

chosen (22 µF) and connected in parallel with

the circuit. Now let us examine this case:

The new impedance value is:



And the new power factor is:

As seen above, this improvement made the power factor closer to 1.Besides, the current is decreased. These two results indicate that the circuitis improved in means of efficiency and economic.

Alternating Current

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