### Power in AC Circuit

Instrument Transformer and Power Management (P1) Course
Chapter 3: Power in AC Circuit
3.1 Inductance (L) :
Is the property of a circuit that opposes any change in current, so when an alternating current (A.C.) voltage applied to an inductor the current through it will take time to build up. For this reason the current through an inductor is said to lag the voltage by 90 degrees (one quarter of cycle).

Capacitance (C) :
Is the property of a circuit that opposes any change in voltage, so when an alternating current (A.C) voltage is applied to capacitor the voltage across it will take time to build up, for this reason the current through the capacitor is said to lead the voltage by 90 degrees ( one quarter of cycle).
In summary, inductors and capacitors both offer position to current flow, and they create a difference angle between the voltage and the current (commonly called phase shift). Because of this this phase shift, their opposition to current flow cant be called resistance. It given a new name, "Reactance", has the symbol X, and is measured in ohms.
- The reactance if an inductor is given by the formula:
XL = 2π F L
Where F is the frequency and L is the inductance in henry.

- The reactance of the capacitor is given by the formula :

Where F is the frequency and C is the capacitance in farads.

Impedance :
The total opposition of current flow in a circuit is given by the "impedance". It is the combination of resistance in the circuit and is measured in ohms.
Impedance has a magnitude associated with it, as well as phase angle. These tow quantities describe the opposition to current flow (the magnitude) and the time delay due to the reactance (phase angle). The symbol for impedance Z

R-X diagram :
Impedance is normally shown graphically on an R-X plot diagram. This is a tow dimension that can display both component of the impedance in one plane. The resistance is shown along the "X-axis", while the reactance is shown along the "Y-axis". Z is found by adding R and X vectorially.
As an example, consider a circuit where 10H coil is connected in series with a 300 ohm resistance. The total impedance of the circuit is the vector sum of the resistance and the reactance.
The resistance is given as 3000 ohm, and the reactance of the 50HZ at 50HZ would be :
XL = 2 x 3.142 x 50 x 10 = 3142 ohm
So, Z = 3000 + j 3142 ohms
Plotting these values on an R-X diagram , one gets :

Because the two vectors from a right angle . the magnitude of Z will be :

І Z І = (R2 + X2) = 4344 ohm.

And the phase angle is given by = tan-1 І X/R І = tan-1 (3000/3142) = 46° . The X-R diagram is very effective tool that is used to analyze complex impedances.
In future protection courses, it will be used to show transmission line impedances. and the relay characteristics for the line protection relays. For this course we will use it to analyze the test circuits.
Phasor diagram :
The polar coordinate system is useful when one wants to represent the relationship between voltage and current in an A.C. circuit. In this case, we are interested in magnitude and relative phase angle, so all we need is a system that shows these two quantities. Generally the current is drawn to one scale and the voltage to another, giving a plot that clearly shows the phase angle between them.
A system that shows vectors in a plan and describe them with magnitude and phase angle is known mathematically as a polar coordinate system. In the electrical field polar plots are given a special name, phasor diagram.
Assume that the previous circuit was energized with 220V. The current would lag the voltage by the angle  of the impedance and would be of the impedance and would be of magnitude ;
I = V/Z  = 220 / 4344 = 51 mA.
The phasor diagram would be as shown

Not that clock-wise angles are normally considered to be lagging as in this example. A  capacitive load with the same reactance would have leading current, and the diagram would appear as follows ;

The phasor diagram is used frequently in the analysis of A.C. power systems.
Power in an A.C. circuit :
When an A.C. voltage is applied to circuit, current will be governed by ohm's law.
The magnitude of the current is determined by dividing the applied voltage by the magnitude of the impedance. Z from DC circuit theory, the product of voltage across and current through a circuit is the power that is flowing. In an A.C. circuit this is not always true. So this type of power is called the apparent power (because when you measure the voltage and current it is the power that appears to be flowing)

Apparent power (VA) = voltage x current

Real power flow in an A.C. circuit is determined by the resistance in the circuit. Only resistors use power from of heat,..etc.) while capacitors and inductors do not (they stored energy in electric and magnetic fields and release it at another time).
Since the voltage across a resistor is always in phase with current through it, power in circuit is, definition, the product of voltage and the component of current that in phase with the voltage. It measured in watts.
The wattmeter is a special test meter that is designed to directly measured the real power flow in a circuit. A typical connection is as follows ;

An easy way to remember the relationship between real, apparent, and reactive power is to remember the power triangle :

It is important to know these relationship in order to be able to understand the measurement power.