Electrical engineering
Oscillating circuit; resonance
Harmonics in electric grid
Harmonics are sinusoidal waves summarized with the fundamental (basic) frequency of 50 Hz (i.e 1t harmonica = 50 Hz, 5th harmonica = 250 Hz).
Any complex form of sinusoidal wave can be decomposed into frequency components, thus complex sinusoidal wave is the sum of a certain number of even and odd harmonics with smaller or large magnitudes.
Harmonics are prolonged disturbance or distortions in the electricity grid, that have different sources and displays such as mpulses, distortions phases, throws and failures that can be categorized as transient disturbances.
The origin of harmonics:
Harmonics are the result of the operation of electrical equipment with nonlinear characteristics. Main contribution for the formation of harmonics belongs to the following equipment:
1. Saturable devices such as transformers, motors, generators, etc. Harmonic amplitude on these devices is usually insignificant compared with elements of power electronics and welding equipment, provided that there is no saturation. Welding, arc equipment: electric arc furnaces, welding machines, lights (mercuryvapour lamp — mercury lamps, fluorescent lamps).
The negative effects of the presence of harmonics in the grid:
• increase of currents and voltage of high harmonics due to the parallel and serial resonances;
• deprecation of electrical equipment insulation and reduction as a consequence of electrical equipment service life;
• false operation of the equipment
• decline of efficiency of the generation processes, transmission, electricity use;
• deterioration of the electric equipment work  increase of heat, noise, energy consumption.
Oscillating circuit; resonance
Oscillating circuit is the circuit containing inductance coil and condenser, in which electric oscillations may occur. If one charges the condenser voltage V0 at definite moment, the energy concentrated in the electric field of the condenser is equal to
Е_{с} = CV_{0}^{2}/2,
Where C  condenser capacity
During the condenser discharge, current I flows in the coil, this current will increase until the condenser is completely discharged. At this point, the electrical energy of Oscillating circuit Ec = 0, and the magnetic energy, which is concentrated in the coil,
E_{L}=LI_{0}^{2}/2,
Where L  coil inductance, I0 – maximum current value
Then current in the coil begins to drop, and the voltage on the condenser increases in absolute value but with the opposite sign. After some time, the current through the inductance will stop and the condenser will be charged to a voltage  V_{0}. The energy of Oscillatory circuit will concentrate again in a charged condenser. Then the process is repeated, but with the opposite direction of the current. The voltage of condenser coating is changed according to the law, V = V_{0} cos w_{0}t and current in the inductance coil I = I_{0} sin w_{0}t, i.e. Own harmonic oscillations of voltage and current with the frequency w_{0} = 2 p/T_{0} are generated in the Oscillatory circuit, where T0 is the period of oscillations equal to T0 = 2p. The energy transfer from the electric field of condenser to the magnetic field of the inductance coil and back is initiated twice for the period in the Oscillatory circuit.
However in real Oscillatory circuit part of the energy is lost. It is spend on heating of the coil wiring, which has resistance, on electromagnetic waves generation to environment, it also includes losses in dielectrics and that leads to decay of oscillations. The oscillation amplitude declines gradually, so that the voltage at the condenser coating is changed according to the law:
V=V_{0}e^{}^{d}^{t}coswt, where the coefficient d = R/2L  index (coefficient) of decay. and w = is decay frequency. Thus, losses lead to changes not only in amplitude, but also in their period T = 2 p/w. The quality of an Oscillating circuit is usually characterized by tuned circuit Qfactor . The value of Q determines the number of oscillations, which will be made by the Oscillating circuit after a single charge of the condenser before the oscillation amplitude will begin to decrease in e times (e  base of natural logarithms).
Tuned circuit Qfactor
The tuned circuit Qfactor is a characteristic of oscillatory system, determining the resonance strip and showing how bigger is the energy store in system than energy losses for one period of oscillations.
The tuned circuit Qfactor is inversely proportional to the rate of oscillation decay in the system. So, the higher Qfactor of oscillatory system is the less energy losses are and the slower is oscillation decay.
Devices with RMS function
Speaking about the values of alternating current, we usually mean the average effective allocated warmth, or rootmeansquare (RMS) value of the current.
This value is the equivalent of direct current value, which will cause the same heating effect as the effect of the measured alternating current. The most common way to measure this "rootmeansquare" value of a current is current rectification with the help of the measuring instrument, determining the average value of rectified signal and multiplying it by a coefficient of 1.1. This coefficient takes into account the constant value that is equal to the ratio between the average and RMS values of ideal sine wave.
There is also measure method at the peak (amplitude) value. If the measured signal has the correct form of a sine wave, then the real value is equal to 0,707 from peak value.
However, if sine wave has deviations from the ideal form these coefficients are no longer applied. That is why, the measuring instrument with average readings often give incorrect results at current measurements in modern power grids.
Method of RMS (root mean square power) signal measuring does not have all above mentioned disadvantages.
Table. The reliability of the measurement depending on the measured signal form (approximate).
Measuring method 
Coefficient (correction) 
Sine wave 
Rectangle 
Run outs 
Cutoff sine wave 
Triangle 



Measuring at peak value  Peak * 0,707 
100% 
82% 
184% 
113% 
121% 
Measuring at average value  Average * 1,1 
100% 
110% 
60% 
84% 
96% 
True RMS  RMS converter 
100% 
100% 
100% 
100% 
100% 
Reactive power
Reactive power is the value that characterizes the workload generated in electrical devices by energy fluctuations of the electromagnetic field in the chain of the sinusoidal alternating current, and it is equal to the product of the operating values of voltage U and current I, multiplied by the phase shift angle φ between them: Q = U * I * sin φ (if the current is lagging behind voltage, phase shift is considered to be positive, if the current is leading  negative).
Or in other words: The consumers of electricity, in which magnetic field is generated (motors, throttles, transformers, induction heaters, welding generators), cause lagging of the current from voltage (phase shift) due to the presence of inductance. Lagging of the current makes current to stay unchanged through the inductive load for some time after the tension became negative. During this time, the current and voltage lead to the formation of negative energy, which is returned back into the grid. During restoring of an identical sign of current and voltage, the same amount of energy is used for creation of a magnetic field in an inductive load. These fluctuations of electromagnetic field in alternate current circuits are called reactive power.
Table 1. Expected effect of electricity saving with compensation of reactive power.
cos(φ)_{1}, without compensation 
cos(φ)_{2}, with compensation  The decrease in current magnitude and general power, %  The decrease in heat loss magnitude, % 

Power coefficient
Power factor or cos φ of electric grid is called ratio of actual power to total load power of the estimated section:
cos φ = P/S,
where
φ — power coefficient
P — actual power W
S — total power WA
Power coefficient can be determined though calculations or measured with special measuring device. Only when load is of active nature completely, cos φ is equal one. Basically, actual power is lesser than total power and that is why power coefficient is less than one. The higher _____ is, the higher is inductance in measured consumer. Cos φ can be equal 0,9 for consumers, which have inductance loads, for example motors.
The power coefficient is equal one for the consumers such as heating elements. For example power coefficient cos φ=1 for the light lines, in which only incandescent light bulbs and halogen lamps are used. If the light lines consists only from fluorescent bulbs, cos φ= 0,92.
cos φ = P/S,