Harmonic studies are used to analyse harmonic situations. They are aimed at detecting resonance and calculating harmonic currents, voltages and distortion levels/factors. This chapter presents a simple approach to the analysis and solution of harmonic problems in industrial power systems. Background material is presented. A practical example is used to illustrate the concepts and outline the data requirements and procedures for a harmonic study. Practical considerations are further provided.
In an electric power system, a harmonic of a voltage or current waveform is a sinusoidal wave whose frequency is an integer multiple of the fundamental frequency. Harmonic frequencies are produced by the action of non-linear loads such as rectifiers, discharge lighting, or saturated electric machines. They are a frequent cause of power quality problems and can result in increased equipment and conductor heating, misfiring in variable speed drives, and torque pulsations in motors and generators.
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Harmonics in power systems are generated by non-linear loads. Semiconductor devices like transistors, IGBTs, MOSFETS, diodes etc are all non-linear loads. Further examples of non-linear loads include common office equipment such as computers and printers, fluorescent lighting, battery chargers and also variable-speed drives. Electric motors do not normally contribute significantly to harmonic generation. Both motors and transformers will however create harmonics when they are over-fluxed or saturated.
Non-linear load currents create distortion in the pure sinusoidal voltage waveform supplied by the utility, and this may result in resonance. The even harmonics do not normally exist in power system due to symmetry between the positive- and negative- halves of a cycle. Further, if the waveforms of the three phases are symmetrical, the harmonic multiples of three are suppressed by delta () connection of transformers and motors as described below.
Power is supplied by a three phase system, where each phase is 120 degrees apart. This is done for two reasons: mainly because three-phase generators and motors are simpler to construct due to constant torque developed across the three phase phases; and secondly, if the three phases are balanced, they sum to zero, and the size of neutral conductors can be reduced or even omitted in some cases. Both these measures results in significant costs savings to utility companies. However, the balanced third harmonic current will not add to zero in the neutral. As seen in the figure, the 3rd harmonic will add constructively across the three phases. This leads to a current in the neutral wire at three times the fundamental frequency, which can cause problems if the system is not designed for it, (i.e. conductors sized only for normal operation.)[2] To reduce the effect of the third order harmonics delta connections are used as attenuators, or third harmonic shorts as the current circulates in the delta the connection instead of flowing in the neutral of a Y- transformer (wye connection).
Voltage harmonics are mostly caused by current harmonics. The voltage provided by the voltage source will be distorted by current harmonics due to source impedance. If the source impedance of the voltage source is small, current harmonics will cause only small voltage harmonics. It is typically the case that voltage harmonics are indeed small compared to current harmonics. For that reason, the voltage waveform can usually be approximated by the fundamental frequency of voltage. If this approximation is used, current harmonics produce no effect on the real power transferred to the load. An intuitive way to see this comes from sketching the voltage wave at fundamental frequency and overlaying a current harmonic with no phase shift (in order to more easily observe the following phenomenon). What can be observed is that for every period of voltage, there is equal area above the horizontal axis and below the current harmonic wave as there is below the axis and above the current harmonic wave. This means that the average real power contributed by current harmonics is equal to zero. However, if higher harmonics of voltage are considered, then current harmonics do make a contribution to the real power transferred to the load.
The even harmonics of a distorted (non-sinusoidal) periodic signal are harmonics whose frequency is a non-zero even integer multiple of the fundamental frequency of the distorted signal (which is the same as the frequency of the fundamental component). So, their order is given by:
The odd harmonics of a distorted (non-sinusoidal) periodic signal are harmonics whose frequency is an odd integer multiple of the fundamental frequency of the distorted signal (which is the same as the frequency of the fundamental component). So, their order is given by:
The fundamental component is an odd harmonic, since when k = 1 {\displaystyle k=1} , the above formula yields h = 1 {\displaystyle h=1} , which is the order of the fundamental component. If the fundamental component is excluded from the odd harmonics, then the order of the remaining harmonics is given by:
The triplen harmonics of a distorted (non-sinusoidal) periodic signal are harmonics whose frequency is an odd integer multiple of the frequency of the third harmonic(s) of the distorted signal. So, their order is given by:
The positive sequence harmonics of a set of three-phase distorted (non-sinusoidal) periodic signals are harmonics that have the same phase sequence as that of the three original signals, and are phase-shifted in time by 120 between each other for a given frequency or order.[7] It can be proven the positive sequence harmonics are harmonics whose order is given by:
The fundamental components of the three signals are positive sequence harmonics, since when k = 1 {\displaystyle k=1} , the above formula yields h = 1 {\displaystyle h=1} , which is the order of the fundamental components. If the fundamental components are excluded from the positive sequence harmonics, then the order of the remaining harmonics is given by:[5]
The negative sequence harmonics of a set of three-phase distorted (non-sinusoidal) periodic signals are harmonics that have an opposite phase sequence to that of the three original signals, and are phase-shifted in time by 120 for a given frequency or order.[7] It can be proven the negative sequence harmonics are harmonics whose order is given by:[5]
The zero sequence harmonics of a set of three-phase distorted (non-sinusoidal) periodic signals are harmonics that are in phase in time for a given frequency or order. It can be proven the zero sequence harmonics are harmonics whose frequency is an integer multiple of the frequency of the third harmonics.[5] So, their order is given by:
Total harmonic distortion, or THD is a common measurement of the level of harmonic distortion present in power systems. THD can be related to either current harmonics or voltage harmonics, and it is defined as the ratio of the RMS value of all harmonics to the RMS value of the fundamental component times 100%; the DC component is neglected.
It is usually the case that we neglect higher voltage harmonics; however, if we do not neglect them, real power transferred to the load is affected by harmonics. Average real power can be found by adding the product of voltage and current (and power factor, denoted by pf here) at each higher frequency to the product of voltage and current at the fundamental frequency, or
where Vk and Ik are the RMS voltage and current magnitudes at harmonic k ( k = 1 {\displaystyle k=1} denotes the fundamental frequency), and P avg , 1 {\displaystyle P_{{\text{avg}},1}} is the conventional definition of power without factoring in harmonic components.
One of the major effects of power system harmonics is to increase the current in the system. This is particularly the case for the third harmonic, which causes a sharp increase in the zero sequence current, and therefore increases the current in the neutral conductor. This effect can require special consideration in the design of an electric system to serve non-linear loads.[9]
Electric motors experience losses due to hysteresis and eddy currents set up in the iron core of the motor. These are proportional to the frequency of the current. Since the harmonics are at higher frequencies, they produce higher core losses in a motor than the power frequency would. This results in increased heating of the motor core, which (if excessive) can shorten the life of the motor. The 5th harmonic causes a CEMF (counter electromotive force) in large motors which acts in the opposite direction of rotation. The CEMF is not large enough to counteract the rotation; however it does play a small role in the resulting rotating speed of the motor.
In the United States, common telephone lines are designed to transmit frequencies between 300 and 3400 Hz. Since electric power in the United States is distributed at 60 Hz, it normally does not interfere with telephone communications because its frequency is too low.
A pure sinusoidal voltage is a conceptual quantity produced by an ideal AC generator built with finely distributed stator and field windings that operate in a uniform magnetic field. Since neither the winding distribution nor the magnetic field are uniform in a working AC machine, voltage waveform distortions are created, and the voltage-time relationship deviates from the pure sine function. The distortion at the point of generation is very small (about 1% to 2%), but nonetheless it exists. Because this is a deviation from a pure sine wave, the deviation is in the form of a periodic function, and by definition, the voltage distortion contains harmonics.
When a sinusoidal voltage is applied to a linear time-invariant load, such as a heating element, the current through it is also sinusoidal. In non-linear and/or time-variant loads, such as an amplifier with a clipping distortion, the voltage swing of the applied sinusoid is limited and the pure tone is polluted with a plethora of harmonics. be457b7860
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