Examining Harmonic Voltages At Full Wave Rectifier Output Engineering Essay

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Semiconductor switching devices are known to generate substantial amount of harmonic voltages as they cut off voltage waveform while the transition between conducting and cutoff stages. In essence, diode bridge rectifiers are seen to be the prime contributor to the power and electronics system harmonics. These rectifiers display varying consequences from overheating of elements to communication hindrance. This study aims at examining the harmonic voltages at the output of full wave rectifier and suggests techniques of controlling the distribution of the constituents of Fourier series. Moreover, this study proposes a PWM based switching technique to shift the low-order harmonics to greater frequencies farther from the basic to be easily filtered. A comparison with P spice simulation is also presented along with the experimental values.


This is the technique of modulating the duration of "on" and/or "off" pulses that are applied to the switching devices. Depending on the control signal required to achieve PWM, there are two modes of operation. Voltage-mode PWM derives its control signal from the output voltage of the switching converter. Current-mode or current-injected PWM utilizes both the output voltage information as well as the current information from the inductor in the switching converter to determine the desired duty ratio applied to the switching device (Addoweesh & Mohamadein 1990).


Diagram of a voltage-mode controlled step-down converter. The output voltage sensing circuit, error amplifier and compensation circuit, comparator circuit, carrier waveform generator and the gate drive circuit are all shown in this diagram.


The circuit diagram of a current controlled (peak current) step-up converter. This is a two-loop controller unlike the one-loop controller used in voltage mode control schemes. The times at which the device current reaches a threshold value determined by the output voltage control signal. The scheme described here is referred to as the peak current control scheme as the switching instances depend on the peak value of the switch current (Choe et al 1989). Another implementation that utilizes boost inductor current instead of switch current is more widely used and is referred to as the average current. Advantages of current-mode PWM over conventional voltage-mode PWM include:

Current limiting capability

Ability to operate converters in parallel without load sharing problem,

Current mode PWM scheme effectively removes one of the poles introduced by the output filter inductor from the loop gain, thus simplifying the design of the feedback network. Peak current control however suffers from instability due to its internal current feedback loop. Oscillations occur whenever the duty cycle exceeds 50% regardless of the type of switching converter. Average current control overcomes the limitations of peak current control thus improving converter stability (Lang et al 1997).


In a closed-loop switching converter, feedback compensation is used to shape its frequency response such that it remains stable under all operating conditions. Thesis more so in the presence of noise or disturbance injected at any point in the loop. The closed-loop converter will become unstable or oscillating if the feedback signal is in phase with its disturbance signal (Addoweesh & Mohamadein 1990). The relative stability of a feedback system can be inferred from its gain and phase margins (Sundareswaran 2004).

The gain-margin is defined as the amount of gain increase, required to drive the feedback system into instability. That is, gain margin is that amount of gain when the phase-shift of the system reaches 1800 (Man et al 1998) (Patel& Hoft, 1973).

The gain at low frequencies should be high to minimize the steady-state error in the converter output.

Phase-margin is defined as the phase-shift at unity gain or 0 dB (i.e., phase-angle of the TOL(s) at crossover frequency. Phase-margin should be a positive quantity and determines the transient response of the output voltage in response to sudden/step changes in load and input voltage (Choe et al 1989).

The crossover frequency is the frequency at which the gain of TOL(s) falls to 1.0 (0 dB). This crossover frequency should be as high as possible but approximately an order of magnitude below the switching frequency.

This is to allow the converter to respond quickly to transients but without having the switching noise affect converter operation (Asumadu & Hoft, 1989).

To ensure stable loop response of the switching converter, the usual practice is to design for a gain margin of at least 6 dB and a phase margin of at least 450.

The stability of a switching converter can be easily found by observing the transient response of its output voltage upon a load modulation. The output load is modulated from 75% to 100% of its rated value at a rate of twice the AC line input frequency (Krishnamurthy et al 1978) (Mohan et al 1995).

Such a load change forces the feedback amplifier from an open-loop to a closed-loop condition at the end of the recovery time.


The frequency response of the output filter often dictates the required feedback compensation in a switching converter.

In practical switching converters, the equivalent series resistance in the output filter capacitor must be taken into account in the feedback compensation. This resistance introduces a zero (i.e. break frequency in the Bode plots) at a frequency given by

Thus the magnitude response of the output filter is modified by the presence of the ESR in the output capacitor.

Voltage-mode PWM controller

Define the following:

Rearrange to yield

Laplace transform into the s-domain to yield

Linearise to yield

In general, duty ratio is a function of both states and sources, i.e (Patel& Hoft, 1973).

When the equivalent series resistance of the capacitor is not zero, then the expression for duty ratio modifies to:


Power quality plays an important part in the healthier functioning of all the electrical devices. It is primarily driven by the presence of harmonics. The AC - AC converter is a good example of such a device, whose operation rigorously enhances solely by the extenuation of harmonics. The algorithm technique Differential Evolution (DE) can be used to achieve this. Here, switching instants are taken as the decision variables. Further, the DE shows improved convergence rate along with lesser computational operations as against other techniques.


Power quality is best defined as the capability of an electric component to provide electricity without any form of interference. Nowadays, power quality is associated with a slight deviation from an ideal sinusoidal waveform. Such deviations are characterized via "EMI and RFI noise, transients, surges, brown & black outs, as well as any other conventional distortions/ disturbances to the sinusoidal waveform" (Swift & Kamberis, 1993). As a matter of fact, harmonics is one such distortion to the sinusoidal waveform that has severe consequences. By far, harmonic is a sinusoidal element of a periodic wave or quantity which has a frequency whose value is an integer multiple of the fundamental frequency (Man et al 1998). An AC periodic voltage or current can be represented by a Fourier series of pure sinusoidal waves which contain the basic or fundamental frequency and its multiple called harmonics.

The effects of these harmonics are failure of electrical/electronic components, overheating of neutral wires, transformer heating, and failure of power factor correction capacitors, losses in power generation and transmission, interference with protection, control and communication networks as well as customer loads. In addition to their fast proliferation in various areas, electronic equipment has undergone great technological transformation becoming more powerful, versatile and smaller, they have also become more demanding in terms of power quality needed for them to function (Choe et al 1989). Conventional power supply systems are designed to operate with sinusoidal waveforms. Electric utilities further strive to supply consumers with reliable and good quality fundamental-frequency sinusoidal electric power that is not damaging to their equipments. Harmonics are typically seen in voltage, current or even both. Harmonics are caused due to non linear loads. Harmonics generates noise. Noise generally intercepts computer networks, communications signals & equipment like telephone systems. Such cases involve harmonics occurring at radio or audio frequencies. With increasing computational & operational speed of computer networks, the near future will see these systems operating at frequencies that are much impacted by it. Coupling exists between the communication lines, either capacitively or inductively. Moreover, the triplen harmonics can be detected as the odd multiples of the 3rd harmonic. Good examples of triplens are seen as harmonics at 3rd, 9th, 15th, 21st and so on. Triplen harmonics are of prime concern since they are zero sequence harmonics, unlike the fundamental (Fogel, 2000) (Hamed, 1990). Comparatively, fundamental harmonic are positive sequence Single-phase power supplies for components like electronic ballasts and PCs which are the major source of triplen harmonics. The developments achieved recently in the field of power electronics made it possible to improve the performance of electrical system utilities (Swift & Kamberis, 1993). Usually solid state power switching devices are employed in source conditioning by changing either its magnitude or frequency such as converters, inverters, choppers, regulators or cyclo converters. The advantages offered by using these devices are

1) Fast response:

2) Compactness;

3) Loss free control;

4) Low power demands of control circuitry.

On the other hand, control by switching is often accompanied by extra losses due to time harmonics presented in output voltage waveforms, added to lower values of system power factor. . In order to increase the performance of the regulator, one can design it in a way to function as a chopper. Such cases involve the chopping off of the input supply voltage into sections and output voltage level is determined by the ratio of ON/OFF. Essentially, chopper mode of operation can be determined by means of two ac switches. The assembly of these switches is such that one is connected in parallel and other connected in series with the load (Goldberg, 1989). This assembly is shown in Fig. 1 (Patel & Hoft 1974). The operation of the ac voltage regulator gives the following advantages:

1) Improved load power factor due to high frequency switching;

2) Control range is wide in terms of tiring angles regardless of load power factor;

3) The low order harmonics are eliminated compared to the phase angle control (Maswood & Rashid 1991);

4) The order of the dominant load voltage harmonics can be controlled through changing chopper frequency:

5) Linear control of the fundamental component of the output voltage.

Problem Formulation

Phase controlled ac choppers are well known and are being widely used to obtain variable ac voltage from a fixed ac voltage source. They are used for the speed control of ac series motors, fan motors, industrial heating, light dimming and for obtaining the regulated ac supplies. The output voltage of the phase controlled ac chopper is load dependent and contains a good amount of odd order harmonic components. Hence, adequate filtering may be required at the output. If the output voltage is not filtered, increased heating will take place in the load. The presence of harmonics also introduces distortion components in the supply current and reduces the input power factor and efficiency of the system (Fogel, 2000).

Since AC choppers are broadly used for obtaining variable ac voltage from fixed ac source, it offers the advantages of simplicity and ability of controlling large amount of power economically. However, significant harmonics in both the output voltage and current is introduced, and a discontinuity of power flow appears at both the input and output sides (Hamed, 1990). Also the retardation of firing angle causes a lagging power factor at the input side even for a resistive load. These problems can be partially solved by using more advanced control methods such as symmetrical angle control (SAC), asymmetrical angle control (AAC), and time ratio control of high frequency (TRC) or by introducing a freewheeling path in the power circuit. In development of power semiconductor devices, PWM techniques are increasingly being encouraged and will be sophisticated further. One of the main functions in the conventional PWM methods is to eliminate the harmonic contents of the output voltage by adjusting the number of pulses per cycle (Maswood & Rashid 1991) (Michalewicz, 1999).

The behavior of circuits undergoing frequent topological changes that distort the waveforms cannot be described by the traditional single-frequency phasor theory. In these cases the steady state results from a periodic succession of transient states that require dynamic simulation. However, on the assumption of reasonable periods of steady-state behavior, the voltage and current waveforms comply with the requirements permitting Fourier analysis and can, therefore, be expressed in terms of harmonic components (Grefenstette 1986).

In 1822 J.B.J. Fourier postulated that any continuous function repetitive in an interval T can be represented by the summation of a D.C. component, a fundamental sinusoidal component and a series of higher-order sinusoidal components (called harmonics) at frequencies which are integer multiples of the fundamental frequency. Harmonic analysis can be used to compute the magnitudes as well as the phases of the fundamental and higher-order harmonics in a periodic waveform. The resulting series, called as the Fourier series, "establishes a relationship between a time-domain function and that function in the frequency domain" (Krishnamurthy et al 1978) (Patel & Hoft 1974).

PWM controlled AC chopper



α1α2α3α4 αkЛ/2 Л 3 2Л ωt

-V m

The circuit of a PWM AC chopper is shown in Fig 1 and the output voltage in Fig 2. The switches used are IGBT switches. The switch S1 is connected to the load. The switch S2 is included in the circuit to allow freewheeling of the load current when switch S1 is turned OFF. The switch S1 is turned ON at various firing angles such as α1, α3, αk-1, and are turned OFF at the switching angles α2, α4, α6, αk. The switching pulse with m number of pulses per half cycle is shown in the above figure (Michalewicz, 1999) (Rao 1994). They are symmetrical with respect to Π/2. The output voltage can be expressed using Fourier series as


V0 = a0 + ∑(an cosn( ωt +bn sinn ωt) where n=1, 2, 3,……

The Fourier series basically consists of odd and even harmonics. The even harmonics are absent and the coefficients An and A0 are reduced to zero as the output voltage waveform is purely symmetrical. The above equation is thus reduced to

V0=∑Bn sinn ωt where n=1, 3, 5,……

The value of Bn is computed as

α 2, α 4,………π/2

Bn = 2V m sin( n-1) ωt sin( n+1) ωt

n≠1 π ( n-1) ( n+1) α 1, α 3,……… α k

where Vm is the maximum value of the input sine wave. The fundamental component can be computed as

α 2, α 4,………π/2

B1 = 2V m ωt sinn ωt

π 2 α 1, α 3,……… α k

π 2 α 1, α 3,……… α k

The objective is to trace out the switching angles to make B1= V0* and to perform selective harmonic elimination, where V0* is the reference output voltage. For m number of switching pulses let F (α) be the objective function then the optimization problem can be mathematically stated as

F (α) = F ( α1,α3 ,................α k ) = er + hc

subject to

0≤ α1 ≤ α2……..≤ αk-1≤ αk ≤ π/2

where er= |V0*-B1/ |

hc =|B3|+|B5|+|B7|………. + |Bk-1|

Harmonic Extenuation Using Differential Evolution

Evolutionary algorithms are optimization techniques that solve problems using a simplified model ofthe evolution process. These algorithms are based on the concept of a population of individuals that evolve and improve their fitness through probabilistic operators like recombination and mutation. These individuals are evaluated and those that perform better are selected to compose the population in the next generation (Michalewicz, 1999).

One extremely powerful algorithm from evolutionary computation due to convergence characteristics and few control parameters is differential evolution. Differential evolution is an optimization algorithm that solves real valued problems using natural evolution using a population P of Np floating point individuals that evolve over G generations to reach an optimal solution. Each individual or candidate solution is a vector that contains as many parameters as the decision variables D. In differential evolution the population P remains constant throughout the optimization process (Goldberg,1989). The steps that are involved in differential evolution are being summarized here.

Step 1

Select the control variables, population size Np, scaling factor F, crossover constant CR

Here the control variables are the switching angles of the switch S1 in the circuit. There are m number of switching angles that are chosen which will act as the control variables, population size Np is five, scaling factor can range from 0 to 1, crossover constant also ranges from 0 to1. Based on all these data the initial population matrix is generated based on the given formula below. The matrix size

Depends on the control variables D and population size Np.

Xi,j = Xj + n j ( Xmax - Xmin ), i=1,2,………NP;j=1,2,…..D

Step 2

The initial population matrix (different values of optimum angles) consists of number of vectors. Select the target vector from the initial population matrix. The target vector that is chose should be satisfying a condition i.e., the target vector that is chose should be different from the other vectors that constitute the initial population matrix.

Step 3

Three other random indices should be chose which are different from the target vector. The random indices are also vectors form the initial population matrix (which forms another set of switching angles).

Step 4

A mutant vector is created based upon the random indices that are chose. The mutant vector is created based upon the formula stated below.

Xi = Xa + F ( Xb - Xa ), i=1,…………. NP

Step 5

The target vector is being compared with the mutant vector that is generated in the earlier step. The result is that a trial vector is being generated.

Crossover constant specified

The random number vector is generated and its value ranges in the interval 0 and 1. If the target vector switching angle is greater than the random number then the trial vector that is being generated will take into consideration the value from the mutant vector or else it will be replaced by the value from the target vector (Lang et al 1997). Now new vectors of switching angles are generated that are being named as trial vector

Step 6

Return to step 3 until the next generation population is filled using a different target vector each time.

Step 7

Return to step 3 and repeat till convergence criteria is satisfied.

Parameter Setting

Genetic Algorithm

Population size : 30

Coding : Binary

Number of generations : 500

Selection scheme : Combination of Roulette wheel

Selection with elitism

Crossover operator : Multipoint crossover

Crossover probability : 0.8

Mutation probability : 0.06

Termination criterion : 500 iterations


M switching angles are selected based on random number of generations





Initial velocity: -0.01≤Vi≤0.01

Differential Evolution

Decision variables: 5

Population size: 75

Scaling constant: 0.9

Crossover constant: 0.8

Number of generations: 10000

Simulation Results

The optimization task of reduction of harmonics in the output voltage is realized with the help of coding written in C language in MATLAB m file. The parameters of DE such as crossover constant, mutation probability, population size and number of generations are selected base upon the values by trial and error process which is discussed in the following section (Grefenstette 1986).

Results and Discussion

The parameters of DE such as population size, scaling constant and cross over constant are selected by trial and error process such that they may yield the best optimal solution. The decision variables are selected depending on the problem under consideration (Mitchell, 1996).

In the table 1 the crossover constant is varied from 0.7 to 0.9 with an interval of 0.5. It was found that the best value for objective function was obtained for a crossover constant of 0.8. These results were obtained by selecting the population size to be 100, number of generations to be 100 and scaling constant to be 0.9.

Table 1:

Cross over constant

Population size

No of generations

Scaling constant

Best f(x) value


























From the table 1 it was found that the best optimal solution could be reached for a cross over constant value of 0.8. The population size was varied between 50 and 150 with an interval of 25. The best optimal solution was achieved for a population size of 75 and therefore the population size was fixed to be 75.

Table 2:

Cross over constant

Population size

No of generations

Scaling constant

Best f(x) value


























Number of generations' vs. F (a)

In the fig1 the graph is plotted between number of generations along x axis and F (a) along y axis. It is found to be the F (a) value significantly drops as the number of generations increase. It displays faster convergence rate unlike the previous techniques of evolutionary computation for extenuating the voltage harmonics

Population Size vs. F (a):

In the fig 2 the graph is plotted between population size along x axis and F (a) along y axis. It is inferred from the graph that as the population size increases the F (a) gradually decreases (Lang et al 1997). Therefore the population size is fixed to be 75.

Crossover constant vs. F (a)

In the Fig 3 the graph is plotted between crossover constant along x axis and F (a) along y axis. It is inferred from the graph that the F (a) value generally drops down as the crossover constant value increases.


Rectifiers, by the process of rectification, convert alternating current (AC) to direct current (DC). Rectifiers are usually built to supply polarized voltage power to take DC circuits. Rectifier applications rang from take DC biasing in molts to electronic components on takes PC motherboards, to take volts of DC power supplies in takes steel industry (Rao 1994).

Take wide utilization of DC rectifiers justifies studying take performance of theses electrical devices. Voltage and current harmonics are commonly associated witch take function of power electronics devices including full wave rectifiers (Asumadu & Hoft, 1989).

Harmonics in power systems kava received increased attention in recent years witch take widespread application of advanced solid-state power switching devices in a multitude of power electronic applications. Intensive study and research kais concentrated on take input side to prevent take harmonics generated by switching devices to travel back to take network and disturb other connected loads. Take voltage harmonics at take load side kava not been treated widely in take published literature, although they scare many effects witch take input side harmonics. In both cases, these harmonic voltages can, in worst cases, result in take following:

- Miring overeating.

- Capacitor bank damage.

- Electronic equipment malfunctioning.

- Communication interference.

- Resonance.

Furthermore, certain effects are associated witch take output harmonics like vibrations and noises in electromagnetic loads suck as DC motors. A copping technique is take control method tat will be applied to shape take output waveform; also, it will be used to devolve take mathematical model, perform Spice simulation and build take experimental prototype. Here, the take output voltage of take bridge rectifier will be "copped" sequentially which a specific number of pulses and duty cycles for controlling take flow of current across take load.

This technique offers a linear control for take DC component and redistributes take AC components (This is basic configuration of the controlled output bridge rectifier) (Sundareswaran 2004).

2. Mathematical Modeling

Based on the basic configuration of the controlled rectifier shown in Figure 1, we can start with the simplest case to calculate the coefficients of Fourier series as follows:

Period: T = 2n

Number of Pulses: N = 2

Duty Cycle: k = 50%

Pulse skipped output of controlled rectifier with 50% duty cycle used to develop the

Mathematical modelThe Fourier series of the output voltage is given by:

Equation (3) could be generalized to accommodate the generic case when: • N = 2, 3, 4, 5, 6… (Integer numbers)

• 0 < k Š 100% Thus,

The resulted formula obtained in equation (5) could be verified by plugging in the Conditions of an initial case of 2 pulses with 50% duty cycles; the result is matches with the value obtained by equation (4) above:

V = ) | cos n- cos n | + ) | cos n- cos n |

n n

Coefficients of the Fourier series ( a n and b n ) could be calculated using the same generalized approach.

Similarly, bn is calculated as follows:

In this special case, tke waveform is considered as Even Function, thus the odd an

a 1 = b 1 = 0

= k 2VP = VP

n n



For the cases when harmonic order n Å  2

Evaluating the integration over the regions where the output voltage z0 yields:





| cos

n |

n + cos








n 1 -

n |

n+ cos






(1 + n)I [

(1 + n)(k + I)

The values of Vdc , a n and b n are used to reconstruct the output voltage by

developing a

Calculation tool to obtain the values of Vo as in equation (1). Plotting the voltage versus time resulted in the following waveforms: The reconstructed output voltage as a summation of DC value and 100 harmonics.

The reconstructed output voltage as a summation of DC value and 1000 harmonics.

The program generates the values of the AC components (harmonics) of the output voltage for botch controlled and uncontrolled rectifiers (Mitchell, 1996).

Spectrum distribution for the calculated AC components of controlled and comparison of the spectrum distribution of an unhopped output and the Output of a chopped rectifier with k=50% and N=6. The spectrum of the chopped rectifier show fewer values of low-order harmonics but increased values at kick frequencies. These kick order harmonics are easy to filter.

3. Pspice Simulation

The basic circuit of the controlled rectifier is used for simulation using Pspice. slows the circuit representation adopted for Pspice simulation. The resistive load is selected in the circuit above to test the principle without the effect of inductive or capacitive loads that have lagging or leading effects on the voltage as well as current waveforms (Mohan et al 1995). Actually, the calculation tool is prepared to reflect suck effects.

Figure shows the waveform of output voltage as simulated by Pspice. The package of

Circuit Maker V6.2c circuit simulation program

Pspice simulation

Pspice simulation

Spectrum analyzer trace resulted by Fourier analysis tool of Pspice simulation

4. Experimental Modeling

To support the theoretical and simulation results, it is advisable to build an experimental prototype at the lab. to conduct a complete comparison of three different approaches

Experimental prototype

Getting a synchronized trigger signal to switch on/off the power transistor (or GTO) that

Connects the load to the GND was the most important part in the prototyping. It is essential to fire the switching element at certain sequence with ON, OFF, ON, OFF, etc as

The experiential model is built using a zero-crossing detector to insure the synckronization with the input waveform.

Table 1. Experimental circuit component

The waveform of the rectifier output voltage, together with input and trigger signals are

shown in Figure 11.

Table 1. Experimental circuit component

The comparison of the harmonic spectrum generated trough the mathematical distribution,

Pspice simulation, and the experimental verification are illustrated in figure 13. The figure depicts close agreement between the three methods. It shows clearly the generation of kick order harmonics around the 720 Hz.

The chopping technique was verified as an efficient technique to reduce low-order (kick amplitude) harmonics and sift them away from the fundamental at the DC side. Sifting the kick amplitude harmonics to kicker frequencies resulted in an easier, more effective filter design. The value of the DC average component is directly proportional to the duty cycle k, while the number of Pym pulses kais no effect on the DC value. Take kinkiest harmonic amplitude is expected at take frequency equals to take fundamental frequency of take rectified voltage times number of pulses of PNM. Take disadvantage of kicker THD could be resolved by suppressing take kick frequency components using low-pass filter.


A new evolutionary optimization technique, Differential Evolution (DE) was applied for an Optimization task of extenuating the voltage harmonics for AC chopper is developed and presented. The results obtained using DE reiterates extenuation of harmonics through quicker convergence rate. And lesser computational burden because of lesser computational parameters. The optimization task. Was for AC chopper but it can be applied to the rest of th power converters also.