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Course Outline

- Signals
- Amplitude
- Phase
- Frequency
- Fourier Analysis of Signals
- Linear, non-linear, Time-varying and Time-invariant Sources
- Frequency Spectra
- Harmonic Distortions
- Harmonics in 3-phase systems
- RMS value of complex waveforms
- Form factor
- Power factor
- Filters
- Ideal low-pass, band-pass, High-pass, band-reject, notch filters
- Frequency response
- Impulse response
- Network characterisation
- Bode diagrams

Analysis of Wave forms

- Time Domain Analysis
- Examines the amplitude vs. time characteristics of a waveform.
- Frequency Domain Analysis
- Replaces the waveform with a group of sinusoids which, when added together, produce a waveform equivalent to the original.
- The relative amplitudes, frequencies, and phases of the sinusoids are examined.

Frequency Domain Analysis vs Time Domain Analysis

- Certain characteristics of a signal are easier to identify and measure using frequency domain analysis than using time domain analysis - This is especially true in machinery and sound-producing equipment, where individual effects can be most easily identified in the frequency domain.
- The effects of processing a signal (e.g., filtering) are easier to determine using frequency domain analysis - This is important for both design and analysis.
- Signals are often more efficiently described in the Frequency Domain
- Power calculations are often easier to perform in the Frequency Domain
- The effects of noise are often easier to handle in the Frequency Domain

Time Domain Representation- Sine Waves –Amplitude & Frequency

F = 1/T = 1/5x10-3hz

A

Amplitude (mm)

T

ω =2πf

Time Domain Representation -Sine Waves – Phase & Delay

A relative measurement defined in relation to an unshifted wave.

Delay, on the other hand, is how far in seconds you must shift a waveform to get it to align with a reference unshifted waveform

ø

Phase differences represent the way the two functions are aligned with respect to each other at all times

Frequency Domain Representation

Amplitude axis has no values less that zero

In this spectral representation, called a magnitude spectrum amplitudes cannot be less than zero--it is not possible to have negative amounts of energy.

Line spectra exactly represent periodic signals like sine waves and square waves that are (at least theoretically) unbounded in time.

Frequency Domain Representation

One of the most convenient features of frequency domain representations is that many different frequencies can be plotted simultaneously on the same figure.

This figure does not show us anything about the phase relationships among the harmonics which were obvious in the time-domain figures earlier.

Frequency Domain Representation

We call spectra like this harmonic spectra rather than line spectra.

Harmonic Spectra Associated with quasiperiodic signals that are bounded in time.

Frequency Domain Representation

Continuous Spectra Associated with aperiodic sounds.

Frequency Spectra

- Every signal has a frequency spectrum.
- The signal defines the spectrum
- The spectrum defines the signal
- We can move back and forth between the time domain and the frequency domain without losing information.

Harmonic Distortion

- When a signal passes through a non-linear device, additional content is added at the harmonics of the original frequencies.
- The total harmonic distortion, or THD is a measurement of the extent of that distortion.

Harmonic Distortion in Three – Phase Systems

- Harmonic distortion is caused by the high use of non-linear load equipment such as:
- Computer power supplies,
- Electronic ballasts,
- Compact fluorescent lamps
- Variable speed drives etc,

which create high current flow with harmonic frequency components.

- High harmonic environments can produce unexpected and dangerous neutral currents.
- Harmonic currents add to the fundamental load current and can affect revenue billing by introducing errors into kilowatt hour metering systems, which will directly increase the net billable kilowatt demand and kilowatt hour consumption charges.
- The commercial effects of harmonic distortion to power quality are dramatically shorter equipment lifetimes, reduced energy efficiency and a susceptibility to nuisance tripping.
- In an electrical distribution system harmonics create:
- Large load currents in the neutral wires of a 3 phase system.
- Overheating of standard electrical supply transformers which shortens the life of a transformer and will eventually destroy it.
- High voltage and current distortions exceeding IEEE Standard 1100-1992
- High neutral-to-ground voltage often greater than 2 volts exceeding IEEE Standard 1100-1992
- Poor power factor conditions that result in monthly utility penalty fees for major users (factories, manufacturing, and industrial) with a power factor less than 0.9.
- Resonance that produces over-current surges
- False tripping of branch circuit breakers

Harmonic Distortion in Three – Phase Systems

- Harmonic current in a 3 phase power distribution system can be expressed as:

Harmonic Distortion in Three – Phase Systems

- The Total Harmonic Distortion of Current is given by:
- In a balanced 3 phase 4 wire power distribution system, the third order harmonic currents in all 3 phases are identical.

)

Harmonic Distortion in Three – Phase Systems

- Since all three phase currents are equal, the third order harmonic currents therefore have the same magnitude and can be written as:

Harmonic Distortion in Three – Phase Systems

- When loads are balanced, harmonic currents of orders which are multiples of 3 are in phase and are added up arithmetically in the neutral line, while the fundamental components and harmonics of orders which are not multiples of 3 cancel one another out.
- The only harmonic current that circulates in the neutral line, in this case are triplen harmonic current.
- Triplen harmonic currents are also known as zero sequence currents.
- In an unbalanced 3 phase 4 wire power distribution system, additional harmonic currents flow in each phase of the system causing high neutral current.

Harmonic Distortion in Three – Phase Systems

- In a balanced system, the relationship between total harmonic distortion and neutral current can be expressed as;
- In an unbalanced system, the 3rd order harmonic distortion is most significant in the neutral line.
- The relationship between 3rd harmonic distortion and 3rd harmonic current in the neutral line can be expressed as;

Harmonic Distortion in Three – Phase Systems/IEEE Model

- Traditional definitions of power, which measured real power (P), reactive power (Q), apparent power (S), and power factor (PF), were adequate for the sinusoidal situation and were universally accepted, for single- and three-phase balanced systems with pure sinusoidal waveforms but are no longer valid for current power systems.
- The electrical energy distribution companies and the Consumers contribute to the total harmonic distortion (THD).
- The THD is defined as the ratio of the rms value of the harmonic components to the rms value of the fundamental components.
- THDR cannot exceed 100% whereas THDF may reach higher values when the spectral energy of the harmonics exceeds that of the fundamental
- Employment of THDR in measurements may yield high errors in significant quantities such as power factor and distortion factor, derived from THD measurements.
- Also, the apparent power can be expressed in the following manner:
- where and are defined as fundamental and non-fundamental apparent power.

Harmonic Distortion in Three – Phase Systems

- Harmonic pollution delivered or absorbed by a load consists of three components.
- The active power is given by:
- The power parameters of a general three-phase network, three or four wires, is determined by using an equivalent single phase system.
- The apparent equivalent power will meet:
- In general, the expressions given for the single-phase case remain true for three phases, with the only exception of the equivalent magnitudes.

Harmonic Distortion in Three – Phase Systems

- The equivalent apparent harmonic power can also be decomposed as:

Harmonic Distortion in Three – Phase Systems/Czarnecki’s Model

- In order to calculate the power in nonsinusoidal conditions, Czarnecki’s definition splits the electrical power system at the PCC on the utility side and the customer side.

Harmonic Distortion in Three – Phase Systems/Czarnecki’s Model

- The power expressions for the characterization of three phase systems are shown below:

Harmonic Distortion in Three – Phase Systems

- When one uses the above models, the rms values of voltage and current fulfill the following relations:
- Where the calculation of the THD (system and client) and the distortion power (system and client) for both voltage and current can be made as follows:
- The active power is obtained from the sum of the following terms:

Harmonic Distortion in Three – Phase Systems

- Electrical distribution system of most facilities was never designed to deal with an abundance of non-linear loads.
- In order to minimise the effects of high levels of harmonics it is recommended that:
- Use double-size neutral wires or separate neutrals for each phase.
- Specify a separate full-size insulated ground wire rather than relying on the conduit alone as a return ground path.
- On a branch circuit use an isolated ground wire for sensitive electronic and computer equipment.
- Segregate sensitive electronic and computer loads on separate branch circuits all the way back to the electrical panel.
- Run a separate branch circuit for every 10 Amps of load.
- Install a comprehensive exterior copper ground ring and multiple deep driven ground rods as part of the grounding system to achieve 5 ohms or less resistance to earth ground.
- Oversize phase wires to minimize voltage drop on branch circuits.
- Shorten the distance on branch circuits from the power panel to minimize voltage drop.
- For high "Power Quality" for your building or facility, it is necessary to treat harmonics.
- Harmonic treatment can be performed by two methods:
- Filtering using a capacitor bank and an induction coil.
- cancellation using harmonic canceling transformers also known as phase-shifting transformers.

Power factor

- The power factor of an AC electric power system is defined as the ratio of the real power flowing to the load to the apparent power
- Real power is the capacity of the circuit for performing work in a particular time.
- Apparent power is the product of the current and voltage of the circuit.
- In a purely resistive AC circuit, voltage and current waveforms are in phase.

Power factor

- In a purely resistive AC circuit, voltage and current waveforms are in phase.

Power factor

- Reactive loads, such as capacitors or inductors, energy storage in the loads result in a time difference between the current and voltage waveforms.
- This stored energy returns to the source and is not available to do work at the load.
- Thus, a circuit with a low power factor will have higher currents to transfer a given quantity of real power than a circuit with a high power factor.
- A linear load does not change the shape of the waveform of the current, but may change the relative timing (phase) between voltage and current.
- Electrical linear loads consume both real power and reactive power.
- The vector sum of real and reactive power is the apparent power.
- The presence of reactive power causes the real power to be less than the apparent power, and so, the electric load has a power factor of less than 1.

Power factor

- AC power flow has the three components:
- Real power (P), measured in watts (W);
- Apparent power (S), measured in volt-amperes (VA);
- Reactive power (Q), measured in reactive volt-amperes (VAr).
- The power factor is defined as:
- In the case of a perfectly sinusoidal waveform, P, Q and S can be expressed as vectors that form a vector triangle such that:
- If φ is the phase angle between the current and voltage, then the power factor is equal to , and:
- The power factor is by definition a dimensionless number between 0 and 1.

Power factor

- When power factor is equal to 0, the energy flow is entirely reactive, and stored energy in the load returns to the source on each cycle.
- When the power factor is 1, all the energy supplied by the source is consumed by the load.
- Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle.
- Inductive loads such as transformers and motors consume reactive power with current waveform lagging the voltage.
- Capacitive loads generate reactive power with current phase leading the voltage.

Power factor

- Instantaneous and average power calculated from AC voltage and current with a zero power factor (φ=90, cosφ=0)

Power factor

- Instantaneous and average power calculated from AC voltage and current with a lagging power factor (φ=45, cosφ=0.71)

Power factor correction of linear loads

- This power factor correction is achieved by switching in or out banks of inductors or capacitors depending on the load.
- Power factor correction may be applied by an electrical power transmission utility to improve the stability and efficiency of the transmission network.
- Correction equipment may be installed by individual electrical customers to reduce the costs charged to them by their electricity supplier.
- The reactive elements can create voltage fluctuations and harmonic noise when switched on or off.
- They will supply or sink reactive power regardless of whether there is a corresponding load operating nearby, increasing the system\'s no-load losses.
- In a worst case, reactive elements can interact with the system and with each other to create resonant conditions, resulting in system instability and severe overvoltage fluctuations.

Power factor correction of linear loads

- An automatic power factor correction unit consisting of a number of capacitors that are switched by means of contactors is used to improve power factor.
- These contactors are controlled by a regulator that measures power factor in an electrical network.
- The regulator uses a Current transformer (CT) to measure the current in one phase.
- Instead of using a set of switched capacitors/inductors, an unloaded synchronous motor referred to as a synchronous condenser can supply reactive power.
- The amount of reactive power supplied is proportional to voltage, not the square of voltage; this improves voltage stability on large networks compared to the capacitor/inductor bank.

Power factor correction of Non-linear loads

- Non-linear loads change the shape of the current waveform from a sine wave to some other form.
- Non-linear loads create harmonic currents in addition to the original AC current.
- Addition of linear components such as capacitors and inductors cannot cancel these harmonic currents, so other methods such as filters or active power factor correction are required to smooth out their current demand over each cycle of alternating current and so reduce the generated harmonic currents.
- In circuits having only sinusoidal currents and voltages, the power factor effect arises only from the difference in phase between the current and voltage, known as "displacement power factor".
- The simplest way to control the harmonic current is to filter out the harmonics using a Passive PFC.
- This filter reduces the harmonic current, which means that the non-linear device now looks like a linear load and inductors or capacitors can be used for correction.

Power factor correction of Non-linear loads

- Active Power Factor Corrector (APFC) is a power electronic system that controls the amount of power drawn by a load in order to obtain a Power factor as close as possible to unity.
- In most applications, the active PFC controls the input current of the load so that the current waveform is proportional to the mains voltage waveform (a sine wave).
- Three common types of APFCs are:
- Boost
- Buck
- Buck-boost

Power factor correction of Non-linear loads - Boost

- A boost converter (step-up converter) is a power converter with an output DC voltage greater than its input DC voltage.
- It is a class of switching-mode power supply (SMPS) containing at least two semiconductor switches (a diode and a transistor) and at least one energy storage element.
- Filters made of capacitors (sometimes in combination with inductors) are normally added to the output of the converter to reduce output voltage ripple.

Power factor correction of Non-linear loads - Boost

- The key principle that drives the boost converter is the tendency of an inductor to resist changes in current.
- When being charged it acts as a load and absorbs energy like a resistor, when being discharged, it acts as an energy source like a battery.
- The voltage it produces during the discharge phase is related to the rate of change of current, and not to the original charging voltage, thus allowing different input and output voltages.
- Boost converter operation can either be continuous mode or discontinuous mode.

Power factor correction of Non-linear loads -Cont

- When a boost converter operates in continuous mode, the current through the inductor (IL) never falls to zero.
- During the On-state, the switch S is closed, which makes the input voltage (Vi) appear across the inductor, which causes a change in current (IL) flowing through the inductor during a time period (t) by the formula:
- At the end of the On-state, the increase of IL is therefore:
- D = the duty cycle
- During the Off-state, the inductor current flows through the load. If we consider zero voltage drop in the diode, and a capacitor large enough for its voltage to remain constant, change in current IL is:

Power factor correction of Non-linear loads - Cont

- Therefore, the variation of IL during the Off-period is:
- For steady-state conditions, the amount of energy stored in each of its components has to be the same at the beginning and at the end of a commutation cycle.
- Therefore, the inductor current has to be the same at the beginning and the end of the commutation cycle.

Output voltage is always higher than the input voltage

Power factor correction of Non-linear loads - Discont

- In Discontinuous mode, the amount of energy required by the load is small enough to be transferred in a time smaller than the whole commutation period.
- The inductor is completely discharged at the end of the commutation cycle.
- As the inductor current at the beginning of the cycle is zero, its maximum value is:
- During the off-period, IL falls to zero after δ.T:
- The load current Io is equal to the average diode current (ID) which is equal to the inductor current during the off-state.
- Therefore the output current can be written as:
- Replacing ILmax and δ by their respective expressions yields:

Power factor correction of Non-linear loads - - Discont

- Therefore, the output voltage gain can be written as flow:
- Compared to the expression of the output voltage for the continuous mode, this expression is much more complicated.
- In discontinuous operation, the output voltage gain not only depends on the duty cycle, but also on the inductor value, the input voltage, the switching frequency, and the output current.

Power factor correction of Non-linear loads - Buck

- A buck converter is a step-down DC to DC converter.
- Its design is similar to the step-up boost converter, and like the boost converter it is a switched-mode power supply that uses two switches (a transistor and a diode) and an inductor and a capacitor.
- The simplest way to reduce a DC voltage is to use a voltage divider circuit - waste energy
- Buck converter efficient and like boost converter, operate in 2 modes.

Power factor correction of Non-linear loads - Cont

- A Buck converter operates in continuous mode if the current through the inductor (IL) never falls to zero during the commutation cycle.
- When the switch pictured above is closed, the voltage across the inductor is VL = Vi − Vo.
- When the switch is opened, the diode is forward biased and the voltage across the inductor is VL = − Vo.
- The energy stored in inductor L is:
- The rate of change of IL can be calculated from:
- The increase in current during the On-state is given by:
- The decrease in current during the Off-state is given by:

Power factor correction of Non-linear loads - Cont

- In steady state, the current IL is the same at t=0 and at t=T
- The output voltage of the converter varies linearly with the duty cycle for a given input voltage.

Power factor correction of Non-linear loads - Discont

- If the amount of energy required by the load is small enough to be transferred in a time lower than the whole commutation period a discontinuous buck converter is used.
- In this case, the current through the inductor falls to zero during part of the period.
- Assuming converter operates in steady state, inductor energy is the same at the beginning and at the end of the cycle i.e. zero
- The average value of the inductor voltage (VL) is zero yielding:
- Current delivered to the load (Io) is constant if capacitor is large enough to maintain a constant voltage across its terminals during a commutation cycle.
- Capacitor current is zero giving us:
- The inductor current waveform has a triangular shape giving us:

Power factor correction of Non-linear loads - Discont

- Substituting for Imax gives:
- Substituting for δ gives:
- Evaluation of expression gives:

Power factor correction of Non-linear loads – Realistic Model

- The Circuits considered so far are ideal with the following assumptions:
- The output capacitor has enough capacitance to supply power to the load (a simple resistance) without any noticeable variation in its voltage.
- The voltage drop across the diode when forward biased is zero
- No commutation losses in the switch nor in the diode
- Real components will cause ripple at the O/P due to
- Switching frequency
- Output capacitance
- Inductor
- Load
- Current limiting features of the control circuitry
- Output voltage ripple is typically a design specification for the power supply and is selected based on several factors.
- Both static and dynamic power losses occur in any switching regulator.
- Static power losses include I2R (conduction) losses in the wires or PCB traces, as well as in the switches and inductors.
- Dynamic power losses occur as a result of switching, and are proportional to the switching frequency.
- The non-ideal behaviour of the components reduces the efficiency of the design

Power factor correction of Non-linear loads – Buck-Boost

- The buck–boost converter is a type of DC-DC converter that has an output voltage magnitude that is either greater than or less than the input voltage magnitude.
- Two different topologies for this type of converter:
- The inverting topology
- The split – pi topology

Power factor correction of Non-linear loads – Inverting

- Polarity of the output voltage is opposite to that of the input

Power factor correction of Non-linear loads – Split-pi

- Split-Pi is a patented Boost (step-up) converter followed by a Buck (step-down) converter that can theoretically produce an output voltage from 0 Volts to infinity.

Form factor

- The form factor of an alternating current waveform is the ratio of the RMS value to the average value.

Deterministic Signals

Deterministic Signals are signals which are completely specified as a function of time.

1.

(Sinosoidal wave)

(Exponential Function)

2.

3.

- (unit step function)

4.

- (impulse function)

- (Rectangle function)

5.

- (Triangle function).

6.

7.

- (Impulse train)

8.

(Signum function)

(Iverson bracketnotation)

9.

Sinc(t)

=

(Sinc Function)

Fourier Analysis of Signals

- Any time domain signal can in principle be made from a sum of sized and delayed cosine waves.
- This means that you could, at least in theory, take a huge number of cosine waves, put them all through a network which adds the voltages (shifted and amplified) together at each point in time, and duplicate any possible time domain signal at the output.
- The Fourier transform, used in many signal analyzers, essentially breaks a time domain waveform into its component cosine waves.
- The transform does this by revealing the size and phase position required of the cosines at each frequency to reconstruct the original waveform.

Example

- As an example of Fourier summation, a square wave contains cosine waves (delayed and sized) only at the frequencies which are odd multiples of the repetition rate.

Fourier analysis uses cosine waves – Why?

- Cosine waves and sine waves are identical except for a phase or time shift of 90 degrees, which means 90/360 (or one fourth) of the basic waveform.
- Sine waves have a value of 0 at time=0, and cosine waves have a value of +1 at time=0.
- Cosines rather than sines are generally used in discussions of Fourier theory Because they do not suppress the DC component of the waveform.

Why Bother to Describe a System as a Collection of Cosine Waves

- The reason for all this trouble to define any signal as a collection of cosine waves is so that unified descriptions can be made of predominantly linear systems such as loudspeakers, which modify signals.
- If we can describe how a linear device modifies cosine waves of any frequency, we need not measure the separate characteristics of square-wave response, triangle wave response, or Beethoven’s Ninth Symphony response.
- The full frequency response including phase and magnitude data, or equivalently the impulse response, contains the information needed to mathematically determine how a system will treat almost any waveform within its dynamic range.

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