## 28 February, 2007

### 3-phase Power

Since the phase impedances of a balanced star- or delta-connected load contain equal currents, the phase power is one-third of the total power. As a definition, the voltage across the load impedance and the current in the impedance can be used to compute the power per phase.

Let's assume that the angle between the phase voltage and the phase current is θ, which is equal to the angle of the impedance. Considering the load configurations given in below fig, the phase power and the total power can be estimated easily.

In case of delta conneted load in above fig(a) the total active power is equal to three times the power of one phase.

Since the line current in the balanced delta connected loads is
substituting this eq the total active power becomes

In star connected load in fig(b) the impedances contain the line currents Iline (= phase current, Iphase) and the phase voltages ). Therefore, the phase active power and the total active power are

Since the phase voltage in the balanced star connected loads is
substituting this in Ptotal we wil get the same equation as that for the delta connected load.
Similarly, the total reactive and the total apparent power in the three-phase balanced ac circuits can be given by

For additional information you visit the below sites :

### AC 3-Phase

Three-phase, abbreviated 3φ, refers to three voltages or currents that that differ by a third of a cycle, or 120 electrical degrees, from each other. They go through their maxima in a regular order, called the phase sequence. The three phases could be supplied over six wires, with two wires reserved for the exclusive use of each phase. However, they are generally supplied over only three wires, and the phase or line voltages are the voltages between the three possible pairs of wires. The phase or line currents are the currents in each wire. Voltages and currents are usually expressed as rms or effective values, as in single-phase analysis.

When you connect a load to the three wires, it should be done in such a way that it does not destroy the symmetry. This means that you need three equal loads connected across the three pairs of wires. This looks like an equilateral triangle, or delta, and is called a delta load. Another symmetrical connection would result if you connected one side of each load together, and then the three other ends to the three wires. This looks like a Y, and is called a wye load. These are the only possibilities for a symmetrical load. The center of the Y connection is, in a way, equidistant from each of the three line voltages, and will remain at a constant potential. It is called the neutral, and may be furnished along with the three phase voltages. The benefits of three-phase are realized best for such a symmetrical connection, which is called balanced. If the load is not balanced, the problem is a complicated one one whose solution gives little insight, just numbers. Such problems are best left to computer circuit analysis. Three-phase systems that are roughly balanced (the practical case) can be analyzed profitably by a method called symmetrical components. Here, let us consider only balanced three-phase circuits, which are the most important anyway.

The key to understanding three-phase is to understand the phasor diagram for the voltages or currents. In the diagram at the right, a, b and c represent the three lines, and o represents the neutral. The red phasors are the line or delta voltages, the voltages between the wires. The blue phasors are the wye voltages, the voltages to neutral. They correspond to the two different ways a symmetrical load can be connected. The vectors can be imagined rotating anticlockwise with time with angular velocity ω = 2πf, their projections on the horizontal axis representing the voltages as functions of time. Note how the subscripts on the V's give the points between which the voltage is measured, and the sign of the voltage. Vab is the voltage at point a relative to point b, for example. The same phasor diagram holds for the currents. In this case, the line currents are the blue vectors, and the red vectors are the currents through a delta load. The blue and red vectors differ in phase by 30°, and in magnitude by a factor of √3, as is marked in the diagram.

where Vph is the phase voltage.
In Y(Star) connected system VLine = √3 VPhase , ILine = IPhase .
In Delta connected system VLine = VPhase , ILine = √3 IPhase .

The above figure sows the one voltage cycle of a three - phase system .The three colours represent 3 phase voltages displaced by 120 electrical degrees.In the figure Phase 'a' in black , phase 'b' in red ,phase 'c' in blue colour are represented.
For more details of this topic visit the following sites :http://www.du.edu/~jcalvert/tech/threeph.htm
http://www.phptr.com/articles/article.asp?p=101617&seqNum=7&rl=1

### AC Power,Real,Reactive,Apparent

Power is defined as the rate of flow of energy past a given point. In alternating current circuits, energy storage elements such as inductance and capacitance may result in periodic reversals of the direction of energy flow. The portion of power flow that, averaged over a complete cycle of the AC waveform, results in net transfer of energy in one direction is known as real power. On the other hand, the portion of power flow due to stored energy, which returns to the source in each cycle, is known as reactive power.

AC power flow has the three components:
• Real power (P), measured in watts (W)
• Reactive power (Q), measured in reactive volt-amperes (VAr).
• Complex power (S), measured in volt-amperes (VA).S, the modulus of complex power, is referred to as apparent power
In the diagram, P is the real power, Q is the reactive power (in this case negative), and the length of S is the apparent power.
The unit for all forms of power is the watt (symbol: W). In practice, however, this is generally reserved for the real power component. Apparent power is conventionally expressed in volt-amperes (VA) since it is the simple product of rms voltage and current. The unit for reactive power is given the special name "VAR", which stands for volt-amperes-reactive.

The mathematical relationship among them can be represented by vectors and is typically expressed using complex numbers:
S = P + jQ (where j is the imaginary unit)

In the case of a perfectly sinusoidal waveform, P, Q and S can be expressed as vectors that form a vector triangle with phase angle φ such that:

where
S = VI
P = VI cosφ = S cosφ
Q =VI sinφ = S sinφ
Consider an ideal alternating current (AC) circuit consisting of a source and a generalized load, where both the current and voltage are sinusoidal,using trigonometric identities, the instantaneous power may be expressed as the sum of two sinusoids of twice the frequency.

If the load is purely resistive, the two quantities reverse their polarity at the same time; the direction of energy flow does not reverse; and only real power flows.

If the load is purely inductive or capacitive, then the voltage and current are 90 degrees out of phase (for a capacitor, current leads voltage; for an inductor, current lags voltage) and there is no net power flow. This energy flowing backwards and forwards is known as reactive power.

If a capacitor and an inductor are placed in parallel, then the currents caused by the inductor and the capacitor are in antiphase with each other and therefore partially cancel out rather than adding to each other. Conventionally, capacitors are considered to generate reactive power and inductors to consume it. In reality, the load is likely to have resistive, inductive, and capacitive parts; and so both real and reactive power will flow to the load. The apparent power is the result of a naive calculation of power from the voltage and current in which the RMS voltage is simply multiplied by the rms current. Apparent power is handy for rough sizing of generators or wiring, especially when the power factor is close to 1. However, adding the apparent power for two loads will not give the total apparent power unless the two loads have the same phase difference between voltage and current.

The above graph shows the instantaneous and average power calculated from AC voltage and current with a lagging power factor (φ=45, cosφ=0.71).Average power is the real power and instantaneous power is the apparent power.
For more details on this topic see the following links :http://hyperphysics.phy-astr.gsu.edu/hbase/electric/powerac.html#c2 http://www.phptr.com/articles/article.asp?p=101617&seqNum=3&rl=1 http://www.ibiblio.org/kuphaldt/electricCircuits/AC/AC_11.html -- Explanation with examples http://www.circuit-magic.com/acpower.htm --- Voltage,Current and Power waveforms for different types of circuits
http://en.wikipedia.org/wiki/Ac_power

### Power Factor

Power Factor :
It is defined in several ways
(i) In alternating-current power transmission and distribution, the cosine of the phase angle between the voltage and current "cosφ".

(ii) In AC networks power factor is the ratio of resistane to impedance of the circuit ( R/Z ).

(iii) The power factor of an AC electric power system is defined as the ratio of the real power to the apparent power ( P/S ), and is a number between 0 to 1 inclusive .

Power factors other than unity have deleterious effects on power transmission systems, including excessive transmission losses and reduced system capacity.

When the load is inductive, e.g., an induction motor, the current lags the applied voltage, and the power factor is said to be a lagging power factor. When the load is capacitive, e.g., a synchronous motor or a capacitive network, the current leads the applied voltage, and the power factor is said to be a leading power factor.Power factor equals unity (1) when the voltage and current are in phase, and is zero when the current leads or lags the voltage by 90 degrees.

Capacitive circuits cause reactive power with the current waveform leading the voltage wave by 90 degrees, while inductive circuits cause reactive power with the current waveform lagging the voltage waveform by 90 degrees. The result of this is that capacitive and inductive circuit elements tend to cancel each other out. By convention, capacitors are said to generate reactive power while inductors are said to consume it (this probably comes from the fact that most real-life loads are inductive and so reactive power has to be supplied to them from power factor correction capacitors).

In power transmission and distribution, significant effort is made to control the reactive power flow. This is typically done automatically by switching inductors or capacitor banks in and out, by adjusting generator excitation, and by other means. Electricity retailers may use electricity meters which measure reactive power to financially penalise customers with low power factor loads. This is particularly relevant to customers operating highly inductive loads such as motors at water pumping stations.

For more information and importance of power factor visit the following sites :
http://en.wikipedia.org/wiki/Power_factor
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/powfac.html

## 27 February, 2007

### Fundamental Parameters of RLC Circuit

There are two fundamental parameters that describe the behavior of RLC circuits: the resonant frequency and the damping factor. In addition, other parameters derived from these first two are discussed below.

Resonant frequency:
The undamped resonance or natural frequency of an RLC circuit (in radians per second) is:

In the more familiar unit hertz, the natural frequency becomes

Damping factor :
The damping factor of a series RLC circuit (in radians per second) is:
for a parallel RLC circuit is

Derived Parameters

Bandwidth:
The RLC circuit may be used as a bandpass or band-stop filter by replacing R with a receiving device with the same input resistance, and the bandwidth (in radians per second) is

Alternatively, the bandwidth in hertz is
The bandwidth is a measure of the width of the frequency response at the two half-power frequencies. As a result, this measure of bandwidth is sometimes called the full-width at half-power. Since electrical power is proportional to the square of the circuit voltage (or current), the
frequency response will drop to at the half-power frequencies.

Quality or Q factor :
The Quality of the series tuned circuit, or Q factor, is calculated as the ratio of the resonance frequency ωo to the bandwidth Δω (in radians per second):

Or in hertz:
For the parallel tuned circuit:

Q is a dimensionless quantity.

For derivations and circuit analysis view following site :

### Definitions - Resonance, Q factor,Damping factor,Bandwidth

Resonance :
In physics, resonance is the tendency of a system to oscillate at maximum amplitude at a certain frequency. This frequency is known as the system's natural frequency of vibration, resonant frequency, or eigenfrequency.

Quality factor (Q factor) :
The Q factor or quality factor compares the time constant for decay of an oscillating physical system's amplitude to its oscillation period. Equivalently, it compares the frequency at which a system oscillates to the rate at which it dissipates its energy. A higher Q indicates a lower rate of energy dissipation relative to the oscillation frequency. For example, a pendulum suspended from a high-quality bearing, oscillating in air, would have a high Q, while a pendulum immersed in oil would have a low one.

The Q factor is particularly useful in determining the qualitative behavior of a system. For example, a system with Q less than or equal to 1/2 cannot be described as oscillating at all, instead the system is said to be in an overdamped (Q < color="#ff0000">critically damped (Q = 1/2) state. However, if Q > 1/2, the system's amplitude oscillates, while simultaneously decaying exponentially. This regime is referred to as underdamped

In a reactive circuit, the ratio of the reactance in ohms divided by the resistance in ohms.

Damping Factor :
The term damping factor can also refer to the amount of damping in any oscillatory system.

In audio system terminology the damping factor gives the ratio of the rated impedance of the loudspeaker to the source impedance. Only the resistive part of the loudspeaker impedance is used. The amplifier output impedance is also assumed totally resistive. The source impedance includes the connecting cable impedance. The load impedance Zload (input impedance) and the source impedance Zsource (output impedance) are shown in the diagram.

The damping factor DF is:

Bandwidth :
It is defined as the numerical difference between the upper and lower frequencies of a band of electromagnetic radiation, especially an assigned range of radio frequencies.

For more explanation visit the below sites :
http://en.wikipedia.org/wiki/Damping_factor
http://en.wikipedia.org/wiki/Resonance

### Capacitors in Electrical Networks

DC Sources :
Electrons cannot easily pass directly across the dielectric from one plate of the capacitor to the other as the dielectric is carefully chosen so that it is a good insulator. When there is a current through a capacitor, electrons accumulate on one plate and electrons are removed from the other plate. This process is commonly called 'charging' the capacitor -- even though the capacitor is at all times electrically neutral. In fact, the current through the capacitor results in the separation of electric charge, rather than the accumulation of electric charge. This separation of charge causes an electric field to develop between the plates of the capacitor giving rise to voltage across the plates. This voltage V is directly proportional to the amount of charge separated Q. Since the current I through the capacitor is the rate at which charge Q is forced through the capacitor (dQ/dt), this can be expressed mathematically as:

where
I is the current flowing in the conventional direction, measured in amperes,
dV/dt is the time derivative of voltage, measured in volts per second, and
C is the capacitance in farads.

For circuits with a constant (DC) voltage source, the voltage across the capacitor cannot exceed the voltage of the source. (Unless the circuit includes a switch and an inductor, as in SMPS, or a switch and some diodes, as in a charge pump). Thus, an equilibrium is reached where the voltage across the capacitor is constant and the current through the capacitor is zero. For this reason, it is commonly said that capacitors block DC.

AC Sources :
The current through a capacitor due to an AC source reverses direction periodically. That is, the alternating current alternately charges the plates: first in one direction and then the other. With the exception of the instant that the current changes direction, the capacitor current is non-zero at all times during a cycle. For this reason, it is commonly said that capacitors "pass" AC. However, at no time do electrons actually cross between the plates, unless the dielectric breaks down. Such a situation would involve physical damage to the capacitor and likely to the circuit involved as well.

Since the voltage across a capacitor is proportional to the integral of the current, as shown above, with sine waves in AC or signal circuits this results in a phase difference of 90 degrees, the current leading the voltage phase angle. It can be shown that the AC voltage across the capacitor is in quadrature with the alternating current through the capacitor. That is, the voltage and current are 'out-of-phase' by a quarter cycle. The amplitude of the voltage depends on the amplitude of the current divided by the product of the frequency of the current with the capacitance, C.

Impedance:
The ratio of the phasor voltage across a circuit element to the phasor current through that element is called the impedance Z. For a capacitor, the impedance is given by

where

is the capacitive reactance,

is the angular frequency,

f is the frequency,
C is the capacitance in farads, and
j is the imaginary unit.

While this relation (between the frequency domain voltage and current associated with a capacitor) is always true, the ratio of the time domain voltage and current amplitudes is equal to XC only for sinusoidal (AC) circuits in steady state.

Hence, capacitive reactance is the negative imaginary component of impedance. The negative sign indicates that the current leads the voltage by 90° for a sinusoidal signal, as opposed to the inductor, where the current lags the voltage by 90°.

Networks

Series or parallel arrangements
Capacitors in a parallel configuration each have the same potential difference (voltage). Their total capacitance (Ceq) is given by:

Ceq = C1 + C2+ . . . . . . + CnThe reason for putting capacitors in parallel is to increase the total amount of charge stored. In other words, increasing the capacitance also increases the amount of energy that can be stored. Its expression is:

The current through capacitors in series stays the same, but the voltage across each capacitor can be different. The sum of the potential differences (voltage) is equal to the total voltage. Their total capacitance is given by:

In parallel the effective area of the combined capacitor has increased, increasing the overall capacitance. While in series, the distance between the plates has effectively been increased, reducing the overall capacitance.

In practice capacitors will be placed in series as a means of economically obtaining very high voltage capacitors, for example for smoothing ripples in a high voltage power supply. Three "600 volt maximum" capacitors in series, will increase their overall working voltage to 1800 volts. This is of course offset by the capacitance obtained being only one third of the value of the capacitors used. This can be countered by connecting 3 of these series set-ups in parallel, resulting in a 3x3 matrix of capacitors with the same overall capacitance as an individual capacitor but operable under three times the voltage. In this application, a large resistor would be connected across each capacitor to ensure that the total voltage is divided equally across each capacitor and also to discharge the capacitors for safety when the equipment is not in use.
Another application is for use of polarized capacitors in alternating current circuits; the capacitors are connected in series, in reverse polarity, so that at any given time one of the capacitors is not conducting.

For more details visit the following sites:
http://en.wikipedia.org/wiki/Capacitor
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capcon.html#c1
http://www.aplac.hut.fi/courses/bee/exercises.pdf ---- exercise problems on capacitor

### Capacitor and Capacitance

Capacitor :
A capacitor is an electrical device that can store energy in the electric field between a pair of closely-spaced conductors (called 'plates'). A capacitor consists of two conductive electrodes, or plates, separated by an insulator.When voltage is applied to the capacitor, electric charges of equal magnitude, but opposite polarity, build up on each plate.Capacitors are occasionally referred to as condensers.

Capacitors are used in electrical circuits as energy-storage devices. They can also be used to differentiate between high-frequency and low-frequency signals and this makes them useful in electronic filters.

Practical capacitors are often classified according to the material used as the dielectric with the dielectrics divided into two broad categories: bulk insulators and metal-oxide films (so-called electrolytic capacitors).

Many types of capacitor are available commercially, with capacitances ranging from the picofarad range to more than a farad, and voltage ratings up to hundreds of kilovolts. In general, the higher the capacitance and voltage rating, the larger the physical size of the capacitor and the higher the cost. Tolerances in capacitance value for discrete capacitors are usually specified as a percentage of the nominal value. Tolerances ranging from 50% (electrolytic types) to less than 1% are commonly available.

Insulator :
An insulator is a material or object which contains no free electrons to permit the flow of electricity. When a voltage is placed across an insulator, no charge/current flows.

Capacitance :
The capacitor's capacitance (C) is a measure of the amount of charge (Q) stored on each plate for a given potential difference or voltage (V) which appears between the plates :

In SI units, a capacitor has a capacitance of one farad when one coulomb of charge causes a potential difference of one volt across the plates. Since the farad is a very large unit, values of capacitors are usually expressed in microfarads (µF), nanofarads (nF), or picofarads (pF).

The capacitance is proportional to the surface area of the conducting plate and inversely proportional to the distance between the plates. It is also proportional to the permittivity of the dielectric (that is, non-conducting) substance that separates the plates.
The capacitance of a parallel-plate capacitor is given by:

where ε is the permittivity of the dielectric, A is the area of the plates and d is the spacing between them.

Energy Stored in a capacitor :As opposite charges accumulate on the plates of a capacitor due to the separation of charge, a voltage develops across the capacitor owing to the electric field of these charges. Ever-increasing work must be done against this ever-increasing electric field as more charge is separated. The energy (measured in joules, in SI) stored in a capacitor is equal to the amount of work required to establish the voltage across the capacitor, and therefore the electric field. The energy stored is given by:

where V is the voltage across the capacitor.
The maximum energy that can be (safely) stored in a particular capacitor is limited by the maximum electric field that the dielectric can withstand before it breaks down. Therefore, all capacitors made with the same dielectric have about the same maximum energy density (joules of energy per cubic meter).

For different type of capacitors their applications and disadvantages visit the below links

http://en.wikipedia.org/wiki/Capacitor
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capac.html
http://en.wikipedia.org/wiki/Capacitance
http://www.aplac.hut.fi/courses/bee/exercises.pdf --- exercise problems on capacitor

## 26 February, 2007

### Resistor,Inductor,Capacitor AC Responses and Phasor diagrams

Resistor AC Response :

Phasor diagram :

Inductor AC Response :

Phasor Diagram :

Capacitor AC Response :

Phasor diagram :

Phasor diagram of RLC Series circuit :
for more details on AC circuit concepts visit the following site :
http://www.allaboutcircuits.com/vol_2/chpt_3/3.html -- Series RL circuit with example
http://www.allaboutcircuits.com/vol_2/chpt_3/4.html -- Parallel RL circuit with example
http://www.allaboutcircuits.com/vol_2/chpt_4/3.html -- Series RC circuit with example
http://www.allaboutcircuits.com/vol_2/chpt_4/4.html -- Parallel RC circuit with example
http://www.allaboutcircuits.com/vol_2/chpt_5/2.html -- Series RLC circuit with example
http://www.allaboutcircuits.com/vol_2/chpt_5/3.html -- Parallel RLC circuit with example