Maximum power Transfer Theorem

As you are probably aware, a normal car battery is rated at 12 V and generally has an open circuit voltage of around 13.5 V. Similarly, if we take 9 pen-torch batteries, they too will have a terminal voltage of 9×1.5 = 13.5 V. However, you would also be aware, that if your car battery is dead, you cannot go to the nearest shop, buy 9 pen-torch batteries and start your car.Why is that ?Because the pen-torch batteries, although having the same open circuit voltage does not have the necessary power (or current capacity) and hence the required current could not be given.Or if stated in different terms, it has too high an internal resistance so that the voltage would drop without giving the necessary current.

This means that a given battery (or any other energy supply, such as the mains) can only give a limited amount of power to a load.The maximum power transfer theorem defines this power, and tells us the condition at which this occurs. 

For example, if we consider the above battery, maximum voltage would be given when the current is zero, and maximum current would be given when the load is short-circuit (load voltage is zero). Under both these conditions, there is no power delivered to the load. Thus obviously in between these two extremes must be the point at which maximum power is delivered. 

The Maximum Power Transfer theorem states that for maximum active power to be delivered to the load, load impedance must correspond to the conjugate of the source impedance (or in the case of direct quantities, be equal to the source impedance).

Compensation Theorem

In many circuits, after the circuit is analysed, it is realized that only a small change need to be made to a component to get a desired result. In such a case we would normally have to recalculate.The compensation theorem allows us to compensate properly for such changes without sacrificing accuracy. 

In any linear bilateral active network, if any branch carrying a current I has its impedance Z changed by an amount ∆Z, the resulting changes that occur in the other branches are the same as those which would have been caused by the injection of a voltage source of (-) I . ∆Z in the modified branch. 

Consider the voltage drop across the modified branch. 

V +∆V = (Z +∆Z)( I +∆I) = Z . I + ∆Z . I + (Z +∆Z) . ∆I 

from the original network, V = Z . I 

∴  ∆V = ∆Z . I + (Z +∆Z) . ∆I

Since the value I is already known from the earlier analysis, and the change required in the 

impedance, ∆Z , is also known, I .∆Z is a known fixed value of voltage and may thus be represented by a source of emf I.∆Z . 

Using superposition theorem, we can easily see that the original sources in the active network give rise to the original current I, while the change corresponding to the emf I.∆Z must produce the remaining changes in the network. 

Reciprocality Theorem

The reciprocality theorem tells us that in a linear passive bilateral network an excitation and the corresponding response may be interchanged. 

In a two port network, if an excitation e(t) at port (1) produces a certain response r(t) at a port (2), then if the same excitation e(t) is applied instead to port (2), then the same response r(t) would occur at the other port (1).

Norton’s Theorem

Norton’s Theorem  states that any linear, active, bilateral network, considered across one of its ports, can be replaced by an equivalent current source (Norton’s current source) and an equivalent shunt admittance (Norton’s Admittance).Since the two sides are identical, they must be true for all conditions. Thus if we compare the current through the port in each case under short circuit conditions, and measure the input admittance of the network with the sources removed (voltage sources short-circuited and current sources open-circuited), then 

Inorton =  Isc

Ynorton =  Yin

Thevenin’s Theorem

The Thevenin’s theorem,basically gives the equivalent voltage source corresponding to an active network.If a linear, active, bilateral network is considered across one of its ports, then it can be replaced by an equivalent voltage source (Thevenin’s voltage source) and an equivalent series impedance (Thevenin’s impedance). 

        Since the two sides are identical, they must be true for all conditions. Thus if we compare the voltage across the port in each case under open circuit conditions, and measure the input impedance of the network with the sources removed (voltage sources short-circuited and current sources open-circuited), then 

Ethevenin=  Voc, and 
Zthevenin=  Zin

What is Kirchoff’s voltage law?

Kirchoff’s voltage law is based on the principle of conservation of energy. This requires that the total work done in taking a unit positive charge around a closed path and ending up at the original point is zero. 

This gives us our basic Kirchoff’s law as the algebraic sum of the potential differences taken round a closed loop is zero.

i.e. around a loop,  ΣVr= 0, where Vrare the voltages across the branches in the loop. 

va+ vb+ vc+ vd– ve= 0 

This is also sometimes stated as the sum of the emfs taken around a closed loop is equal to the sum of the voltage drops around the loop. 

What is Kirchoff’s current law?

Kirchoff’s current law is based on the principle of conservation of charge.This requires that the algebraic sum of the charges within a system cannot change. Thus the total rate of change of charge must add up to zero. Rate of change of charge is current.

This gives us our basic Kirchoff’s current law as the algebraic sum of the currents meeting at a point is zero. 
i.e. at a node, ΣIr= 0,  where Ir are the currents in the branches meeting at the node.

This is also sometimes stated as the sum of the currents entering a node is equal to the sum of the current leaving the node. The theorem is applicable not only to a node, but to a closed system. 

What is Ohm’s Law?

Ohm’s Law states that the voltage v(t)across a resistor R is directly proportional to the current  i(t)flowing through it. 

v(t) ∝ i(t)

v(t) =R .i(t)

This general statement of Ohm’s Law can be extended to cover inductance's and capacitors as well under alternating current conditions and transient conditions. This is then known as the Generalized Ohm’s Law. This may be stated as 

v(t) = Z(p) . i(t),  where p = d/dt = differential operator 

Z(p)is known as the impedance function of the circuit, and the above equation is the 

differential equation governing the behavior of the circuit.

For a resistor,  Z(p) = R 

For an inductor Z(p) = Lp 

For a capacitor,  Z(p) = 1/pC

In the particular case of alternating current,  p = jω so that the equation governing circuit 

behavior may be written as 

V = Z(jω). I 

For a resistor,  Z(jω) = R 

For an inductor Z(jω) = jωL

For a capacitor,  Z(jω) = 1/C jω

Synchronous condensers in Power factor Improvement

Synchronous motors are used for an  improved power factor which is obtained by adjusting the field excitation of the motors. Synchronous motors used in this way are termed synchronous condensers. The use of this method is however limited to cases where the synchronous motors are in constant use to provide the required field excitation at all times. The high cost of such motors is also a limiting factor to its widespread use.

What is the impact on power quality while adding capacitor for power factor improvement?

A properly designed capacitor application should not have an adverse affect on end user equipment or power quality. However, despite the significant benefits that can be realized using power factor correction capacitors, there are a number of power quality-related concerns that should be considered before capacitors are installed. Potential problems include increased harmonic distortion and transient over voltages.

Harmonic distortion on power systems can most simply be described as noise that distorts the sinusoidal wave shape. Harmonics are caused by nonlinear loads (e.g., adjustable-speed drives, compact fluorescent lighting, induction furnaces, etc.) connected to a facility's power system. These loads draw non sinusoidal currents (e.g., on a 60 Hz system, the 5th harmonic is equal to 300 Hz), which in turn react with the system impedance to produce voltage distortion. Generally, the harmonic impedances are low enough that excessive distortion levels do not occur. However, power factor correction capacitors can significantly alter this impedance and create what is known as a "resonance" condition. High voltage distortion can occur if the resonant frequency is near one of the harmonic currents produced by the nonlinear loads. Indications that a harmonic resonance exists include device overheating, frequent circuit breaker tripping, unexplained fuse operation, capacitor failures, and electronic equipment malfunction. Ways to avoid excessive distortion levels include altering (or moving) the capacitor size to avoid a harmful resonance point (e.g., 5th, 7th), altering the size (or moving) of the nonlinear load(s), or adding reactors to the power factor correction capacitors to configure them as harmonic filters.

Transient over voltages can be caused by a number of power system switching events; however, utility capacitor switching often receives special attention due to the impact on customer equipment. Each time a utility switches a capacitor bank a transient over voltage occurs.  Generally, these overvoltages are low enough that they do not affect the system. However, high overvoltages can occur when customers have power factor correction capacitors. This phenomenon is often referred to as "voltage magnification". Magnification occurs when the transient oscillation initiated by the utility capacitor switching excites a resonance (refer to previous definition for hresonance) formed by a step-down transformer and low voltage power factor correction capacitors. Magnified overvoltages can be quite severe and the energy associated with these events can be damaging to power electronic equipment and surge protective devices (e.g., transient voltage surge suppressors). Adjustable-speed drives have been found to be especially susceptible to these transients and nuisance tripping can result even when overvoltage levels are not severe.

How to improve power factor ?

Low power factor is generally solved by adding power factor correction capacitors to a facility's electrical distribution system. Power factor correction capacitors supply the necessary reactive portion of power (kVAr) for inductive devices. By supplying its own source of reactive power, a facility frees the utility from having to supply it. This generally results in a reduction in total customer demand and energy charges.

Power factor correction requirements determine the total amount of capacitors required at low voltage buses. These capacitors can be configured as harmonic filters if necessary. The power factor characteristics of plant loads typically are determined from billing information, however, in the case of a new installation, typical load power factors will determine the required compensation.

Importance of Powerfactor

In AC circuits, the power factor is the ratio of the real power that is used to do work and the apparent power that is supplied to the circuit and is a dimensionless number in the closed interval of -1 to 1. A power factor of less than one means that the voltage and current wave forms are not in phase, reducing the instantaneous product of the two wave forms (V x I). 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. Due to energy stored in the load and returned to the source, or due to a non-linear load that distorts the wave shape of the current drawn from the source, the apparent power will be greater than the real power. A negative power factor occurs when the device (which is normally the load) generates power, which then flows back towards the source, which is normally considered the generator.

Power factor is a measurement of how efficiently a facility uses electrical energy. A high power factor means that electrical capacity is being utilized effectively, while a low power factor indicates poor utilization of electric power. However, this is not to be confused with energy efficiency or conservation which applies only to energy. Improving the efficiency of electrical equipment reduces energy consumption, but does not necessarily improve the power factor.

Power factor involves the relationship between these two types of power. Active Power is measured in kilowatts (kW) and Reactive Power is measured in kilovolt-amperes-reactive (kVAr). Active power and reactive power together make up Apparent Power, which is measured in kilovolt-amperes (kVA). This relationship is often illustrated using the familiar "power triangle" that is shown in the following figure.

Power Factor Compensation

Power factor is the ratio between active power and apparent power. Active power does work and reactive power produces an electromagnetic field for inductive loads. Using the values in the power triangle example shown above, the facility is operating at 400 kW (Active Power) with an 80% power factor, resulting in a total load of 500 kVA.

Lightly-loaded or varying-load inductive equipment such as HVAC systems, arc furnaces, molding equipment, presses, etc., are all examples of equipment that can have a poor power factor. One of the worst offenders is a lightly loaded induction motor (e.g., saws, conveyors, compressors, grinders, etc.).

End users should be concerned about low power factor because it means that they are using a facility's electrical system capacity inefficiently. It can cause equipment overloads, low voltage conditions, greater line losses, and increased heating of equipment that can shorten service life. Most importantly, low power factor can increase an electric bill with higher total demand charges and cost per kWh.

How circuit breakers are classified?

There are different ways of classifying circuit breakers.However,the general way is by the medium used for extinction of arc.While circuit breaker opens, an arc is formed between fixed and open contacts.The circuit breaker is considered to be of good quality if it extinguishes the arc fast.The following are the types of circuit breakers.

Oil Circuit Breakers - Uses insulating oil as arc extinguisher

Air Blast Circuit Breaker- Uses high pressure air blast as arc extinguisher

SF6  Circuit Breaker - Uses SF6( sulphur hexa  fluoride gas ) as arc extinguisher

VCB(Vacuum Circuit Breaker)  - Uses vacuum for arc extinction

what is current?

Electricity is the flow of free electrons in a conductor from one atom to the next atom in the same general direction. This flow of electrons is referred to as current and is designated by the symbol I.Electrons move through a conductor at different rates and electric current has different values.

Current is determined by the number of electrons that pass through a cross-section of a conductor in one second.  Current is measured in amperes which is abbreviated  as amps.The symbol for amps is the letter A. 

Working principle of Circuit Breaker(C.B)

A circuit breaker consists of two contacts - a fixed contact and moving contact.Under normal operating conditions,these contacts remain closed.But when a fault occurs,trip coil of C.B get activated and moving contact is pulled apart by a mechanism,thus opening the circuit.

Arcing Phenomenon

As soon as the moving contact is separated from fixed contact, an arc of current will be formed between the contacts.This arc is  quenched with  in short period of time so that circuit is braked completely. 

How a circuit breaker detect a fault with relay?

Primary winding of C.T is connected to the circuit which to be protected and secondary winding to the relay operating coil.There is a tripping circuit which consists of  a source of supply,trip coil and relay stationary circuit.

Under normal condition,current flowing in secondary winding of C.T is low  and current flowing in relay operating coil is small to close the relay contacts.When fault occurs,large current flows through C.T primary,which causes large current in secondary of C.T.This will cause sufficient current for relay operating coil to energize and relay contact will be closed.This will cause tripping coil to energize and the circuit breaker will be tripped.

What is a C.T?

C.T is short form of Current Transformer.In a C.T, current in the primary is stepped down in secondary.It is mainly used in instrumentation and fault detection.It is expressed in ratio. For example,100/1 C.T means if  100 A current flows in primary,then secondary current will be 1 A. 

What is a relay?

A relay is a device which detect the fault and supplies information to the circuit breaker for tripping the circuit.

What is a fuse ?

A fuse is a short piece of wire,or thin strip which melts when excessive current flows through it for a sufficient time.

What is the difference between a circuit breaker and a switch?

A circuit breaker can open or close an electrical circuit under full load, no load and fault condition.But a switch can open or close an electrical circuit under full load and  no load condition only.

What is a circuit breaker ?

A circuit breaker is used to open or close an electrical circuit under full load, no load and fault condition.

What is a switch?

A switch is used to open or close an electrical circuit under full load and no load condition.

What are the equipment related to switchgear?

Switches,fuses,circuit breakers and relays are some of the equipment related to switchgear.

Types of Switchgear according to voltage class

• Low voltage Switchgear-switchgear with voltage rating less than 1 kV AC          
• Medium voltage Switchgear-switchgear with voltage rating from 1 kV AC through to approximately 75 kV AC
• High voltage Switchgear-switchgear with voltage rating from 75 kV to about 230 kV AC
• Extra high voltage Switchgear-switchgear with voltage rating more than 230 kV