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Three Phase Induction Motor:

The three phase induction motors are the most widely used electric motors in the industry.They run at essentially constant speed from no-load to full-load.However, the speed is frequency dependent and consequently, these motors are not easily adapted to speed control.We usually prefer dc motors when large speed variations are required. Nevertheless, the 3-phase induction motors are simple, rugged, low-priced, easy to maintain and can be manufactured with characteristics to suit most industrial requirements.In this, we shall discuss working principle of 3-phase induction motors.

Like any electric motor, a 3-phase induction motor has a stator and a rotor.The stator carries a 3-phase winding (called stator winding) while the rotor carries a short-circuited winding (called rotor winding). Only the stator winding is fed from 3-phase supply.The rotor winding derives its voltage and power from the externally energized stator winding through electromagnetic induction and hence the name.The induction motor may be considered to be a transformer with a rotating secondary and it can, therefore, be described as a “transformer type” ac machine in which electrical energy is converted into mechanical energy.

Construction of Three Phase Induction Motor:

Figure 8.1 shows the construction of three phase induction motor.A 3 phase induction motor has two main parts (i) stator and (ii) rotor. The rotor is separated from the stator by a small air-gap which ranges from 0.4 mm to 4 mm, depending on the power of the motor.

1. Stator :

FIG.(8.1)

It consists of a steel frame which encloses a hollow, cylindrical core made up of thin laminations of silicon steel to reduce hysteresis and eddy current losses. A number of evenly spaced slots are provided on the inner periphery of the laminations.[See Fig.(8.1)].The insulated connected to form a balanced 3-phase star or delta connected circuit. The 3-phase stator winding is wound for a definite number of poles as per requirement of speed.Greater the number of poles, lesser is the speed of the motor and vice-versa.When 3-phase supply is given to the stator winding, a rotating magnetic field(See Sec. 8.3) of constant magnitude is produced.This rotating field induces currents in the rotor by electromagnetic induction.

2. Rotor:

The rotor, mounted on a shaft, is a hollow laminated core having slots on its outer periphery. The winding placed in these slots (called rotor winding) may be one of the following two types:

(i) Squirrel cage type (ii) Wound type

(i) Squirrel cage rotor: It consists of a laminated cylindrical core having parallel slots on its outer periphery.One copper or aluminum bar is placed in each slot.All these bars are joined at each end by metal rings called end rings.This forms a permanently short circuited winding which is indestructible. The entire construction (bars and end rings) resembles a squirrel cage and hence the name.The rotor is not connected electrically to the supply but has current induced in it by transformer action from the stator. Those induction motors which employ squirrel cage rotor are called squirrel cage induction motors.Most of 3 phase induction motors use squirrel cage rotor as it has a remarkably simple and robust construction enabling it to operate in the most adverse circumstances.However, it suffers from the disadvantage of a low starting torque.It is because the rotor bars are permanently short-circuited and it is not possible to add any external resistance to the rotor circuit to have a large starting torque.

(ii) Wound rotor: It consists of a laminated cylindrical core and carries a 3-phase winding, similar to the one on the stator [See Fig. (8.3)].The rotor winding is uniformly distributed in the slots and is usually star-connected.The open ends of the rotor winding are brought out and joined to three insulated slip rings mounted on the rotor shaft with one brush resting on each slip ring. The three brushes are connected to a 3-phase star-connected rheostat as shown in Fig. (8.4).At starting, the external resistances are included in the rotor circuit to give a large starting torque.These resistances are gradually reduced to zero as the motor runs up to speed.

The external resistances are used during starting period only. When the motor attains normal speed, the three brushes are short-circuited so that the wound rotor runs like a squirrel cage rotor.

Working Principle of Three Phase Induction Motor:

Speed of rotating magnetic field:

The speed at which the rotating magnetic field revolves is called the synchronous speed (Ns).Referring to Fig. (8.6 (ii)), the time instant 4 represents the completion of one-quarter cycle of alternating current Ix from the time instant 1.During this one quarter cycle, the field has rotated through 90°. At a time instant represented by 13 or one complete cycle of current Ix from the origin, the field has completed one revolution. Therefore, for a 2-pole stator winding, the field makes one revolution in one cycle of current.In a 4-pole stator winding, it can be shown that the rotating field makes one revolution in two cycles of current.In general, fur P poles, the rotating field makes one revolution in P/2 cycles of current.

The speed of the rotating magnetic field is the same as the speed of the alternator that is supplying power to the motor if the two have the same number of poles.Hence the magnetic flux is said to rotate at synchronous speed.

Three Phase Induction Motor Advantages:

(i) It has simple and rugged construction.

(ii) It is relatively cheap.

(iii) It requires little maintenance.

(iv) It has high efficiency and reasonably good power factor.

(v) It has self starting torque.

Three Phase Induction Motor Disadvantages:

(i) It is essentially a constant speed motor and its speed cannot be changed

easily.

(ii) Its starting torque is inferior to dc shunt motor.

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Principles of Power Systems By V.K Mehta ebook pdf Free Download

Good Evening Friends.Are you looking for "Principles of Power Systems By V.K Mehta ebook" in pdf format for free download ?. If yes then you are on the right place . Glad to see you on my site.I am Amarnath Reddy from Hyderabad.Thank you for browsing my site.Now I am going to provide you "Principles of Power Systems By V.K Mehta ebook" to download for free.

Power systems is a very interesting subject for electrical engineering students because this subject is seen real time in practical as we see transmission lines,electrical poles,distributation lines,protection equipments,relays,switchgear equipments,current transformer and potential transformers daily. You will find here all the concepts which are in original "Principles of Power Systems By V.K Mehta" book .This pdf book is as per JNTU,OU and all other states semester syllabus.It also has objective questions or MCQ's of Power systems at the end of each chapter. Also we provide all free downloads of electrical engineering ebooks,electrical engineering lecturer notes,electrical engineering free pdf.We always try to provide quality content and downloads.This ebook "Principles of Power Systems By V.K Mehta ebook" is already produced on internet for free. We are re-producing this book for free without any copyright issues.We always respect the copyrights of this book and sharing the download link for the benefit of electrical engineering students who are searching for it on internet.

Contents in Principles of Power Systems By V.K Mehta ebook:

Introduction

Generating stations

Variable load on power stations

Economics power generation

Tariff

Power factor improvement

Supply systems

Mechanical design of overhead lines

Electrical design of overhead lines

Performance of transmission lines

Underground cables

Distribution system general

D.C Distribution

A.C Distribution

Voltage control

Introduction to switchgear

Symmetrical fault calculations

Unsymmetrical fault calculations

Circuit breakers

Fuses

Protective relays

Protection of Alternators and transformers

Protection of busbars and lines

Protection againts overvoltages

Substations

Neutral Grounding

Principles of Power Systems By V.K Mehta ebook pdf Free Download

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Transfer Function Of Synchros:

The term synchro is a generic name for a family of inductive devices which works on the principle of a rotating transformer(Induction motor). The trade names for synchros are Selsyn, Autosyn and Telesyn.Basically they are electro-mechanical devices or electromagnetic transducers which produces an output voltage depending upon angular position of the rotor.

A synchro system is formed by interconnection of the devices called the synchro transmitter and the synchro control transformer.They are also called Synchro pair.The synchro pair measures and compares two angular displacements and its output voltage is approximately linear with angular difference of the axis of both the shafts.They can be used in the following two ways,

1.To control the angular position of load from a remote place/long distance.

2.For automatic correction of changes due to disturbance in the angular position of the load.

Must Read:

What is Transformer ?

Construction of Synchro Transmitter:

The constructional features, electrical circuit and a schematic symbol of synchro transmitter are shown in the below figure.The two major parts of synchro transmitter are stator and rotor. The stator is identical to the stator of three phase alternator. It is made of laminated silicon steel and slotted on the inner periphery to accommodate a balanced three phase winding.The stator winding is concentric type with the axis of three coils 120° apart.The stator winding is star connected ( Y-connection).

The rotor is of dumb bell construction with a single winding. The ends of rotor winding are terminated on two slip rings. A single phase ac excitation voltage is applied to rotor through slip rings.

Working principle of Synchro Transmitter:

When the rotor is excited by ac voltage, the rotor current flows, and a magnetic field is produced. The rotor magnetic field induces an emf in the stator coils by transformer action.The effective voltage induced in any stator coil depends upon the angular position of the coil's axis with respect to rotor axis.

Let,

er = Instantaneous value of ac voltage applied to rotor.

es1,es2,es3 = Instantaneous value of emf induced in stator coils S1 ,S2,S3 with respect to neutral respectively.

Er = Maximum value of rotor excitation voltage.

ω = Angular frequency of rotor excitation voltage.

Kt = Turns ratio of stator and rotor windings.

Kc = Coupling coefficient.

θ = Angular displacement of rotor with respect to reference.

Let, the instantaneous value of rotor excitation voltage, er = Er sin ωt

Let the rotor rotates in anticlockwise direction. When the rotor rotates by an angle,θ emfs are induced in stator coils. The frequency of induced emf is same as that of rotor frequency. The magnitude of induced emfs are proportional to the turns ratio and coupling coefficient. The turns ratio, Kt is a constant, but coupling coefficient, K is a function of rotor angular position.

Induced emf in stator coil = Kt Kc Er sin ωt

Let es2 be reference vector.With reference to the above figure, when θ = 0, the flux linkage of coil S2 is maximum and when θ = 90°, the flux linkage of coil S2 is zero. Hence the flux linkage of coil S2 is function of cosθ (i.e., kc= Kt cosθ for coil S2). The flux linkage of coil s3 will be maximum after a rotation of 120° in anticlockwise direction and that of S1 after a rotation of 240°.

Coupling coefficient,Kc for S2 = K1 cos θ

Coupling coefficient,Kc for S3 = K1 cos (θ-120°)

Coupling coefficient,Kc for S1 = K1 cos (θ-240°)

Hence the emfs of stator coils with respect to neutral can be expressed as follows.

es2 = Kt K1 cos θ Er sin ωt = K Er cos θ sin ωt

es3 = Kt K1 cos (θ-120°) Er sin ωt = K Er cos (θ-120°) sin ωt

es1 = Kt K1 cos (θ-240°) Er sin ωt = K Er cos (θ-240°) sin ωt

With reference to the below figure by Kirchoffs voltage law the coil-to-coil emf can be expressed as ,

eS1S2 = eS1 - eS2 = √3 KEr sin (θ + 240° )sin ωt

eS2s3 = eS2 - es3 = √3 KEr sin (θ + 120° )sin ωt

eS3s1 = eS3 - es1 = √3 KEr sin θ sin ωt

Must Read:

Examples of Open & Closed Loop Control Systems

When θ = 0, from equation the above equation we can say that maximum emf is induced in coil S2.But from equation above ,it is observed that the coil-to-coil voltage es3s1 is zero.This position of the rotor is defined as the electrical zero of the transmitter.The electrical zero position is used as reference for specifying the angular position of the rotor. The input to the synchro transmitter is the angular position of its rotor shaft and the output is a set of three stator coil-to-coil voltages.By measuring and identifying the set of voltages at the stator terminals.

It is possible to identify the angular position of the rotor [A device called synchro/digital converter is available to measure the stator voltages and to calculate the angular measure and then display the direction and angle of rotation of the rotor].

Construction of Synchro Control Transfromer:

The constructional features of synchro control transformer is similar to that of synchro transmitter. except the shape of rotor.The rotor of the control transformer is made cylindrical so that the air gap is practically uniform.This feature of the control transformer minimizes the changes in the rotor impedance with the rotation of the shaft. The constructional features, electrical circuit and a schematic symbol of control transformer are shown in the below figure.

Working of Synchro Control Transfromer:

The generated emf of the synchro transmitter is applied as input to the stator coils of control transformer. The rotor shaft is connected to the load whose position has to he maintained at the desired value. Depending on the current position of the rotor and the applied emf on the stator, an emf is induced on the rotor winding.This emf can be measured and used to drive a motor so that the position of the load is corrected.

Synchro as error detector:

The synchro error detector is formed by interconnection of a synchro transmitter and synchro control transformer. In this arrangement, the stator leads of the transmitter are directly connected to the stator leads of the control transformer. The angular position of the transmitter-rotor is the reference input (or the input corresponding to the desired output) and the rotor is excited by ac supply with frequency, ω.

The control transformer rotor is connected to a servo motor and to the shaft of the load, whose position is the desired output. The induced emf (error voltage) available across the rotor slip rings of control transformer is measured by a signal conditioning ciruit.The output of signal conditioning circuit is used to drive motor so.that desired load position is achieved. A simple schematic diagram of synchros as error detector is shown in the below figure.

Initially the shafts of transmitter and control transformer are assumed to be in aligned position. In this position the transmitter rotor will be in eletrical zero position and the control transformer rotor will be in null position and the angular separation of both rotor axis in aligned position is 90°.The null position of a control transformer in a servo system is defined as position of its rotor for which the output voltage on the rotor winding is zero with the transmitter in its electrical zero position.

When, the transmitter rotor is excited, the rotor flux is set-up and emfs are induced in stator coils, These induced emfs are impressed on the stator coils of control transformer.The currents in the stator coils set up flux in control transformer. Due to the similarity in the magnetic construction, the flux patterns produced in the two synchros will be the same if all losses are neglected. The flux patterns are shown in the below figure.

Let the rotor of the transmitter rotate through an angle θ from its electrical zero position. Now the rotor of the control transformer will rotate in the same direction through an angle α from its null position.The net angular separation of the two rotors is equal to (90-θ+α) and the voltage induced in the control transformer rotor is proportional to the cosine of this angle. The error voltage is amplified and used to drive a servo motor.The motor drives the shaft of the synchro control transformer until it comes to a new aligned position at which the error voltage is zero.

Must Read:

Transfer Function of Armature Controlled DC Motor

Voltage across slip rings of control transformer (modulated error voltage)

em = K' Er cos(90 - θ + α) sin ωt

= K' Er cos(90 - (θ - α)) sin ωt

= K' Er sin(θ - α) sin ωt

where K' is a proportional constant

Let φ(t) = θ - α

For small values of φ(t), sin(θ - α) = sin φ(t) ≈ φ(t)

em = K' Er φ(t) sin ωt

From the above equation we can say that the output voltage of the synchro error detector is a modulated signal with carrier frequency, ω (which is same as supply frequency of the transmitter rotor). The magnitude of the modulated carrier wave is proportional to φ(t) and the phase reversals of the modulated wave depend on the sign of φ(t).The signal conditioning circuit demodulates the voltage available across slip rings and develops a demodulated and amplified error voltage to drive the motor.

The demodulated error voltage, e = Ks φ(t)

where K = Sensitivity of synchro error detector in Volts/deg.

On taking Laplace transform of equation above we get,

E(s) = Ks φ(s)

∴ E(s)/φ(s) = Ks

The equation above is the transfer function of the Synchro error detector.

Conclusion:

In this post we have learnt Transfer Function & Construction , Working Principle Of Synchros transmitter.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected] below if you have any queries!

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Transfer Function of Pneumatic System:

Pneumatic system uses compressible fluid as working medium usually air. In pneumatic systems,compressibility effects of gas cannot he neglected and hence dynamic equations are obtained using conservation of mass.In this we will derive the transfer function of pneumatic system.

In pneumatic systems, change in fluid inertia energy and the fluid's internal thermal energy are assumed negligible. In pneumatic system which employs compressible fluid as working fluid, the mass and volume flow rates are not readily interchangeable and the analysis of gas flow is more complicated.

The pneumatic devices involve the flow of gas or air, through connected pipe tines and pressure vessels.Hence the variables of pneumatic system are mass flow rate, qm and pressure, P. The mass now rate is through variable and it is analogous to current.The pressure is across variable and it is analogous to voltage. The two basic elements of pneumatic system are the resistance and capacitance.The restrictions in the pipes and valves offers resistance to gas flow. The gas flow resistance, R is defined as the rate of change in gas pressure difference for a change in gas flow rate.

Must Read:

Transfer Function & Mathematical Model of Hydraulic System

The pneumatic capacitance is defined for a pressure vessel and depends on the type of expansion process involved. The capacitance of a pressure vessel may be defined as the ratio of change in gas stored for a change in gas pressure.

Pneumatic devices are employed in guided missiles,air craft systems, automation of production machinery and in many other fields as automatic controllers.

Advantages of pneumatic system:

The advantages of pneumatic system are, 1)The air or gas used is non-inflammable and so it offers safety from fire hazards. 2)The air or gas has negligible viscosity, compared to the high viscosity of hydraulic fluids. 3)No return pipelines are required since air can be let out, at the end of device work cycle.

The disadvantage in pneumatic system is that the response is slower than that of hydraulic systems, because of the compressibility of the working fluid.

Example of a Pneumatic System:

A simple pneumatic system is shown in the below figure and it consists of a pneumatic bellows in line with the restriction.The pneumatic bellows consists of a hollow chamber with thin pneumatic walls.The side walls of bellows are corrugated and the input and output surface are flat.An increase in pressure within the bellows results in an increase in separation between the input and output surfaces.

Let Pi = Steady state value of input air pressure pi = Increase in the pressure air source P = Steady State value or pressure inside the bellows p =Increase in pressure inside the bellow Qm = Steady state value of air flow rate qm = Area of each flat surface of the bellows R = Resistance of the restriction C = Capacitance of the bellows x= Displacement of the movable surface of the bellows

Let the pressure of air source be increased from its steady state value by an amount pi.This results in an increase in air flow by qm and increase in the pressure inside the bellows by p.Due to increase in pressure, there will be a displacement of the movable surface of the bellows, by an amount x. Here, the terms pi, qm, p and x are all functions of time, t and so can be expressed as pi(t), qm(t), p(t) and x(t). Must Read:

Transfer Function and Mathematical Model of Thermal System

The force exerted on the movable surface of the bellows is proportional to increase in pressure inside the bellows, i.e, fb ∝ p(t). The force exerted on the movable surface of the bellow, fb=A p(t) The force opposing the movement of the flat surface of bellow walls is proportional to displacement i.e, fo ∝ x(t).

Force opposing the motion, fo= K x(t) where K Constant representing stiffness of bellows. At steady state the above two forces are balanced, fb = fo A p(t) = K x(t)

On equating the two equations of qm(t) we get,

From above equation we get , p(t) = K/A x(t) Must Read:

Transfer Function of Field Controlled DC Motor

On differentiating above equation w.r.t t ,we get,

On substituting for p(t) and dp(t)/dt , we get,

On taking Laplace Transform with zero initial conditions , we get,

where t = RC = Time constant of the system The above equation is the required transfer function of the pneumatic system. Conclusion: In this post we have learnt Transfer Function of Pneumatic System.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected] below if you have any queries!

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Mathematical Model of Thermal System:

List of symbols used in thermal system.

q = Heat flow rate, Kcal/sec.

θ1 = Absolute temperature of emitter, °K.

θ2 = Absolute temperature of receiver, °K.

∆θ = Temperature difference, °C.

A = Area normal to heat flow, m²

K = Conduction or Convection coefficient; Kcal/sec-°C.

Kr = Radiation coefficient, Kcal/sec-°C.

H = K/A=Convection coefficient, Kcal/m2-sec-°C.

H = Thermal conductivity , Kcal/m-sec-°C.

∆X = Thickness of conductor,m

R = Thermal resistance, °C-sec/Kcal

C = Thermal capacitance, Kcal/°C

Heat Flow Rate:

Thermal systems are those that involve the transfer of heat from one substance to another.There are three difference ways of heat flow from one substance to another.They are conduction, convection and radiation.

For conduction,

Heat flow rate = q = K ∆θ = KA/∆X

For convection,

Heat flow rate, q= K ∆θ = HA ∆θ

For radiation,

Heat flow rate, q = Kr(θ1⁴ - θ2⁴)

If θ1 » θ2 then, q = Kr θ⁴

Where θ4 = (θ1⁴ - θ2⁴)¹/⁴

Note : θ4 is called effective temperature difference of the emitter and receiver.

Basic elements of thermal system:

The model of thermal systems are obtained by using thermal resistance and capacitance which are the basic elements of the thermal system.The thermal resistance and capacitance are distributed in nature.But for simplicity in analysis umped parameter models are used.

In lumped parameter model it is assumed that the substances that are characterized by resistance to heat flow have negligible heat capacitance and the substances that are characterized by heat capacitance have negligible resistance to heat flow.The thermal resistance, R for heat transfer between two substances is defined as the ratio of change in temperature and change in heat flow rate.

The thermal resistance ,R for heat transfer between two substances is defined as the ratio of change in temperature and change in heat flow rate.

For conduction or convection,

Heat flow rate, q = K ∆θ

On differentiating we get,

dq = K d(∆θ)

d(∆θ)/dq = 1/K

But thermal resistance , R = d(∆θ)/dq

Thermal resistance, R = 1/K for conduction

= 1/K = 1/HA = for convection

For radiation,

Heat flow rate, q = Krθ⁴

On differentiating we get

dq = Kr 4 θ³ dθ

dθ/dq = 1/Kr4 θ³

But thermal resistance, R = dθ/dq

Thermal resistance, R = 1/4Kr θ³ (for radiation)

Thermal Capacitance, C is defined as the ratio of change in heat stored and change in temperature.

Let M = Mass of substance considered, Kg

c = Specific heat of substance, Kcal/Kg -°C

Now, Thermal capacitance, C = Mcp

Examples of Thermal system:

Consider a simple thermal system shown in the below figure.Let us assume that the tank is insulated to eliminate heat loss to the surrounding air, there is no heat storage in the insulation and liquid in the tank is kept at uniform temperature by perfect mixing with the help of a stirrer.Thus, a single temperature is used to describe the temperature of the liquid in the tank and of the outflowing liquid. The transfer function of this system can be derived as shown below.

Let θ1 = Steady state temperature of inflowing liquid, °C

θ0 = Sandy state temperature of outflowing liquid, °C

G = Steady state liquid flow rate, Kg/sec

M = Mass of liquid in tank, Kg

c = Specific heat of liquid, Kcal/Kg °C

R = Thermal resistance, °C - sec/Kcal

C = Thermal capacitance, Kcal/°C

Q = Steady state heat input rate, Kcal/Sec

Let us assume that the temperature of inflowing liquid is kept constant. Let the heat input rate to the system supplied by the heater is suddenly changed from Q to Q + q1.Due to this, the heat output flow rate will gradually change from Q to Q + q0.The temperature of the outflowing liquid will also be changed from θ0 to θ0 + θ.

For this system the equation for q0, C and R are obtained as follows,

Thermal capacitance, C = Mass, M x Specific heat of liquid, c = Mc

On substituting for qo from equation (1) in equation (3) we get,

In this system,rate of change of temperature is directly proportional to change in heat input rate.

The constant of proportionality is capacitance C of the system.

Equation (5) is the differential equation governing system.Since equation (5) is of first order equation ,the system is first order system.

From equation (3) , R = θ/q0

q0 = θ/R

On substituting for q0 from equation (6) in equation (5) we get,

Let, L {θ} = θ(s); L{dθ/dt} = sθ(s) ; L{q1} = Q1(s)

On taking Laplace transform of equation (7)

RC s θ(s) + θ(s) = R Q1(s)

θ(s) [sRC + 1] = R Q1(s)

θ(s)/Q1(s) is the required transfer function of the system.

Conclusion:

In this post we have learnt Transfer Function and Mathematical Model of Thermal System.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected] below if you have any queries!

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Signal Flow Graph in Control Systems:

Signal flow graph is used to represent the control system graphically and it was developed by S.J.Mason.Signal flow graph is a diagram that represents a set of simultaneous linear algebraic equations.By differential equations governing a control system can be algebraic equations in s-domain.The signal flow graph of the system can be constructed using these equations.

It should be noted that the signal flow graph approach and the block diagram approach yield the same information.The advantage in signal flow graph method is that, using Mason's gain formula the overall gain of the system can be computed easily.This method is simpler than the tedious block diagram reduction techniques.

The signal flow graph depicts the flow of signals from one point of a system to another and give, relationships among the signals. A signal flow graph consists of a network in which nodes are connected by directed branches.Each node represents a system variable and each branch connected between two nodes acts as a signal multiplier.Each branch has a gain or transmittance.When the signal pass through, a branch, it gets multiplied by the gain of the branch.

Must Read:

What is Control Systems and Types of Control systems

In a signal flow graph, the signal flows in only one direction.The direction of signal flow indicated by an arrow placed on the branch and the gain (multiplication factor) is indicated along the branch.

Explanation of terms used in Signal Flow Graph:

Node : A node is a point representing a variable or signal.

Branch : A branch is directed line segment joining two nodes. The arrow on the branch indicates the direction of signal flow an the gain of a branch. is the transmittance.

Transmittance : The gain acquired by the signal when it travels from one node to another is called transmittance.The transmittance can he real or complex.

Input node ( Source) : It is a node that has only outgoing branches.

Output node ( Sink ) : It is a node that has only incoming branches.

Mixed node : It is a node that has both incoming and outgoing branches.

Path : A path is

(vii) The signals travel along branches only in the marked direction and then it gets multiplied by the gain or transmittance of the branch.

(viii) The signal flow graph of system is not unique.By rearranging the system equations different types of signal flow graphs can be drawn for a given system.

Must Read:

Examples of Open & Closed Loop Control Systems

Signal Flow Graph Algebra:

Signal flow graph for a system can be reduced to obtain the transfer function on of the system using the following rules.The guideline in developing the rules for signal flow graph algebra is that the signal at a anode is given by sum of all incoming signals.

Rule 1 : Incoming signal to a node through a branch is given by the product of a signal at previous node and the gain of the branch.Check below solved examples.

Rule 2 : Cascaded branches can be combined to give a single branch whose transmittance is equal to the product of individual branch transmittance.Check below solved examples.

Rule 3 : Parallel branches may be represented by single branch whose transmittance is the sum of individual branch transmittance.Check below solved examples.

Ruled 4 : A mixed node can be eliminated by multiplying the transmittance of outgoing branch (from the mixed node) to the transmittance of all incoming branches to the mixed node.Check below solved examples.

Rule 5 : A loop may be eliminated by writing equations at the input and output node and rearranging the equations to find the ratio of output to input.This ratio gives the gain of resultant branch.Check below solved examples.

Signal Flow Graph Reduction:

The signal flow graph of a system can he reduced either by using the rules of a signal flow graph algebra or by. using Mason's gain formula. For signal flow graph reduction using the rules of signal flow graph, write equations at every node and then rearrange these equations to get the ratio of output and input (transfer function).

The signal flow graph reduction by above method will be time consuming and tedious.S.J.Mason has.developed a simple procedure to determine the transfer function of the system represented as a signal flow graph.He has developed a formula called by his name Mason's gain formula which can be directly used to find the transfer function of the system.

Mason's Gain Formula:

The Mason's gain formula is used to determine the transfer function of the system from the signal flow graph of the system.

Let R(s) = Input to the system

C(s) = Output of the system

Now, Transfer function of the system, T(s) = C(s)/R(s)

Mason's Gain formula states the overall gain of the system [transfer function] as follows,

where, T = T(s) = Transfer function of the system

PK = Forward path gain of Kth forward path

K = Number of forward paths in the signal flow graph

Δ = 1 — (Sum of individual loop gains) + (Sum of gain products of all possible -combinations of two non-touching loops) - (Sum of gain products of all possible combinations of three non - touching loops)

ΔK = Δ for that part of the graph which is not touching Kth forward path

Constructing Signal Flow Graph for Control Systems:

A control system can be represented diagrammatically by signal flow graph.The differential equations governing the system are used to construct the signal flow graph.The following procedure can be used to construct the signal flow graph of a system.

1)Take Laplace transform of the differential equations governing the system in order to convert them to algebraic equations in s-domain.

2. The constants and variables of the s-domain equations are identified.

3. From the working knowledge of the system, the variables are identified as input, output and intermediate variables.

4. For each variable anode is assigned in signal flow graph and constants are assigned as the gain or transmittance of the branches connecting the nodes.

5. For each equation a signal flow graph is drawn and then they are interconnected to give overall signal flow graph of the system.

Must Read:

Transfer Function of Field Controlled DC Motor

Procedure for converting block diagram to signal flow graph:

The signal flow graph and block diagram of a system provides the same information but there is no standard procedure for reducing the block diagram to find the transfer function of the system.Also the block diagram reduction technique will be tedious and it is difficult to choose the rule to be applied for simplification.

Hence it will be easier if the block diagram is converted to signal flow graph and Mason's gain formula is applied to find the transfer function.The following procedure can be used to convert block diagram to signal flow graph.

1. Assume nodes at input, output,at every summing point, at every branch point and in between cascaded blocks.

2. Draw the nodes separately as small circles and number the circles in the order 1, 2, 3, 4, ..... etc.

3. From the block diagram find the gain between each node in the main forward path and connect all the corresponding circles by straight line and mark the gain between the nodes.

4. Draw the feed forward paths between various nodes and mark the gain of feed forward path along with sign.

5. Draw the feedback paths between various nodes and mark the gain of feedback paths along with sign.

Conclusion:

In this post we have learnt Signal Flow Graph in Control Systems with Solved Examples.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected] below if you have any queries!

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Transfer Function of Field Controlled DC Motor:

The speed of a DC motor is directly proportional to armature voltage and inversely proportional to flux. In field controlled DC motor the armature voltage is kept constant and the speed is varied by varying the flux of the machine. Since flux is directly proportional to field current, the flux is varied by varying field current.

The speed control system is an electro-mechanical control system.The electrical system consists of armature and field circuit but for analysis purpose, only field circuit is considered because the armature is excited by a constant voltage.The mechanical system consists of the rotating part of the motor and the load connected to the shaft of the motor.The field controlled DC motor speed control system is shown in the below figure.For this field controlled DC motor we shall find transfer function.

Let Rf = Field resistance

Lf = Field inductance

if = Field current

Vf= Field voltage

T = Torque developed by motor

Kt = Torque constant

J = Moment of inertia of rotor and load

B = Frictional coefficient of rotor and load

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Speed Control of D.C. Series & Shunt Motors Methods

The equivalent circuit of field is shown in the below figure.

By Kirchoff 's voltage law, we can write

The torque of DC motor is proportional to product of flux and armature current. Since armature current is constant in this system, the torque is proportional to flux alone, but flux is proportional to field current.

T ∝ ir

Torque , T = Ktf ir

The mechanical system of the motor is shown in the below figure.

The differential equation governing the mechanical system of the motor is given by,

The Laplace transform of various time domain signals involved in this system are shown below.

L{if} = If(s) ; L{T} = T(s) ; L{vf} = Vf(s) ; L{θ} = θ(s)

The differential equations governing the field controlled DC motor are,

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What is Control Systems and Types of Control systems

On taking Laplace transform of the above equations with zero initial condition we get,

Rf If(s) + Lf s If(s) = Vf(s) => (1)

T(s) = Ktf If(s) => (2)

J s² θ(s) + B s (s) = T(s) => (3)

Equating equations (2) & (3) we get,

=> (4)

The equation (1) can be written as

(Rf + sLf) If(s) = Vf(s) => (5)

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Examples of Open & Closed Loop Control Systems

On substituting to If(s) from equation (4) in equation (5) we get,transfer function of field controlled dc motor

where Km = Ktf/Rf B = Motor gain constant

Tf = Lf/Rf = Field time constant

Tm = J/B = Mechanical time constant

Conclusion:

In this post we have learnt Transfer Function of Field Controlled DC Motor.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected] below if you have any queries!

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Transfer Function of Armature Controlled DC Motor:

The speed of DC motor is directly proportional to armature voltage and inversely proportional to flux in field winding.In armature controlled DC motor the desired speed is obtained by varying the armature voltage.This speed control system is an electro-mechanical control system.

The electrical system consists of the armature and the field circuit but for analysis purpose, only the armature circuit is considered because the field is excited by a constant voltage.The mechanical system consist of the rotating part of the motor and load connected to the shaft of the motor.The armature controlled DC motor speed control system is shown in the below figure.

Let Ra = Armature resistance,

La = Armature inductance

Ia = Armature current

Va = Armature voltage

eb = Back emf

Kt = Torque constant

T = Torque developed by motor

θ = Angular displacement of shaft

J = Moment of inertia of motor and load

B = Frictional coefficient of motor and load

Kb= Back emf constant

Must Read:

Speed Control of D.C. Series & Shunt Motors Methods

The equivalent circuit of armature is shown in the below figure.

By Kirchoff's voltage law, we can write,

Torque of DC motor is proportional to the product of flux and current.Since flux is constant in this system, the torque is proportional to ia alone.

The mechanical system of the motor is shown in the below figure.

The differential equation governing the mechanical system of motor is given by

The back emf of DC machine is proportional to speed (angular velocity) of shaft.

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What is Control Systems and Types of Control systems

The Laplace transform of various time domain signals involved in this system are shown below.

L{va} = Va(s); L{eb} = Eb(s) ; L{T}= T(s) ; L{ia} = la (s) ; L{θ} = θ(s)

The differential equations governing the armature controlled DC motor speed control system are

On taking Laplace transform of the above equations with zero initial conditions we get,

Ia(s) Ra + La sIa(s) + Eb(s) = Va(s) => (1)

T(s) = Kt Ia(s) => (2)

J s θ(s) + B s θ(s) = T(s) => (3)

Eb(s) = Kb s θ(s) => (4)

On equating the above equations (2),(3) we get

=> (5)

Equation (1) can be written as,

(Ra + sLa) Ia(s) + Eb(s) = Va(s) => (6)

Substituting for Eb(s) & Ia(s) from (4),(5) respectively in equation (6)

The required transfer function is θ(s)/Va(s)

=> (7)

Must Read:

Examples of Open & Closed Loop Control Systems

The transfer function of armature controlled dc motor can be expressed in another standard form as shown below. From the equation (7) we get,

where La/Ra = T = Electrical time constant

J/B = Tm = Mechanical time constant

Conclusion:

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Example 1 : Electrical Furnace

Open Loop System:

The electric furnace shown in the below is an open loop system.The output in the system is the desired temperature.The temperature of the system is raised by heat generated by the heating element.The output temperature depends on the time during which the supply to heater remains ON.The ON and OFF of the supply is governed by the time setting of the relay.

The temperature is measured by a sensor, which gives art analog voltage corresponding to the temperature of the furnace.The analog signal is converted to digital signal by an Analog to digital converter (AD converter).The digital signal is given to the digital display device to display the temperature.In this system if there is any change in output temperature then the time setting of the relay is not altered automatically.

Closed Loop System:

The electric furnace shown in the below figure is a closed loop system. The output of the system is the desires temperature and it depends on the time during which the supply to heater remains ON.

The switching ON and OFF of the relay is controlled by a controller which is a digital system or computer. The desired temperature is input to the system through keyboard or as a signal corresponding to desired temperature via ports. The actual temperature is sensed by Sensor and converted to digital signal by the A/D converter. The computer reads the actual temperature and compares with desired temperature. If it finds any difference then it sends signal to switch ON or OFF the relay through D/A converter and amplifier. Thus the system automatically corrects any changes in output.Hence it is a closed loop system.

Example 2:Traffic Control System

Open Loop System:

Traffic control by means of traffic signals operated on a time basis constitutes an open loop control system.The sequence of control signals are based on a time slot given for each signal.The time slots are decided based on a traffic study. The system will not measure the density of the traffic before giving the signals. Since the time slot does not changes according to traffic density, the system is open loop system.

Closed Loop System:

Traffic control system can be made as a closed loop system if the time slots of the signals are decided based on the density of traffic. In closed loop traffic control system, the density of the traffic is measured on all the sides and the information is fed to a computer.The timings of the control signals are decided by the computer based on the density of traffic Since the closed loop system dynamically changes the timings, the flow of vehicles will be better than open loop system.

Example 3:Numerical Control Sytem

Open Loop System:

Numerical control is a method of controlling the motion of machine components using of numbers. Here, the position of work head tool is controlled by the binary information contained in a disk.A magnetic disk is prepared in binary form representing the desired part P (P is the metal part to be machined). The tool will operate on the desired part P.To start the system, the disk is fed through the reader to the D/A converter.

The D/A converter converts the FM(frequency modulated) output of the reader to a analog signal. It is amplified and fed to servomotor which positions the cutter on the desired part P. The position of the cutter head is controlled by the angular motion of the servomotor.This is an open loop system since no feedback path exists between the output and input.The system positions the tool for a given input command.Any deviation in the desired position is not checked and corrected automatically.

Closed Loop System:

A magnetic disk is prepared in binary form representing the desired part P (P is the metal part to be machined). To start the system, the disk is loaded in the reader.The controller compares the frequency modulated input pulse signal with the feedback pulse signal.The controller is a computer or microprocessor system. The controller carries out mathematical operations on the difference in the pulse signals and generates an error signal.The D/A converter converts the controller output pulse (error signal) into an analog signal.The amplified analog signal rotates the servomotor to position the tool on the job.

The position of the cutterhead is controlled according to the input of the servomotor.The transducer attached to the cutterhead converts the motion into an electrical signal.The analog electrical signal is converted to the digital pulse signal by the A/D converter.Then this signal is compared with the input pulse signal. If there is any difference between these two, the controller sends a signal to the servomotor to reduce it.Thus the system automatically corrects any deviation in the desired output tool position.An advantage of numerical control is that complex parts can be produced with uniform tolerances at the maximum milling speed.

Example 4:Position Control System Using Servomotor

The position control system shown in the below is a closed loop system. The system consists of a Servomotor powered by a generator.The load whose position has to be controlled is connected to motor shaft through gear wheels. Potentiometers are used to convert the mechanical motion to electrical signals. The desired load position (θR) is set on the input potentiometer and the actual load position (θc) is fed to feedback potentiometer.

The difference between the two angular positions generates an error signalnwhich is amplified and fed to generator field circuit. The induced emf of the generator drives the motor. The rotation of the motor stops when the error signal is zero, i.e. when the desired load position is reached.This type of control systems are called servomechanisms.The servo or servomechanisms are feedback control systems in which the output is mechanical position (or time derivatives of position e.g. velocity and acceleration).

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Measurement of Capacitance:

A circuit for measurement of capacitance By Transformer Ratio Bridge is given in the below figure.An unknown capacitance C is measured in comparison with a standard capacitance C, which is assumed to be perfect.Resistance R represents the loss of the capacitor.Since for balance, the magnitude and phase of the currents passing through the detector should be the same, a variable standard resistance is connected in parallel with the standard capacitor.

Must Read:

Construction and Working principle of Transformer

Measurement of Phase Angle:

A circuit for measurement of small phase angles by ratio transformers bridge is shown in the below figure.An RC circuit is used where the capacitance is variable in order to get phase shift. The value of resistance should be large in order that there are no loading effects on the ratio transformer. The capacitance is changed till the detector indicates null.

Phase angle ∮ = tan⁻¹(-ωRC)

The magnitude of in-phase component is

Transformer Double Ratio Bridges:

The transformer ratio bridges described till now in this chapter are similar to the conventional a.c. bridges in the sense that the balance conditions are indicated when the voltage across the detector is zero and therefore, the current through the detector is zero.

An alternate way of indication of balance is that the current flowing through the unknown impedance is equal and opposite to that through the standard or known impedance.The detector indicates this condition of equality. The arrangement of Transformer Double Ratio Bridges is shown in the below figure.It consists of an ideal voltage transformer having a secondary winding of N1 turns.The secondary winding is tapped at N2 turns.The resulting voltages E1 and E2 are applied across the unknown impedance, Z, and standard impedance, Zs, respectively.

The two voltages E1 and E2 produce currents I1 and I2 respectively. These currents flow in the windings n1 and n2 of a current comparator.The two mmfs n1I1 and n2I2 produce fluxes in the core of the comparator.The two fluxes oppose each other and when there is no flux in the core, no emf is induced in the secondary winding of this comparator and the detector D indicates null condition.

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At balance, the two mmfs, n1I1 = n2I2

The balance is obtained by changing the setting of N2.

If the unknown impedance is a pure resistance R, a standard resistance Rs is required for obtaining balance

If the unknown impedance is a pure capacitance C, a standard capacitor Cs is required for balance,

Above equation can also be converted to a form similar to that of Eqn. 16.78. This is done by using conductance in place of resistance.

Unknown conductance

where Gs is the standard conductance.

In practice, the capacitors are not perfect, the losses being represented by a conductance in parallel with the capacitance.Therefore, balance has to be obtained both for resistance as well as for capacitance.In other words, mmf equality for both magnitude as well as for phase has to be obtained in order to get balance. A circuit used of measurement of an imperfect capacitor in terms of standard capacitor and standard conductance is shown in the below figure.

Under balance,

It is clear from the above equations that the transformer double ratio bridge allows the reactive and resistive components of an unknown to be adjusted independently of each other (capacitive balance can be had by changing N21 and N22 while resistive balance can be obtained by changing N'21 and N'22).This property of independent balance facilitates the use of this bridge in a very wide range of applications.

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Measurement of Inductance:

The circuit for measurement of inductance is the same as that for measurement of capacitance except that the unknown capacitance is replaced by the unknown inductance.

In order that independent balance of the resistive and reactive mmfs can be obtained any inductive circuit must be considered as a two terminal network, the components of which are in parallel. The mmf due to the resistive component of the unknown may then be opposed by that through the conductance standard.On the other hand, the mmf due to inductive component of the unknown must be balanced due to capacitance standard.

In order to obtain mmf balance due to unknown inductance and the standard capacitance, it is necessary to reverse the comparator winding connected to the standard capacitor.This condition of bridge operation is normally indicated by addition of a negative sign to the display of capacitance Cm.Suppose an inductance Lp in parallel with a resistance Rp represents the actual inductor having an inductance L in series with a resistance R.

Measurement of Components in 'SITU':

One of the greatest advantages of double ratio transformer bridges is their capability to measure the values of the components while they remain connected in the circuit.The below figure shows a circuit for measurement of an unknown impedance Z.

If the ratio transformers are considered ideal the values of impedances Ze and Zi can be assumed to be zero and the voltage applied to the unknown impedance Z is not affected by current flowing in impedance Z1. Also all the current flowing through Z would flow into winding as :

The value of unknown approximately given by :

In practice impedances Ze and Zi have small finite values and corrections must be applied for this.In order that corrections be applied Ze and Zi must be determined.

Conclusion:

In this post we have learnt Measurement of Capacitance,Inductance,phase angle By Transformer Ratio Bridges.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected]. Comment below if you have any queries!

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What are Transformer Ratio Bridges:

The transformer Ratio Bridges are becoming increasingly popular and are being used for a wide range of applications. This is on account of versatility and accuracy of Ratio Transformers, which are used in the transformer ratio bridges.In fact, transformer ratio bridges are replacing the conventional a.c. bridges at a rapid rate.

A transformer ratio bridge consists of voltage transformer whose performance approaches that of an ideal transformer. An ideal transformer is one that has no resistance, no core loss and no leakage flux (i.e., there is perfect coupling between the windings).The ratio transformer is provided with a number of tappings in order to obtain voltage division.

Voltage appearing across the windings of a transformer is :

E = 4 Kf N φm f volt

where N = number of turns

φm = maximum value of flux ; Wb,

f = frequency ; Hz,

Kf = form sector ,(Its value is 1.11 for sinusoidal flux).

For a given value of K flux φm and frequency f,

E = K1N Must Read:

Construction and Working principle of Transformer

The below figure shows an autotransformer provided with tappings. Suppose an alternating voltage E is applied across the winding.Assuming that the autotransformer is ideal type, the division of applied voltage E into output voltages E1 and E2 is : E1 = E. N1/N and E2 = E. N2/N

Different values of E1 and E2 may be had by changing the position of wiper on the tappings. However, in practice it is impossible to construct an ideal transformer. But the ideals of zero winding resistance, zero core loss and perfect coupling can be closely achieved if the design features similar to those for instrument transformers am used. The material used for construction of core should be such that it gives the smallest core losses at the desired operating frequency. The magnetizing current is reduced by using a Toroidal Core.The added advantage of toroidal core is that winding put on it has minimum leakage reactance giving an almost perfect coupling.The leakage reactance can be reduced further by using a special type of construction for the windings as shown in the below figure.This winding takes the form of a Multiconductor Rope. In order to obtain a decade of voltage division,the multiconductor rope has ten wires with successive sets of turns connected in series and a tapping is taken from each joint.

The resistance of the windings can be reduced by using copper wire of heavy cross-section.A 4-decade ratio transformer is shown.The successive decades are obtained by using an arrangement similar to that in a Kelvin Varley slide.This transformer arrangement gives a ratio error of less than 1 part in 10⁴.

Must Read:

Ideal Transformer | Elementary Theory of an Ideal Transformer

Applications and Features of Ratio Transformers:

Applications of Ratio Transformers:

The ratio transformers can be used for :

(i) Measurement of resistance capacitance and inductance in comparison with standard resistance, standard capacitance and standard inductance respectively, (ii) Measurement of amplifier gain and phase shift, and (iii) Measurement of transformer ratios.

Features of Ratio Transformers :

The ratio transformers have the following features : (i) They can be used on a.c. only. (ii) They have very small ratio errors. (iii) They have a wide frequency range extending from 50 Hz to 50 kHz. (iv) They have high input impedance and low input impedance. Thus the loading effects in them are small.

Measurement of Resistance By Transformer Ratio Bridges:

The circuit used for measurement of an unknown resistance, R, in comparison with a standard resistance, Rs is shown in the below figure. The position of the wiper is adjusted till the detector D shows null.(The detectors used are the same as for conventional a.c. bridges.)

Under balance conditions, the current through the detector is zero i.e., I1 = 12.

A circuit used for measurement of low resistance is shown in the below figure.This circuit is similar to that of Kelvin's double bridge.

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By using two transformers a form of Kelvin's double bridge for measurement of low resistances may be devised as shown in the below figure, where

If the impedances of leads i.e., Z1, Z2, Z3, Z4, and Z5 are small and the resistances R and Rs are of the same order, the unknown resistance is given by : R = (N1/N2)Rs This form of Kelvin's bridge can, of course, only be used on ac but by plotting the values R against frequency, an extrapolation may be used to find the d.c. value of R with an accuracy of a few parts in a million. Conclusion: In this post we have learnt Transformer Ratio Bridges Working Principle.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected]. Comment below if you have any queries!

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Measurement of Frequency by Wien's Bridge:

Some bridges have balance equations which involve frequency directly even if the performance of individual bridge elements is independent of frequency. Wien's bridge may be used for determination of frequency in terms of values of various bridge elements.We shall describe here the Wien's bridge, which is the most important, one for determination of frequency.

Wien's Bridge:

The Wien's bridge is primarily known as a frequency determining bridge and is described here not only for its use as an ac bridge to measure frequency but also for its application in various other useful circuits.A Wien's bridge, for example, may be employed in a harmonic distortion analyzer, where it is used as notch filter, discriminating against one specific frequency. The Wien's bridge also finds applications in audio and HF oscillators as the frequency determining device.The below figure shows a Wien's bridge under balance conditions. Must Read:

Measurement of Mutual Inductance By Heaviside Bridge

Must Read:

Measurement of Capacitance By Schering Bridge

In most Wien bridges, the components are so chosen that

R1 = R2 = R and C1 = C2 = C

Then first equation reduces to,

R4/R3=2

and second equation reduces to

f =1/2πRC

Switches for resistors R1 and R2 are mechanically linked so as to fulfil the condition R1 = R2.

As long as C1 and C2 are fixed capacitors equal in value and R4 =2 R3; the Wien's bridge may be used as a frequency determining device, balanced by a single control.This control may be directly calibrated in terms of frequency as is evident from the above equation.Wien's bridge is suitable for measurement of frequencies from 100 Hz to 100 kHz. It is possible to obtain an accuracy of 0.1 to 0.5 per cent.

Because of its frequency sensitivity, the Wien's bridge may be difficult to balance unless the waveform of the applied voltage is sinusoidal.Wien's bridge is not balanced for any harmonics present in the applied voltage, so that these harmonics will sometimes produce an output voltage marking the true balance point.This difficulty can be overcome by connecting a filter in series with the null detector.A Wien's bridge may be used for measurement of capacitance also. Must Read:

Measurement of Capacitance By De Sauty's Bridge

Conclusion: In this post we have learnt Measurement of Frequency by Wien's Bridge.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected] below if you have any queries!

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Heaviside Mutual Inductance Bridge:

Heaviside Bridge (shown in the below figure) measures mutual inductance in terms of a known self- inductance.The same bridge, slightly modified, was used by Campbell to measure a self-inductance in terms of a known mutual inductance.

Let M = unknown mutual inductance,

L1 = self-inductance of secondary of mutual inductance,

L2 =known self-inductance, and

R1,R2,R3,R4 = non-inductive resistors.

At balance voltage drop between b and c must equal the voltage drop between d and c. Also the voltage drop across a-b-c must equal the voltage drop across a-d-c, Thus we have the following equations at balance

It is clear from the above equation, that L1, the self inductance of the secondary of the mutual inductor must be known in order that M be measured by this method.

In case R3 = R4

we get M = (L2-L1)/2 and R1= R2

This method can be used for measurement of self-inductance. Supposing L2 is the self-inductance to be determined. From above equations, we get

Campbell's Modification of Heaviside Bridge:

The below figure shows a modified Heaviside bridge.This modification is due to Campbell.This is used to measure a self-inductance in terms of a mutual inductance. In this case an additional balancing coil R is included in arm ad in series with inductor under test.An additional resistance r is put in arm ab. Balance is obtained by varying M and r. A short circuiting switch is placed across the coil R2, L2 under measurement.Two sets of readings are taken one with switch being open and the other with switch being closed. Let values of M and r, be M1 and r1 with switch open, and M2 and r2 with switch closed.

With switch open from above equation we have,

Hence L2 = (M1-M2)(1 +R4/R3)

With switch open, we have, R2+ R=(R1+r1)R4/R3

and with switch closed R=(R1+r2)R4/R3 R2 = (r1-r2) R4/R3

This method is a good example of the methods adopted to eliminate the effects of leads etc. When we have equal ratio arms R3 = R4 and therefore from above equations we get, L2 = (M1-M2) and R2 = (r1-r2)

Heaviside Campbell Equal Ratio Bridge:

The use of balancing coil in the above method reduces the sensitivity of the bridge.The below figure shows Heaviside Campbell equal ratio bridge. This is a better arrangement which improves sensitivity and also dispenses with the use of a balancing coil. In this method the secondary of the mutual inductor is made up of two equal coils each having a self-inductance. One of the coils is connected in arm ab and the other in arm ad. The primary of mutual inductance reacts with both of them. L2, R2 is the coil whose self-inductance and resistance is to be determined. The resistances R3 and R4 are made equal.Balance is obtained by varying the mutual inductance and resistance r.

At balance I1R3 = I2R4 but R3 = R4 and therefore I1 = I2 = I/2 as I = I1 + I2 Writing the other equation for balance

Equating the real and imaginary terms

R2 = R1 + r and L2 =2(Mx+My)=2M

Thus the magnitude of inductance which can be measured with this method is twice the range of the mutual inductor.The values calculated above include the effects of leads etc. In order to eliminate these effects, we take two readings with switch open circuited and another with switch closed.Let M1, r1 be the readings of M,r with open circuit and M2, r2 with short circuit. R1 = (r1-r2) and L1 = 2(M1-M2) Conclusion:

In this post we have learnt Measurement of Mutual Inductance By Heaviside Mutual Inductance Bridge.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected]. Comment below if you have any queries!

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Measurement of Capacitance of Schering Bridge:

The connection and phasor diagram of the bridge under balance conditions are shown in the below figure.

Let

C1 = capacitor whose capacitance is to be determined,

r1 = a series resistance representing the loss in the capacitor

C2 = a standard capacitor.This capacitor is either an air or a gas capacitor and hence is loss-free.However, if necessary, a correction may be made for the loss angle of this capacitor,

R3 = a non-inductive resistance,

C4 = a variable capacitor, and

R4 = a variable non-inductive resistance in parallel with variable capacitor C4.

Equating the real and imaginary terms, we obtain

r1 = R3C4/C2

and C1 = C2 ( R4/R3)

Two independent balance equations are obtained if C4 and R4 are chosen as the variable elements.

Dissipation factor,

D1 = tan δ = ωC1r1 =ω(C2R4/R3) x (R3C4/C2) = ωC4R4

Therefore values of capacitance C1, and its dissipation factor are obtained from the values of bridge elements at balance.

Permanently set up Schering bridges are some times arranged so that balancing is done by adjustment of R2 and C4 with C2 and R4 remaining fixed. Since R3 appears in both the balance equations and therefore there is some difficulty in obtaining balance but it has certain advantages as explained below :

The equation for capacitance is C1 = (R4/R3) C2 and since R4 and C2 are fixed, the dial of resistor R3 may be calibrated to read the capacitance directly.

Dissipation factor D1 = ωC4R4 and in case the frequency is fixed the dial of capacitor C4 can be calibrated to read the dissipation factor directly.

Let us say that the working frequency is 50 Hz and the value of R4 is kept fixed at 3,180 Ω.

Dissipation factor,

D1 = 2π x 50 x 3180 x C4 = C4 X 10⁶.

Since C4 is a variable decade capacitance box, its setting in tiF directly gives the value of the dissipation factor. .

It should, however, be understood that the calibration for dissipation factor holds good for one-particular frequency, but may be used at another frequency if correction is made by multiplying by the ratio of frequencies.

Conclusion:

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Measurement of Capacitance By DeSauty's Bridge:

DeSauty's Bridge is the simplest method of comparing two capacitances. The connections and the phasor diagram of this bridge are shown in the below figure.

Let C1 = capacitor whose capacitance is to be measured,

C2 = a standard capacitor, and

R3, R4 = non-inductive resistors.

The balance can be obtained by varying either R3 or R4.The advantage of DeSauty's Bridge is its simplicity.But this advantage is nullified by the fact that it is impossible to obtain balance if both the capacitors are not free from dielectric loss.Thus with DeSauty's Bridge only loss-less capacitors like air capacitors can be compared. In order to make measurements on imperfect capacitors (i.e., capacitors having dielectric loss), DeSauty's Bridge is modified as shown in the below figure.This modification is due to Grover.

Resistors R1 and R2 are connected in series with C1and C2 respectively. r1 and r2 are resistances representing the loss component of the two capacitors.-

The balance may be obtained by variation of resistances R1, R2, R3, R4.The above figure (b) shows the phasor diagram of the DeSauty's Bridge under balance conditions.The angles δ1 and δ2 are the phase angles of capacitors C1 and C2 respectively.

Dissipation factors for the capacitors are :

D1 = tan δ1 = ωC1r1 and

D2 = tan δ2 = ωC2r2

From above equation, we have

Therefore, if the dissipation factor of one of the capacitors is known, the dissipation factor for the other can be determined. This method does not give accurate results for dissipation factor since its value depends on difference of quantities R1 R4 I R3 and R2. These quantities are moderately large and their difference is very small and since this difference cannot be known with a high degree accuracy, the dissipation factor cannot be determined accurately.

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Measurement of Self Inductance by Owen's Bridge :

Owen's bridge may be used for measurement of an inductance in terms of capacitance.The below figure shows the connections and phasor diagrams, for Owen's bridge, under balance conditions.

Let

L1 = unknown self-inductance of resistance

R2 = variable non-inductive resistance,

R3 = fixed non-inductive resistance,

C2 = variable standard capacitor,

C4 = fixed standard capacitor

At balance,

Advantages of Owen's Bridge:

1)Examining the equations for balance, We find that we obtain two independent equations in case C2 and R2 are made variable.Since R2 and C2, the variable elements,are in the same arm,convergence to balance conditions is much easier.

2)The balance equations are quite simple and do not contain any frequency component.

3)Owen's Bridge can be used over a wide range of measurement of inductances.

Disadvantages of Owen's Bridge:

1)Owen's bridge requires a variable capacitor which is an expensive item and also its accuracy is about 1 percent.

2)The value of capacitance C2 tends to become rather large when measuring high Q coils.

Conclusion:

In this post we have learnt Measurement of Self Inductance by Owen's Bridge.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected]. Comment below if you have any queries!

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Measurement of Self Inductance By Anderson's Bridge:

Anderson's Bridge, in fact, is a modification of the Maxwell's inductance capacitance bridge.In Anderson's Bridge, the self-inductance is measured in terms of a standard capacitor.Anderson's Bridge is useful applicable for precise measurement of self-inductance over a very wide range of values.The below figure shows the connections and the phasor diagram of the bridge for balanced conditions.

Let L1 = self-inductance to be measured,

R1 = resistance of self-inductor,

r = resistance connected in series with self-inductor,

r, R2, R3, R4 = known non-inductive resistances, and

C = fixed standard capacitor

At balance,

An examination of balance equations reveals that to obtain easy convergence of balance, alternate adjustments of r1 and r should be done as they appear in only one of the two balance equations.

Advantages of Anderson's Bridge:

1. In case adjustments are carried out by manipulating control over r1 and r, they become independent of each other. This is a marked superiority over sliding balance conditions met with low Q coils when measuring with Maxwell's bridge. A study of convergence conditions would reveal that it is much easier to obtain balance in the case of Anderson's bridge than in Maxwell's bridge for low Q-coils.

2. A fixed capacitor can be used instead of a variable capacitor as in the case of Maxwell's bridge.

3. Anderson's Bridge may be used for accurate determination of capacitance in terms of inductance.

Disadvantages of Anderson's Bridge :

1. The Anderson's bridge is more complex than its prototype Maxwell's bridge. The Anderson's bridge has more parts and is more complicated to set up and manipulate. The balance equations are not simple and in fact are much more tedious.

2. An additional junction point increases the difficulty of shielding the bridge.

Conclusion:

In this post we have learnt Measurement of Self Inductance By Anderson's Bridge.You can download this article as pdf,ppt.If you have queries you can mail us @ [email protected]. Comment below if you have any queries!

#IFTTT #Blogger

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