Beta & Gamma functions

 Beta & gamma functions:-

Properties of beta functions:-

Proof of beta functions properties:-

Properties of Gamma functions:-

Proof of Gamma functions properties:-

Here, gamma of (n+1) =n! 

            n gamma(n) =n! 

          It's also called it reduction formula of Gamma function

 • Relation between Beta & Gamma functions:-   (most important topic )

TWO-WATTMETER METHOD [BALANCED LOAD]

POWER IN THREE-PHASE CIRCUITS

 It may be noted that power measured by the two voltmeter at any instant, is the instantaneous power absorbed by the three loads connected in three phases. This power is the average power drawn by the load since the wattmeters read the average power because of the inertia of their moving system.

Test of convergence by Testing ( All Methods )

1. Ratio test:-

2. P test:-

3. Comparison test:-

4. Necessary condition :-

5. Absolute or conditionally convergent :-

Sequence and Infinite series

Introduction of sequence :-

 Types of sequence :-

1. Convergent :-

2. Divergent:-

3. Oscillatory :-

Test for convergence & divergence:-

#Introduction of infinite series :-

Alternating series:-

03.Reflection and Transmission of Waves at a Boundary

 Reflection and Transmission of Waves at a Boundary

 An incident wave travelling along string meets discontinuity in Impedance at x=0,

some part of incident wave will be reflected and part of it will be transmitted.

• Tension(T) is same on both side of x=0.
• Linear density is different on both side P1 & P2.
• Velocity is different on both side c1 & c2.
• So, Impedance will be different on both side.
• Assume amplitude A1 of incident wave with velocity c1 and displacement yi .
• and B1 amplitude of reflected wave with velocity c1 (as same) and displacement yr .
• and A2 amplitude  of transmitted wave with velocity c2 and displacement y t.

Conditions

1. A Geometrical condition that the displacement is the same immediately to the left and right of x=0, so that no discontinuity of displacement.
   
    
    
   2. A Dynamical condition that there is a continuity of the transverse force at x=0 so a continuous slope.
   
    

   

02.Characterstic Impedance of a String

 Impedance (Z) 

How much opposition is there to oppose change in the string, so the presence of loss mechanism will introduce Impedance.

01. The Wave Equation

 We take a very small part of transverse wave from x to x+dx, 

 at x we assume tension is T so it will be downside at angle theta and upside at x=dx with angle (theta + d theta)

Three phase balanced circuit

Three phase balanced star circuit 

 

Relation between line current and phase current :-

 

Relation between line voltage and phase voltage :-

Digarm:-

.                                             

Phase Sequence  

•  There are three voltages having same magnitude and frequency dis-         placed by an angle of 120° electrical.

 

•  The emfs induced in the three coils or phases attain their positive maximum value in the order of a 1 a 2 , b 1 b 2 , and c 1 c 2 ; therefore, the phase sequence is a, b, and c.

 

•  The order in which the voltages (or emfs) in the three phases attain their 

         maximum positive value is called the Phase Sequence.  

 

 Important Points-

 

•  The direction of rotation of three-phase induction motors depends upon the   phase sequence of three-phase supply.  

 

•  To reverse the direction of rotation, the phase sequence of the supply       given to the motor has to be changed.

 

• The parallel operation of three-phase alternators and transformers is only possible if phase sequence is known.

Three Phase AC

 Three Phase AC Introduction

 Consider three identical coils a1a2, b1b2, c1c2,

 We can see phase difference of 120° electrical 

 is maintained between the corresponding start

 terminals a1, b1, and c1.

 

Phasor Diagram