12. A 480-V, 60-Hz, 50-hp, three-phase induction motor is drawing 60 A at 0.85 PF lagging. The stator copper losses are 2 kW, and the rotor copper losses are 700 W. The friction and windage losses are 600 W, the core losses are 1800 W and the stray losses are negligible. Find the following quantities: (a) The air-gap power P_AG (b) The power converted P_coav  (c) The output power P_out (d) The efficiency of the motor.

Thevenin's theorem is a fundamental concept in circuit analysis that simplifies complex linear circuits into a simple equivalent circuit at a pair of terminals. This equivalent circuit consists of a single voltage source (Thevenin Voltage, V_Th) in series with a single resistor (Thevenin Resistance, R_Th). This simplification is particularly useful for analyzing how a specific load component behaves when connected to the complex circuit.

To determine the Thevenin's equivalent circuit at terminal X-Y, follow these steps:

    
•         Step 1: Identify the terminals (X-Y) where the Thevenin equivalent circuit is to be determined. Disconnect any load connected across these terminals.     
    
•         Step 2: Calculate the Thevenin Voltage (V_Th) or Open-Circuit Voltage (V_OC).         
  This is the voltage across the terminals X-Y when the load is disconnected.         
  To find V_Th, analyze the circuit with the terminals open. You can use various circuit analysis techniques such as Kirchhoff's Voltage Law (KVL), Kirchhoff's Current Law (KCL), nodal analysis, or mesh analysis.     
    
•         Step 3: Calculate the Thevenin Resistance (R_Th) or Equivalent Resistance (R_EQ).         
  This is the equivalent resistance looking into the terminals X-Y with all independent voltage sources short-circuited (replaced by a wire) and all independent current sources open-circuited (removed from the circuit).         
  If there are dependent sources in the circuit, a test voltage (V_test) or test current (I_test) source must be applied to the terminals X-Y, and R_Th is calculated as V_test/I_test (or I_test/V_test if a current source is used). All independent sources are still turned off during this step.         
  For circuits with only independent sources, simplify the resistor network seen from terminals X-Y using series and parallel resistance combinations.     
    
•         Step 4: Construct the Thevenin Equivalent Circuit.         
  Draw the equivalent circuit consisting of the calculated Thevenin Voltage (V_Th) source connected in series with the Thevenin Resistance (R_Th) at terminals X-Y.     
    
•         Step 5: Reconnect the Load (Optional, for analysis).         
  If there was a load originally connected, reconnect it to the Thevenin equivalent circuit to easily find the voltage across or current through the load using simple Ohm's law.     

Maximum Power Transfer Theorem :

In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals.

For t<0, the switch is closed.

Capacitor acts as open to DC.

Voltage across the capacitor,

     v(0^-)= {(12||4)×V}÷{(12||4)+6} = 8VFor t=0, the switch is opened,Voltage across the Capacitor cannot change instantaneously,So, v(0^-)= v(0) = 8VAt t>0, The capacitor is discharging.R_th = (12||4) = 3 ohmC= 1/6 FTime constant, Π = R_th×C = 0.5 So, v(t) = v(0)e^-t/Π  = 8e^-t/0.5 =8e^-2t V (Ans.)