A 400 V DC motor takes an armature current of 100 A when its speed is 1000 rpm. If the armature resistance is 0.25 ohm, calculate the torque produced in Nm. 

উত্তরঃ

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.     

উত্তরঃ

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.

উত্তরঃ

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.)

উত্তরঃ

এখানে প্রদত্ত বর্গাকার ভোল্টেজ তরঙ্গ থেকে গড় (average) এবং কার্যকর (effective) মান নির্ণয় করা হলো:

প্রদত্ত চিত্র থেকে পাই:

    
• সর্বোচ্চ ভোল্টেজ, \(V_m = 10 \text{ V}\)
    
• তরঙ্গটি \(1 \text{ ms}\) এর জন্য \(+10 \text{ V}\) এবং \(1 \text{ ms}\) এর জন্য \(-10 \text{ V}\) থাকে।
    
• সুতরাং, পর্যায়কাল, \(T = 1 \text{ ms} + 1 \text{ ms} = 2 \text{ ms}\)

১. গড় মান (Average Value) নির্ণয়:

কোনো ভোল্টেজ তরঙ্গের গড় মান নির্ণয়ের সূত্র হলো:

\[ V_{avg} = \frac{1}{T} \int_0^T v(t) dt \]

যেহেতু তরঙ্গটি \(0\) থেকে \(1 \text{ ms}\) পর্যন্ত \(+10 \text{ V}\) এবং \(1 \text{ ms}\) থেকে \(2 \text{ ms}\) পর্যন্ত \(-10 \text{ V}\) থাকে, তাই:

\[ V_{avg} = \frac{1}{2 \times 10^{-3}} \left[ \int_0^{1 \times 10^{-3}} 10 dt + \int_{1 \times 10^{-3}}^{2 \times 10^{-3}} (-10) dt \right] \] \[ V_{avg} = \frac{1}{2 \times 10^{-3}} \left[ [10t]_0^{1 \times 10^{-3}} + [-10t]_{1 \times 10^{-3}}^{2 \times 10^{-3}} \right] \] \[ V_{avg} = \frac{1}{2 \times 10^{-3}} \left[ (10 \times 1 \times 10^{-3} - 0) + (-10 \times 2 \times 10^{-3} - (-10 \times 1 \times 10^{-3})) \right] \] \[ V_{avg} = \frac{1}{2 \times 10^{-3}} \left[ 10 \times 10^{-3} - 20 \times 10^{-3} + 10 \times 10^{-3} \right] \] \[ V_{avg} = \frac{1}{2 \times 10^{-3}} \left[ 0 \right] \] \[ V_{avg} = 0 \text{ V} \]

সুতরাং, ভোল্টেজ তরঙ্গের গড় মান \(0 \text{ V}\)।

২. কার্যকর মান (Effective Value) বা RMS মান (Root Mean Square Value) নির্ণয়:

কোনো ভোল্টেজ তরঙ্গের RMS মান নির্ণয়ের সূত্র হলো:

\[ V_{rms} = \sqrt{\frac{1}{T} \int_0^T v(t)^2 dt} \]

যেহেতু তরঙ্গটি \(0\) থেকে \(1 \text{ ms}\) পর্যন্ত \(+10 \text{ V}\) এবং \(1 \text{ ms}\) থেকে \(2 \text{ ms}\) পর্যন্ত \(-10 \text{ V}\) থাকে, তাই \(v(t)^2\) সবসময় \(10^2 = 100 \text{ V}^2\) হবে:

\[ V_{rms} = \sqrt{\frac{1}{2 \times 10^{-3}} \left[ \int_0^{1 \times 10^{-3}} (10)^2 dt + \int_{1 \times 10^{-3}}^{2 \times 10^{-3}} (-10)^2 dt \right]} \] \[ V_{rms} = \sqrt{\frac{1}{2 \times 10^{-3}} \left[ \int_0^{1 \times 10^{-3}} 100 dt + \int_{1 \times 10^{-3}}^{2 \times 10^{-3}} 100 dt \right]} \] \[ V_{rms} = \sqrt{\frac{1}{2 \times 10^{-3}} \left[ [100t]_0^{1 \times 10^{-3}} + [100t]_{1 \times 10^{-3}}^{2 \times 10^{-3}} \right]} \] \[ V_{rms} = \sqrt{\frac{1}{2 \times 10^{-3}} \left[ (100 \times 1 \times 10^{-3} - 0) + (100 \times 2 \times 10^{-3} - 100 \times 1 \times 10^{-3}) \right]} \] \[ V_{rms} = \sqrt{\frac{1}{2 \times 10^{-3}} \left[ 100 \times 10^{-3} + 200 \times 10^{-3} - 100 \times 10^{-3} \right]} \] \[ V_{rms} = \sqrt{\frac{1}{2 \times 10^{-3}} \left[ 200 \times 10^{-3} \right]} \] \[ V_{rms} = \sqrt{100} \] \[ V_{rms} = 10 \text{ V} \]

সুতরাং, ভোল্টেজ তরঙ্গের কার্যকর মান (RMS) \(10 \text{ V}\)।