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Energy Dissipated in Resistor: RC Network: ECE 524: Transients in Power Systems Session 6 Page 1/2 Spring 2018

1) The document discusses energy dissipation in a resistor-capacitor (RC) circuit. It derives an equation for the energy (Ediss) dissipated in the resistor in terms of the initial voltages (VC1(0) and VC2(0)) across the two capacitors and the capacitance values. 2) Through algebraic manipulations using the principles of conservation of energy and charge, the equation is simplified to express Ediss as proportional to the square of the difference between the initial capacitor voltages, divided by the sum of the capacitances. 3) The key result is a simple final expression for the energy dissipated in the resistor in the RC circuit in terms of the initial conditions

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125 views2 pages

Energy Dissipated in Resistor: RC Network: ECE 524: Transients in Power Systems Session 6 Page 1/2 Spring 2018

1) The document discusses energy dissipation in a resistor-capacitor (RC) circuit. It derives an equation for the energy (Ediss) dissipated in the resistor in terms of the initial voltages (VC1(0) and VC2(0)) across the two capacitors and the capacitance values. 2) Through algebraic manipulations using the principles of conservation of energy and charge, the equation is simplified to express Ediss as proportional to the square of the difference between the initial capacitor voltages, divided by the sum of the capacitances. 3) The key result is a simple final expression for the energy dissipated in the resistor in the RC circuit in terms of the initial conditions

Uploaded by

Atiq_2909
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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ECE 524: Session 6; Page 1/2

Transients in Power Systems Spring 2018

Energy Dissipated in Resistor: RC Network


100ohm
I IS

VC1 VC2

U
U(0)

U(0)
2507.17uF
+

+
1253.58uF

 Initial energy in circuit

 C1   VC1 ( 0)    C2   VC2 ( 0) 
1 2 1 2
E ( 0) =
2 2

 Final energy in the circuit

VC1 ( ∞) = VC2 ( ∞)

  C1  C2    VC1 ( ∞) 
1 2
E ( ∞) =
2

 Energy is conserved:

E ( 0) = E ( ∞)  Ediss
 Rewriting

Ediss = E ( 0)  E ( ∞)

1 2  1 2
Ediss =   C1   VC1 ( 0)    C2   VC2 ( 0)       C1  C2    VC1 ( ∞)  
2 1
( 1)
2 2  2 
ECE 524: Session 6; Page 2/2
Transients in Power Systems Spring 2018

 Now try to simplify the expression.

 Recall that charge is also conserved:

q ( 0) = q ( ∞)

C1 VC1 ( 0)  C2 VC2 ( 0) =  C1  C2  VC1 ( ∞)

VC1 ( ∞) =
 C1 VC1 (0)  C2 VC2 (0)
C1  C2

 Substitute into equation (1)


   C1 VC1 ( 0)  C2 VC2 ( 0)   
2
 2  1
Ediss =   C1   VC1 ( 0)    C2   VC2 ( 0)      C1  C2   
1 2 1

2 2   2  C1  C2 

1   C1  C2   C1  VC1 ( 0)   C2   VC2 ( 0)     C1  VC1 ( 0)  C2  VC2 ( 0) 


2 2 2
Ediss = 
2  C1  C2  C1  C2

1   C1  VC1 ( 0)  C1  C2 VC1 ( 0)  C1  C2 VC2 ( 0)  C2   VC2 ( 0)  C1  VC1 ( 0)  2 C1 C2  VC1 ( 0)  VC2 ( 0)  C2  VC2 ( 0) 
2 2 2 2 2 2 2 2 2 2

Ediss =    
 2   C1  C2 C1  C2 

Ediss =
1  C1  C2 
  2 2

1  C1  C2 
  VC1 ( 0)  2 VC1 ( 0)  VC2 ( 0)  VC2 ( 0) =  
2  C1  C2 
   VC1 ( 0)  VC2 ( 0) 
2  C1  C2 
2

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