[<< wikibooks] Circuit Theory/Complex Frequency Examples/example15
Find everything give that:

  
    
      
        
          V
          
            s
          
        
        =
        10
        ∗
        
          e
          
            −
            100
            t
          
        
        cos
        ⁡
        (
        6000
        t
        )
      
    
    {\displaystyle V_{s}=10*e^{-100t}\cos(6000t)}
  Terminal Relations:

  
    
      
        
          
            V
          
          
            2
          
        
        =
        
          R
          
            1
          
        
        ∗
        
          
            I
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {V} _{2}=R_{1}*\mathbb {I} _{2}}
  

  
    
      
        
          
            I
          
          
            3
          
        
        =
        
          C
          
            1
          
        
        ∗
        s
        ∗
        
          
            V
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {I} _{3}=C_{1}*s*\mathbb {V} _{3}}
  

  
    
      
        
          
            V
          
          
            1
          
        
        =
        
          L
          
            1
          
        
        ∗
        s
        ∗
        
          
            I
          
          
            1
          
        
      
    
    {\displaystyle \mathbb {V} _{1}=L_{1}*s*\mathbb {I} _{1}}
  

  
    
      
        
          
            V
          
          
            4
          
        
        =
        
          R
          
            2
          
        
        ∗
        
          
            I
          
          
            1
          
        
      
    
    {\displaystyle \mathbb {V} _{4}=R_{2}*\mathbb {I} _{1}}
  Junction Equation:

  
    
      
        
          
            I
          
          
            1
          
        
        −
        
          
            I
          
          
            2
          
        
        −
        
          
            I
          
          
            3
          
        
        =
        0
      
    
    {\displaystyle \mathbb {I} _{1}-\mathbb {I} _{2}-\mathbb {I} _{3}=0}
  Loop Equations:

  
    
      
        
          
            V
          
          
            3
          
        
        +
        
          
            V
          
          
            1
          
        
        +
        
          
            V
          
          
            4
          
        
        −
        
          
            V
          
          
            s
          
        
        =
        0
      
    
    {\displaystyle \mathbb {V} _{3}+\mathbb {V} _{1}+\mathbb {V} _{4}-\mathbb {V} _{s}=0}
  

  
    
      
        
          
            V
          
          
            3
          
        
        −
        
          
            V
          
          
            2
          
        
        =
        0
      
    
    {\displaystyle \mathbb {V} _{3}-\mathbb {V} _{2}=0}
   (trivial loop)Complex Frequency:

  
    
      
        s
        =
        −
        100
        +
        6000
        j
      
    
    {\displaystyle s=-100+6000j}
  Complex frequency source:

  
    
      
        
          
            V
          
          
            s
          
        
        =
        10
      
    
    {\displaystyle \mathbb {V} _{s}=10}
  Solving in the complex frequency domain: