[<< wikibooks] Circuit Theory/2Source Excitement/Node and Mesh
The first step is to convert everything to phasors and impedances in symbolic form if possible:

  
    
      
        
          
            V
          
          
            1
          
        
        =
        5
        ∗
        (
        
          
            3
          
        
        +
        j
        )
      
    
    {\displaystyle \mathbb {V} _{1}=5*({\sqrt {3}}+j)}
  

  
    
      
        
          
            I
          
          
            1
          
        
        =
        
          
            
              1
              −
              j
            
            
              2
              ∗
              
                
                  2
                
              
            
          
        
      
    
    {\displaystyle \mathbb {I} _{1}={\frac {1-j}{2*{\sqrt {2}}}}}
  

  
    
      
        
          Z
          
            C
            1
          
        
        =
        
          
            1
            
              j
              ω
              
                C
                
                  1
                
              
            
          
        
        =
        −
        10
        j
      
    
    {\displaystyle Z_{C1}={\frac {1}{j\omega C_{1}}}=-10j}
  

  
    
      
        
          Z
          
            C
            2
          
        
        =
        
          
            1
            
              j
              ω
              
                C
                
                  2
                
              
            
          
        
        =
        −
        5
        j
      
    
    {\displaystyle Z_{C2}={\frac {1}{j\omega C_{2}}}=-5j}
  

  
    
      
        
          Z
          
            L
          
        
        =
        j
        ω
        
          L
          
            1
          
        
        =
        
          L
          
            2
          
        
        =
        1
        j
      
    
    {\displaystyle Z_{L}=j\omega L_{1}=L_{2}=1j}
  


== Node Analysis ==

  
    
      
        
          
            
              
                
                  V
                
                
                  1
                
              
              −
              
                
                  V
                
                
                  a
                
              
            
            
              
                R
                
                  1
                
              
              +
              j
              ω
              
                L
                
                  1
                
              
            
          
        
        +
        
          
            I
          
          
            1
          
        
        −
        
          
            
              
                
                  V
                
                
                  a
                
              
              −
              
                
                  V
                
                
                  b
                
              
            
            
              1
              
                j
                ω
                
                  C
                  
                    1
                  
                
              
            
          
        
        =
        0
      
    
    {\displaystyle {\frac {\mathbb {V} _{1}-\mathbb {V} _{a}}{R_{1}+j\omega L_{1}}}+\mathbb {I} _{1}-{\frac {\mathbb {V} _{a}-\mathbb {V} _{b}}{\frac {1}{j\omega C_{1}}}}=0}
  

  
    
      
        
          
            
              
                
                  V
                
                
                  a
                
              
              −
              
                
                  V
                
                
                  b
                
              
            
            
              1
              
                j
                ω
                
                  C
                  
                    1
                  
                
              
            
          
        
        −
        
          
            
              
                V
              
              
                b
              
            
            
              R
              
                3
              
            
          
        
        −
        
          
            
              
                
                  V
                
                
                  b
                
              
              −
              
                
                  V
                
                
                  c
                
              
            
            
              1
              
                j
                ω
                
                  C
                  
                    2
                  
                
              
            
          
        
        =
        0
      
    
    {\displaystyle {\frac {\mathbb {V} _{a}-\mathbb {V} _{b}}{\frac {1}{j\omega C_{1}}}}-{\frac {\mathbb {V} _{b}}{R_{3}}}-{\frac {\mathbb {V} _{b}-\mathbb {V} _{c}}{\frac {1}{j\omega C_{2}}}}=0}
  

  
    
      
        
          
            
              
                
                  V
                
                
                  b
                
              
              −
              
                
                  V
                
                
                  c
                
              
            
            
              1
              
                j
                ω
                
                  C
                  
                    2
                  
                
              
            
          
        
        −
        
          
            I
          
          
            1
          
        
        −
        
          
            
              
                V
              
              
                c
              
            
            
              j
              ω
              
                L
                
                  2
                
              
            
          
        
        =
        0
      
    
    {\displaystyle {\frac {\mathbb {V} _{b}-\mathbb {V} _{c}}{\frac {1}{j\omega C_{2}}}}-\mathbb {I} _{1}-{\frac {\mathbb {V} _{c}}{j\omega L_{2}}}=0}
  Results using matlab:

  
    
      
        
          
            V
          
          
            a
          
        
        =
        −
        6.5
        −
        7
        i
        ⇒
        
          V
          
            a
          
        
        (
        t
        )
        =
        9.55
        c
        o
        s
        (
        1000
        t
        −
        2.32
        )
      
    
    {\displaystyle \mathbb {V} _{a}=-6.5-7i\Rightarrow V_{a}(t)=9.55cos(1000t-2.32)}
  

  
    
      
        
          
            V
          
          
            b
          
        
        =
        −
        3.08
        −
        3.31
        i
        ⇒
        
          V
          
            b
          
        
        (
        t
        )
        =
        4.52
        c
        o
        s
        (
        1000
        t
        −
        2.32
        )
      
    
    {\displaystyle \mathbb {V} _{b}=-3.08-3.31i\Rightarrow V_{b}(t)=4.52cos(1000t-2.32)}
  

  
    
      
        
          
            V
          
          
            c
          
        
        =
        0.327
        +
        0.385
        i
        ⇒
        
          V
          
            c
          
        
        (
        t
        )
        =
        0.506
        c
        o
        s
        (
        1000
        t
        +
        0.866
        )
      
    
    {\displaystyle \mathbb {V} _{c}=0.327+0.385i\Rightarrow V_{c}(t)=0.506cos(1000t+0.866)}
  


== Mesh Analysis ==

  
    
      
        
          
            I
          
          
            1
          
        
        
          R
          
            2
          
        
        +
        
          
            
              
                
                  I
                
                
                  1
                
              
              +
              
                
                  I
                
                
                  2
                
              
            
            
              j
              ω
              
                C
                
                  1
                
              
            
          
        
        +
        
          
            
              
                
                  I
                
                
                  1
                
              
              +
              
                
                  I
                
                
                  3
                
              
            
            
              j
              ω
              
                C
                
                  2
                
              
            
          
        
        −
        
          
            V
          
          
            I
            s
          
        
        =
        0
      
    
    {\displaystyle \mathbb {I} _{1}R_{2}+{\frac {\mathbb {I} _{1}+\mathbb {I} _{2}}{j\omega C_{1}}}+{\frac {\mathbb {I} _{1}+\mathbb {I} _{3}}{j\omega C_{2}}}-\mathbb {V} _{Is}=0}
  

  
    
      
        
          
            V
          
          
            1
          
        
        +
        
          
            I
          
          
            2
          
        
        
          R
          
            1
          
        
        +
        
          
            
              
                
                  I
                
                
                  2
                
              
              +
              
                
                  I
                
                
                  1
                
              
            
            
              j
              ω
              
                C
                
                  1
                
              
            
          
        
        +
        (
        
          
            I
          
          
            2
          
        
        −
        
          
            I
          
          
            3
          
        
        )
        
          R
          
            3
          
        
        +
        
          
            I
          
          
            2
          
        
        j
        ω
        
          L
          
            1
          
        
        =
        0
      
    
    {\displaystyle \mathbb {V} _{1}+\mathbb {I} _{2}R_{1}+{\frac {\mathbb {I} _{2}+\mathbb {I} _{1}}{j\omega C_{1}}}+(\mathbb {I} _{2}-\mathbb {I} _{3})R_{3}+\mathbb {I} _{2}j\omega L_{1}=0}
  

  
    
      
        
          
            
              
                
                  I
                
                
                  3
                
              
              +
              
                
                  I
                
                
                  1
                
              
            
            
              j
              ω
              
                C
                
                  2
                
              
            
          
        
        +
        
          
            I
          
          
            3
          
        
        j
        ω
        
          L
          
            2
          
        
        +
        (
        
          
            I
          
          
            3
          
        
        −
        
          
            I
          
          
            2
          
        
        )
        
          R
          
            3
          
        
        =
        0
      
    
    {\displaystyle {\frac {\mathbb {I} _{3}+\mathbb {I} _{1}}{j\omega C_{2}}}+\mathbb {I} _{3}j\omega L_{2}+(\mathbb {I} _{3}-\mathbb {I} _{2})R_{3}=0}
  Results using Matlab:

  
    
      
        
          
            I
          
          
            2
          
        
        =
        −
        0.239
        +
        0.234
        i
        ⇒
        
          I
          
            2
          
        
        (
        t
        )
        =
        0.334
        c
        o
        s
        (
        1000
        t
        +
        2.37
        )
      
    
    {\displaystyle \mathbb {I} _{2}=-0.239+0.234i\Rightarrow I_{2}(t)=0.334cos(1000t+2.37)}
  

  
    
      
        
          
            I
          
          
            3
          
        
        =
        −
        0.238
        +
        0.235
        i
        ⇒
        
          I
          
            3
          
        
        (
        t
        )
        =
        0.334
        c
        o
        s
        (
        1000
        t
        +
        2.36
        )
      
    
    {\displaystyle \mathbb {I} _{3}=-0.238+0.235i\Rightarrow I_{3}(t)=0.334cos(1000t+2.36)}
  

  
    
      
        
          
            V
          
          
            I
            s
          
        
        =
        175
        −
        179
        i
        ⇒
        
          V
          
            I
            s
          
        
        (
        t
        )
        =
        250
        c
        o
        s
        (
        1000
        t
        −
        0.795
        )
      
    
    {\displaystyle \mathbb {V} _{Is}=175-179i\Rightarrow V_{Is}(t)=250cos(1000t-0.795)}