[<< wikibooks] Electronics Handbook/Circuits/Summary of RLC Series
== Normal ==
RL and RC both pocess the same character

Differential equation of first ordered in the form
  
    
      
        
          
            d
            
              d
              t
            
          
        
        f
        (
        t
        )
        +
        
          
            1
            T
          
        
        =
        0
      
    
    {\displaystyle {\frac {d}{dt}}f(t)+{\frac {1}{T}}=0}
  Which can only has one real root in the form
  
    
      
        f
        (
        t
        )
        =
        A
        
          e
          
            (
          
        
        −
        
          
            t
            T
          
        
        )
      
    
    {\displaystyle f(t)=Ae^{(}-{\frac {t}{T}})}
  

  
    
      
        A
        =
        
          e
          
            c
          
        
      
    
    {\displaystyle A=e^{c}}
  LC and RLC both pocess the same character

Differential equation of second ordered in the form
  
    
      
        
          
            
              d
              
                2
              
            
            
              d
              
                t
                
                  2
                
              
            
          
        
        f
        (
        t
        )
        +
        α
        
          
            d
            
              d
              t
            
          
        
        f
        (
        t
        )
        +
        β
        =
        0
      
    
    {\displaystyle {\frac {d^{2}}{dt^{2}}}f(t)+\alpha {\frac {d}{dt}}f(t)+\beta =0}
  Which can only has one real root in the form
  
    
      
        f
        (
        t
        )
        =
        A
        (
        
          e
          
            ω
          
        
        t
        +
        
          e
          
            −
          
        
        ω
        t
        )
      
    
    {\displaystyle f(t)=A(e^{\omega }t+e^{-}\omega t)}
  

  
    
      
        A
        =
        
          e
          
            α
          
        
        t
      
    
    {\displaystyle A=e^{\alpha }t}
  

  
    
      
        ω
        =
        
          
            
              α
              
                2
              
            
            −
            
              β
              
                2
              
            
          
        
      
    
    {\displaystyle \omega ={\sqrt {\alpha ^{2}-\beta ^{2}}}}
  


== Resonance ==
At resonance

LC will generate Oscillation of Standing Sin Wave of frequency
  
    
      
        ω
        =
        
          
            
              1
              
                L
                C
              
            
          
        
      
    
    {\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}
  RLC will act as Resonant Tuned Selected Bandpass Filter
  
    
      
        ω
        =
        
          
            
              1
              
                L
                C
              
            
          
        
        a
        t
        I
        =
        
          
            V
            R
          
        
      
    
    {\displaystyle \omega ={\sqrt {\frac {1}{LC}}}atI={\frac {V}{R}}}
  

  
    
      
        δ
        ω
        =
        
          ω
          
            2
          
        
        −
        
          ω
          
            1
          
        
        a
        t
        I
        =
        
          
            V
            
              2
              R
            
          
        
      
    
    {\displaystyle \delta \omega =\omega _{2}-\omega _{1}atI={\frac {V}{2R}}}