Exercise 3.80: A 
series RLC circuit
consists of a resistor, a capacitor, and an inductor connected in series, as
shown in Figure 3.36.  If 
  R
, 
  L
, and 
  C
 are the resistance,
inductance, and capacitance, then the relations between voltage 
  
    (
    v
    )
  
 and
current 
  
    (
    i
    )
  
 for the three components are described by the equations

  
    
      
        
          v
          R
        
      
      
        =
      
      
        
          i
          R
        
        R
        ,
      
    
    
      
        
          v
          L
        
      
      
        =
      
      
        L
        
        
          
            
              d
              
                i
                L
              
            
            
              d
              t
            
          
        
        ,
      
    
    
      
        
          i
          C
        
      
      
        =
      
      
        C
        
        
          
            
              d
              
                v
                C
              
            
            
              d
              t
            
          
        
        ,
      
    
  

and the circuit connections dictate the relations

  
    
      
        
          i
          R
        
      
      
        =
      
      
        
          i
          L
        
        =
        −
        
          i
          C
        
        ,
      
    
    
      
        
          v
          C
        
      
      
        =
      
      
        
          v
          L
        
        +
        
          v
          R
        
        .
      
    
  

Combining these equations shows that the state of the circuit (summarized by

  
    v
    C
  
, the voltage across the capacitor, and 
  
    i
    L
  
, the current in
the inductor) is described by the pair of differential equations

  
    
      
        
          
            
              d
              
                v
                C
              
            
            
              d
              t
            
          
        
      
      
        =
      
      
        −
        
          
            
              i
              L
            
            C
          
        
        
        ,
      
    
    
      
        
          
            
              d
              
                i
                L
              
            
            
              d
              t
            
          
        
      
      
        =
      
      
        
          
            1
            L
          
        
        
        
          v
          C
        
        −
        
          
            R
            L
          
        
        
        
          i
          L
        
        .
      
    
  

The signal-flow diagram representing this system of differential equations is
shown in Figure 3.37.