WHAT IS SERIES RC CIRCUITS?

WHAT IS SERIES RC CIRCUITS? 

Consider RC series circuit is shown in the figure (i), a pure resistance R and a pure
capacitance C are connected in series.

r. m. s value of applied voltage = V
Current through R and L = I (Same current )
Reference phasor = I (Same current )
Voltage drop across the resistance = VR = I R (in − phase with current )
Voltage drop across the capacitor = VC = I XC (I leads the V by 90°
)
Voltage V = Vector sum of VR and VC
Phase angle difference between V & I = θ
From phasor diagram, V 2 = VR 2 + VC2
V = √VR 2 + VC 2 = √(IR) 2 + (IXC)2
= I√R2 + XC 2
 V
---   = √R2 + XC2
 I

Z = √R2 + XC 2

              XC
tan θ = ---------
               R


                    XC
θ = tan−1 (---------)
                      R


             XC
tan θ =-----
             R


Average Power:
Instantaneous current (v) = Vm sin ωt
Instantaneous voltage (i) = Im sin(ωt + θ)
Phase angle between v & i = θ
                                                               XC
Power factor (cos θ) = cos [tan−1 (-------) ]
                                                               R

Relationship between v & i = Current leads the voltage by θ
Instantaneous power : p = v x i
Instantaneous power : p = Vm sin ωt x Im sin(ωt + θ)
p = Vm Im [sin ωt x sin(ωt + θ)]

                      cos(−θ) + cos(2ωt + θ)
p = Vm Im [ ---------------------------------------]
                                        2

      Vm Im                 Vm Im 
p = ----------- cos θ − ------------- cos(2ωt + θ) 
         2                            2



The average power over one
                                            complet  cycle = Pav

                                            1   2π
Average power : Pav =---------∫ p dθ
                                        2π    0
 
             1    2π Vm Im                   Vm Im
Pav =--------∫  ----------- cos θ − -----------cos(2ωt + θ)dθ
           2π   0           2                           2





                                      2π
             1        Vm Im
Pav =--------   -----------    ∫ cos θ − cos(2ωt + θ) dθ
           2π .         2
                                     0

           Vm Im
Pav =------------- cos θ
                2


          Vm     Im
Pav =------   -------  cos θ
          √2      √2



Pav = V I cos θ

Where, V and I are rms values