AC circuit contains ohmic resistance , capacitor and inductive coil connected in series ( RLC – circuit )

Impedance

In an electric circuit containing an AC power supply together with inductive coils , capacitors and resistors , the AC current would be opposed by reactance ( inductive reactance or capacitive reactance ) in addition to the resistance of resistors and wires , The resistance and reactance together are called impedance and given the symbol ( Z ) and measured in Ohm ( Ω ) .

The impedance ( Z ) is the equivalent to the resistance , capacitive reactance and inductive reactance together in an AC circuit , When the impedance of RLC circuit = 300 Ω , It means that the total opposition to the electric current in this circuit due to the resistance and the reactance of both the coil and capacitor = 300 Ω .

AC circuit contains ohmic resistance and inductive coil connected in series ( RL – circuit )

It is almost impossible practically to construct an inductive coil with zero resistance , because any coil must have a resistance due to the wires used in its fabrication .

When electric circuit contains inductive coil , ohmic resistance and AC source connected in series , we noticed that the current passing through each of the resistance and the induction coil is the same in value and phase since they are connected in series .

But voltage in the coil ( VL) leads current ( I ) by ¼ cycle ( phase angle 90° ) in the induction coil , voltage in the resistance ( VR ) is in phase with current ( I ) in ohmic resistance .

The potential difference across a coil ( VL) leads the potential difference across the resistance ( VR ) by a phase angle 90° , thus the total potential difference is out of phase with the current intensity ( I ) .

Total voltage can be found using the vectors from the relation :

V² = V²R + V²L

V = I Z  , VR = I R  , VL = I XL

( I Z )² = I² R² + I² X² = I² ( R² + X²L )

Z ² = ( R² + X²L )

The phase angle can be determined between the total voltage ( V ) and the resistor voltage ( VR ) from the relation :

tan θ = VL / VR  , tan θ = I XL / I R  , So ,  tan θ = XL / R           

AC circuit contains ohmic resistance and capacitor connected in series ( RC-circuit )

When electric circuit contains capacitor , ohmic resistance and AC source connected in series , We notice that the current passing through each of the resistance and the capacitor is the same in value and phase since they are connected in series .

But voltage in the capacitor ( VC ) lags current ( I ) by ¼ cycle ( phase angle 90° ) in the capacitor , Voltage in the resistance ( VR ) is in phase with current ( I ) in ohmic resistance .

The potential difference across the capacitor ( VC ) lags the potential difference across the resistance ( VR ) by phase angle 90° , thus the total potential difference ( V ) is out of phase with current intensity ( I ) .

Total voltage can be found using the vectors from the relation :

V² = V²R + V²C

V = I Z    ,  VR = I R   , VC = I XC

I² Z² = I² R² + I² X²C

I² Z² = I² ( R² + X²C )

Z² = ( R² + X²C )

The phase angle ( θ ) can be determined between the total voltage difference ( V ) and the resistor voltage ( VR ) from the relation :

tan θ = −VC / VR  , tan θ = −I XC / I R  , So ,  tan θ = −XC / R   

The negative sign means that the potential difference ( VC ) lags the potential difference ( VR ) by an angle 90° .

AC circuit

AC circuit

AC circuit contains ohmic resistance , induction coil and capacitor connected in series ( RLC – circuit )   

When electric circuit contains capacitor , ohmic resistance , induction coil and AC source connected in series , We noticed that the current passing through each of the resistance , the induction coil and the capacitor is the same in value and phase since they are connected in series , But : 

  • In the induction coil , voltage ( VL ) leads current ( I ) by phase angle 90° .
  • In the ohmic resistance , voltage ( VR) is in phase with current ( I ) .
  • In the capacitor , voltage ( Vc ) lags current ( I ) by phase angle 90° .

The coil voltage ( VL ) leads the resistance voltage ( VR ) by angle 90° and the capacitor voltage ( Vc ) lags the resistance voltage ( VR ) by angle 90° and thus the phase difference between ( VL ) and ( Vc ) is 180° .

Total voltage ( V ) can be found using the vectors from the relation :

V² = V²R + ( VL− Vc

Where : V = I Z       ,  VR = I R   ,   VL = I XL   , VC = I XC

I² Z² = ( I R )² + ( I XL − I XC)² = I² [ ( R )² + ( XL − XC)² ]

Z² = ( R )² + ( XL − XC

From the previous figure , the phase angle can be determined from the relation :

tan θ = ( VL − VC ) / VR

tan θ = ( XL − XC ) / R

The phase angle θ is affected by changing the values of capacitive and inductive reactance then when :

  • VL > VC , XL > XC , The phase angle ( θ ) is positive , the total voltage ( V ) leads the current ( I ) by angle ( θ ) , That means inductive circuit behavior .
  • VL = VC , XL = XC , The phase angle ( θ ) equals zero , the total voltage ( V ) is in phase with the current ( I ) , That means ohmic resistance behavior .
  • VL < VC , XL < XC , The phase angle ( θ ) is negative , the total voltage ( V ) lags the current ( I ) by angle ( θ ) , That means capacitive circuit behavior .

The real consumed power ( Pw ) in AC circuit , either RL , RC or RLC is only the power that consumed in the ohmic resistance in the form of heat energy .

Pw = I² R = V²R / R

Capacitive reactance of capacitors network , AC current & AC voltage in a capacitor circuit

The oscillating circuit , Tuning or resonant circuit in the wireless radio receivers

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