# 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 ( V _{L}) leads current ( I ) by ¼ cycle ( phase angle 90° ) in the induction coil , voltage in the resistance ( V_{R} ) is in phase with current ( I ) in ohmic resistance .**

**The potential difference across a coil ( V _{L}) leads the potential difference across the resistance ( V_{R} ) 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 = V² _{R} + V²_{L}**

**V = I Z , V _{R }= I R , V_{L} = I X_{L}**

**( I Z )² = I² R² + I² X² _{L } = I² ( R² + X²_{L} ) **

**Z ² = ( R² + X² _{L} )**

**The phase angle can be determined between the total voltage ( V ) and the resistor voltage ( V _{R} ) from the relation :**

**tan θ = V _{L} / V_{R} , tan θ = I X_{L} / I R , So , tan θ = X_{L} / 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 ( V _{C} ) lags current ( I ) by ¼ cycle ( phase angle 90° ) in the capacitor , Voltage in the resistance ( V_{R} ) is in phase with current ( I ) in ohmic resistance .**

**The potential difference across the capacitor ( V _{C} ) lags the potential difference across the resistance ( V_{R} ) 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 = V² _{R} + V²_{C
}**

**V = I Z , V _{R} = I R , V_{C} = I X_{C}**

**I² Z² = I² R² + I² X² _{C}**

**I² Z² = I² ( R² + X² _{C} )**

** Z² = ( R² + X² _{C} )**

**The phase angle ( θ ) can be determined between the total voltage difference ( V ) and the resistor voltage ( V _{R} ) from the relation :
**

**tan θ = −V _{C} / V_{R} , tan θ = −I X_{C} / I R , So , tan θ = −X_{C} / R **

**The negative sign means that the potential difference ( V _{C }) lags the potential difference ( V_{R }) by an angle 90° .**

**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 ( V**_{L }) leads current ( I ) by phase angle 90° .**In the ohmic resistance , voltage ( V**_{R}) is in phase with current ( I ) .**In the capacitor , voltage ( V**_{c}) lags current ( I ) by phase angle 90° .

**The coil voltage ( V _{L} ) leads the resistance voltage ( V_{R }) by angle 90° and the capacitor voltage ( V_{c} ) lags the resistance voltage ( V_{R }) by angle 90° and thus the phase difference between ( V_{L} ) and ( V_{c} ) is 180° .**

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

**V² = V² _{R} + ( V_{L}− V_{c} )²**

**Where : V = I Z , V _{R }= I R , V_{L} = I X_{L} , V_{C} = I X_{C}**

**I² Z² = ( I R )² + ( I X _{L} − I X_{C})² = I² [ ( R )² + ( X_{L} − X_{C})² ]**

**Z² = ( R )² + ( X _{L} − X_{C})²**

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

**tan θ = ( V _{L} − V_{C }) / V_{R}**

**tan θ = ( X _{L} − X_{C }) / R**

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

**V**_{L}> V_{C }, X_{L}> X_{C}, The phase angle ( θ ) is positive , the total voltage ( V ) leads the current ( I ) by angle ( θ ) , That means inductive circuit behavior .**V**_{L}= V_{C }, X_{L}= X_{C}, The phase angle ( θ ) equals zero , the total voltage ( V ) is in phase with the current ( I ) , That means ohmic resistance behavior .**V**_{L}< V_{C }, X_{L}< X_{C}, 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 ( P _{w }) in AC circuit , either RL , RC or RLC is only the power that consumed in the ohmic resistance in the form of heat energy .**

**P _{w} = I² R = V²_{R }/ 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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