What is RC Series Circuit? Circuit Diagram, Phasor Diagram, Derivation & Formula

Figure (1): RC Series Circuit.

An RC series circuit consisting of a resistor ‘R’ in series with a capacitor ‘C’ connected to an A.C supply as shown in figure (l).

Application of Kirchhoff’s voltage law to the circuit shown in figure (l), we get,

\[V={{V}_{R}}+{{V}_{C}}\]

Where,

V_R = Voltage drop across resistor = IR

V_C = Voltage drop across capacitor = IX_c

Figure (2): RC Series Circuit Phasor Diagram.

The phasor diagram is drawn taking current ‘I’ as the reference. The voltage drop V_R will be in phase with current I and voltage drop V_C will lagging current I by 90º. The phasor diagram is as shown in figure (2).

From the phasor diagram, we have,

\[{{V}^{2}}=V_{R}^{2}+V_{C}^{2}\]

\[={{\left( IR \right)}^{2}}+{{\left( I{{X}_{C}} \right)}^{2}}\]

\[={{I}^{2}}\left( {{R}^{2}}+X_{C}^{2} \right)\]

\[V=\sqrt{{{I}^{2}}\left( {{R}^{2}}+X_{C}^{2} \right)}\]

\[=L\sqrt{\left( {{R}^{2}}+X_{C}^{2} \right)}=IZ\]

Where

\[Z=\sqrt{\left( {{R}^{2}}+X_{C}^{2} \right)}\]

Figure (3): RC Series Circuit Impedance Triangle.

The phase angle difference $\phi$ is given by,

\[\tan \phi =\frac{{{V}_{C}}}{{{V}_{R}}}=\frac{I{{X}_{C}}}{IR}=\frac{{{X}_{C}}}{R}\]

\[\phi ={{\tan }^{-1}}\left( \frac{{{X}_{C}}}{R} \right)\]

Power factor,

\[\cos \phi =\frac{{{V}_{R}}}{V}=\frac{IR}{IZ}=\frac{R}{Z}\]

Total power,

P = VI cosϕ watts