15.3 Rlc series circuits with ac

The ac circuit shown in [link] , called an RLC series circuit , is a series combination of a resistor, capacitor, and inductor connected across an ac source. It produces an emf of

Since the elements are in series, the same current flows through each element at all points in time. The relative phase between the current and the emf is not obvious when all three elements are present. Consequently, we represent the current by the general expression

where $I_0$ is the current amplitude and $\varphi$ is the phase angle    between the current and the applied voltage. The phase angle is thus the amount by which the voltage and current are out of phase with each other in a circuit. Our task is to find $I_0\text{and}\varphi .$

A phasor diagram involving $i(t),v_R(t),v_C(t),\text{and}{v}_L(t)$ is helpful for analyzing the circuit. As shown in [link] , the phasor representing $v_R(t)$ points in the same direction as the phasor for $i(t);$ its amplitude is $V_R=I_{0}R.$ The $v_C(t)$ phasor lags the i ( t ) phasor by $\pi \text{/}2$ rad and has the amplitude $V_C=I_0{X}_C.$ The phasor for $v_L(t)$ leads the i ( t ) phasor by $\pi \text{/}2$ rad and has the amplitude $V_L=I_0{X}_L.$

At any instant, the voltage across the RLC combination is $v_R(t)+v_L(t)+v_C(t)=v(t),$ the emf of the source. Since a component of a sum of vectors is the sum of the components of the individual vectors—for example, ${\left(A+B\right)}_y=A_y+B_y$ —the projection of the vector sum of phasors onto the vertical axis is the sum of the vertical projections of the individual phasors. Hence, if we add vectorially the phasors representing $v_R(t),v_L(t),\text{and}{v}_C(t)$ and then find the projection of the resultant onto the vertical axis, we obtain

The vector sum of the phasors is shown in [link] . The resultant phasor has an amplitude $V_0$ and is directed at an angle $\varphi$ with respect to the $v_R\left(t\right),$ or i ( t ), phasor. The projection of this resultant phasor onto the vertical axis is $v(t)=V_0\text{sin}\omega t.$ We can easily determine the unknown quantities $I_0$ and $\varphi$ from the geometry of the phasor diagram. For the phase angle,

and after cancellation of $I_0,$ this becomes

Furthermore, from the Pythagorean theorem,

The current amplitude is therefore the ac version of Ohm’s law:

where

is known as the impedance    of the circuit. Its unit is the ohm, and it is the ac analog to resistance in a dc circuit, which measures the combined effect of resistance, capacitive reactance, and inductive reactance ( [link] ).