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15.3: RLC Series Circuits with AC

The ac circuit shown in Figure \(\PageIndex{1}\), 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

\[v(t) = V_0 \space sin \space \omega t.\]

 Figure a shows a circuit with an AC voltage source connected to a resistor, a capacitor and an inductor in series. The source is labeled V0 sine omega t. Figure b shows sine waves of AC voltage and current on the same graph. Voltage has a greater amplitude than current and its maximum value is marked V0 on the y axis. The maximum value of current is marked I0. The two curves have the same wavelength but are out of phase. The voltage curve is labeled V parentheses t parentheses equal to V0 sine omega t. The current curve is labeled I parentheses t parentheses equal to I0 sine parentheses omega t minus phi parentheses.

Figure \(\PageIndex{1}\): (a) An RLC series circuit. (b) A comparison of the generator output voltage and the current. The value of the phase difference \(\phi\) depends on the values of R, C, and L.

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

\[i(t) = I_0 \space sin (\omega t - \phi),\]

where \(I_0\) is the current amplitude and \(\phi\) 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 \(I_0\) and \(\phi\).

A phasor diagram involving \(i(t)\), \(v_R(t)\), \(v_C(t)\), and \(v_L(t)\) is helpful for analyzing the circuit. As shown in Figure \(\PageIndex{2}\), 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 \(\pi/2\) rad and has the amplitude \(V_C = I_0 X_C\). The phasor for \(v_L(t)\) leads the i(t) phasor by \(\pi/2\) rad and has the amplitude \(V_L = I_0 X_L\).

Figure shows the coordinate axes, with four arrows starting from the origin. Arrow V subscripts R points up and right, making an angle omega t minus phi with the x axis. Its y intercept is V subscript R parentheses t parentheses. Arrow I0 is along arrow V subscript R, but shorter than it. Arrow V subscript L points up and left and is perpendicular to V subscript R. It makes a y intercept V subscript L parentheses t parentheses. Arrow V subscript C points down and right. It is perpendicular to V subscript R. It makes a y intercept V subscript C parentheses t parentheses. Three arrows labeled omega are each perpendicular to V subscript R, V subscript L and V subscript C, shown near their tips.

Figure \(\PageIndex{2}\): The phasor diagram for the RLC series circuit of Figure \(\PageIndex{1}\).

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, \((A + B)_y = A_y + B_y \) - 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 \(v_R(t),\space v_L(t)\), and \(v_C(t)\) and then find the projection of the resultant onto the vertical axis, we obtain

\[v_R(t) + v_L(t) + v_C(t) = v(t) = V_0 \space sin \space \omega t.\]

The vector sum of the phasors is shown in Figure \(\PageIndex{3}\). The resultant phasor has an amplitude\(V_0\) and is directed at an angle \(\phi\) with respect to the \(v_R(t)\), or i(t), phasor. The projection of this resultant phasor onto the vertical axis is \(v(t) = V_0 \space sin \space \omega t\). We can easily determine the unknown quantities \(I_0\) and \(\phi\) from the geometry of the phasor diagram. For the phase angle,

\[\phi = tan^{-1} \frac{V_L - V_C}{V_R} = tan^{-1} \frac{I_0X_L - I_0X_C}{I_0R},\]

and after cancellation of \(I_0\), this becomes

\[\phi = tan^{-1}\frac{X_L - X_C}{R}.\]

Furthermore, from the Pythagorean theorem,

\[V_0 = \sqrt{V_R^2 + (V_L - V_C)^2} = \sqrt{(I_0R)^2 + (I_0X_L - I_) X_C)^2} = I_0 \sqrt{R^2 + (X_L - X_C)^2}.\]

Three arrows start from the origin on the coordinate axis. Arrow V subscript R points up and right, making an angle omega t minus phi with the x axis. Arrow V0 points up and right, making an angle omega t with the x axis. It makes an angle phi with the arrow V subscript R. It makes a y intercept labeled V0 sine omega t. The third arrow is labeled V subscript L minus V subscript C. It points up and left and is perpendicular to arrow V subscript R. Dotted lines indicate that the rectangle formed with its longer side being V subscript R and shorter side being V subscript L minus V subscript C, would have the arrow V0 as a diagonal. An arrow labeled omega is shown near the tip of V subscript R, perpendicular to it.

Figure \(\PageIndex{3}\): The resultant of the phasors for \(v_L(t)\), \(v_C(t)\), and \(v_R(t)\) is equal to the phasor for \(v_R(t) = V_0 \space sin \space \omega t\). The i(t) phasor (not shown) is aligned with the \(v_R(t)\) phasor.

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

\[I_0 = \frac{V_0}{\sqrt{R^2 + (X_L - X_C)^2}} = \frac{V_0}{Z}, \label{eq1}\]

where

\[Z = \sqrt{R^2 + (X_L - X_C)^2} \label{eq2}\]

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 (Figure \(\PageIndex{4}\)).

Photograph of power capacitors at a power station.

Figure \(\PageIndex{4}\): Power capacitors are used to balance the impedance of the effective inductance in transmission lines.

The RLC circuit is analogous to the wheel of a car driven over a corrugated road (Figure \(\PageIndex{5}\)). The regularly spaced bumps in the road drive the wheel up and down; in the same way, a voltage source increases and decreases. The shock absorber acts like the resistance of the RLC circuit, damping and limiting the amplitude of the oscillation. Energy within the wheel system goes back and forth between kinetic and potential energy stored in the car spring, analogous to the shift between a maximum current, with energy stored in an inductor, and no current, with energy stored in the electric field of a capacitor. The amplitude of the wheel’s motion is at a maximum if the bumps in the road are hit at the resonant frequency, which we describe in more detail in Resonance in an AC Circuit.

Figure shows one wheel of a car. Arrows show the up-down motion of its shock absorber spring.

Figure \(\PageIndex{5}\): On a car, the shock absorber damps motion and dissipates energy. This is much like the resistance in an RLC circuit. The mass and spring determine the resonant frequency.

Problem-Solving Strategy: AC Circuits

To analyze an ac circuit containing resistors, capacitors, and inductors, it is helpful to think of each device’s reactance and find the equivalent reactance using the rules we used for equivalent resistance in the past. Phasors are a great method to determine whether the emf of the circuit has positive or negative phase (namely, leads or lags other values). A mnemonic device of “ELI the ICE man” is sometimes used to remember that the emf (E) leads the current (I) in an inductor (L) and the current (I) leads the emf (E) in a capacitor (C).

Use the following steps to determine the emf of the circuit by phasors:

  1. Draw the phasors for voltage across each device: resistor, capacitor, and inductor, including the phase angle in the circuit.
  2. If there is both a capacitor and an inductor, find the net voltage from these two phasors, since they are antiparallel.
  3. Find the equivalent phasor from the phasor in step 2 and the resistor’s phasor using trigonometry or components of the phasors. The equivalent phasor found is the emf of the circuit.

Example \(\PageIndex{1}\): An RLC Series Circuit

The output of an ac generator connected to an RLC series combination has a frequency of 200 Hz and an amplitude of 0.100 V. If \(R = 4.00 \space \Omega, \space L = 3.00 \times 10^{-3} H\), and \(C = 8.00 \times 10^{-4}F\), what are (a) the capacitive reactance, (b) the inductive reactance, (c) the impedance, (d) the current amplitude, and (e) the phase difference between the current and the emf of the generator?

Strategy

The reactances and impedance in (a)–(c) are found by substitutions into [link], [link], and Equation \ref{eq1}, respectively. The current amplitude is calculated from the peak voltage and the impedance. The phase difference between the current and the emf is calculated by the inverse tangent of the difference between the reactances divided by the resistance.

Solution

  1. From [link], the capacitive reactance is \[X_C = \frac{1}{\omega C} = \frac{1}{2\pi (200 \space Hz)(8.00 \times 10^{-4}F)} = 0.995 \space \Omega.\]
  2. From [link], the inductive reactance is \[X_L = \omega L = 2\pi(200 \space Hz)(3.00 \times 10^{-3}H) = 3.77 \space \Omega.\]
  3. Substituting the values of R, \(X_C\), and \(X_L\) into Equation \ref{1}, we obtain for the impedance \[Z = \sqrt{(4.00 \space )^2 + (3.77 \space \Omega - 0.995 \space \Omega)^2} = 4.87 \space \Omega.\]
  4. The current amplitude is \[I_0 = \frac{V_0}{Z} = \frac{0.100 \space V}{4.87 \space \Omega} = 2.05 \times 10^{-2} A.\]
  5. From Equation, the phase difference between the current and the emf is \[\phi = tan^{-1}\frac{X_L - X_C}{R} = tan^{-1}\frac{2.77 \space \Omega}{4.00 \space \Omega} = 0.607 \space rad.\]

Significance

The phase angle is positive because the reactance of the inductor is larger than the reactance of the capacitor.

Exercise \(\PageIndex{1}\)

Find the voltages across the resistor, the capacitor, and the inductor in the circuit of Figure \(\PageIndex{1}\) using \(v(t) = V_0 \space sin \space \omega t\) as the output of the ac generator.

Solution

\(v_R = (V_0R/Z) \space sin \space (\omega t - \phi)\); \(v_C = (V_0X_C/Z) \space sin \space (\omega t - \phi + \pi2/) = - (V_0X_C/Z) \space cos \space (\omega t - \phi)\);

\(v_L = (V_0X_L/Z) \space sin \space (\omega t - \phi + \pi/2) = (V_0X_L/Z) \space cos \space (\omega t - \phi)\)

Contributors

Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).