# RLC Series Circuit: At Large and Small Frequencies; Phasor Diagram

## Response of an RLC circuit depends on the driving frequency—at large enough frequencies, inductive (capacitive) term dominates.

#### Key Points

• RLC circuits can be described by the (generalized) Ohm's law. As for the phase, when a sinusoidal voltage is applied, the current lags the voltage by a 90º phase in a circuit with an inductor, while the current leads the voltage by 90∘ in a circuit with a capacitor.

• At large enough frequencies <equation contenteditable="false">$(\nu \gg \frac{1}{\sqrt{2\pi LC}})$, the circuit is almost equivalent to an AC circuit with just an inductor. Therefore, the rms current will be Vrms/XL, and the current lags the voltage by almost 90∘.

• At small enough frequencies $(\nu \ll \frac{1}{\sqrt{2\pi LC}})$, the circuit is almost equivalent to an AC circuit with just a capacitor.  Therefore, the rms current will be given as Vrms/XC, and the current leads the voltage by almost 90.

#### Terms

• A law of electromagnetic induction that states that an electromotive force, induced in a conductor, is always in such a direction that the current produced would oppose the change that caused it; this law is a form of the law of conservation of energy.

• The increase in the amplitude of an oscillation of a system under the influence of a periodic force whose frequency is close to that of the system's natural frequency.

• Root mean square: a statistical measure of the magnitude of a varying quantity.

#### Figures

1. ##### Series RLC Circuit

A series RLC circuit: a resistor, inductor and capacitor (from left).

In previous Atoms we learned how an RLC series circuit, as shown in Figure 1, responds to an AC voltage source. By combining Ohm’s law (Irms=Vrms/Z; Irms and Vrms are rms current and voltage) and the expression for impedance Z, from:

$Z = \sqrt{R^2 + (X_L - X_C)^2}$ $(X_L = 2\pi \nu L, X_C = \frac{1} {2\pi \nu C})$

we arrived at: $I_{rms} = \frac{V_{rms}}{\sqrt{R^2 + (X_L - X_C)^2}}$

From the equation, we studied resonance conditions for the circuit. We also learned the phase relationships among the voltages across resistor, capacitor and inductor: when a sinusoidal voltage is applied, the current lags the voltage by a 90º phase in a circuit with an inductor, while the current leads the voltage by 90 in a circuit with a capacitor. Now, we will examine the system's response at limits of large and small frequencies.

### At Large Frequencies

At large enough frequencies $(\nu \gg \frac{1}{\sqrt{2\pi LC}})$, XL is much greater than XC. If the frequency is high enough that XL is much larger than R as well, the impedance Z is dominated by the inductive term.  When $Z \approx X_L$, the circuit is almost equivalent to an AC circuit with just an inductor. Therefore, the rms current will be Vrms/XL, and the current lags the voltage by almost 90. This response makes sense because, at high frequencies, Lenz's law suggests that the impedance due to the inductor will be large.

### At Small Frequencies

The impedance Z at small frequencies $(\nu \ll \frac{1}{\sqrt{2\pi LC}})$  is dominated by the capacitive term, assuming that the frequency is high enough so that XC is much larger than R. When $Z \approx X_C$, the circuit is almost equivalent to an AC circuit with just a capacitor.  Therefore, the rms current will be given as Vrms/XC, and the current leads the voltage by almost 90

#### Key Term Glossary

AC
Alternating current.
##### Appears in these related concepts:
atom
The smallest possible amount of matter which still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.
##### Appears in these related concepts:
capacitor
An electronic component capable of storing an electric charge, especially one consisting of two conductors separated by a dielectric.
##### Appears in these related concepts:
circuit
A pathway of electric current composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. T
##### Appears in these related concepts:
current
The time rate of flow of electric charge.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
frequency
The quotient of the number of times n a periodic phenomenon occurs over the time t in which it occurs: f = n / t.
##### Appears in these related concepts:
impedance
A measure of the opposition to the flow of an alternating current in a circuit; the aggregation of its resistance, inductive and capacitive reactance. Represented by the symbol Z.
##### Appears in these related concepts:
inductor
A passive device that introduces inductance into an electrical circuit.
##### Appears in these related concepts:
Inductor
A device or circuit component that exhibits significant self-inductance; a device which stores energy in a magnetic field.
##### Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
##### Appears in these related concepts:
ohm
in the International System of Units, the derived unit of electrical resistance; the electrical resistance of a device across which a potential difference of one volt causes a current of one ampere; symbol: Ω
##### Appears in these related concepts:
phase
Any one point or portion in a recurring series of changes, as in the changes of motion of one of the particles constituting a wave or vibration; one portion of a series of such changes (in distinction from a contrasted portion) as the portion on one side of a position of equilibrium (in contrast with that on the opposite side).
##### Appears in these related concepts:
resistor
An electric component that transmits current in direct proportion to the voltage across it.
##### Appears in these related concepts:
resonance
The increase in the amplitude of an oscillation of a system under the influence of a periodic force whose frequency is close to that of the system's natural frequency.
##### Appears in these related concepts:
rms
Root mean square: a statistical measure of the magnitude of a varying quantity.
##### Appears in these related concepts:
rms current
the root mean square of the current, Irms=I0/√2 , where I0 is the peak current, in an AC system
##### Appears in these related concepts:
series
A number of things that follow on one after the other or are connected one after the other.
##### Appears in these related concepts:
sinusoidal
In the form of a wave, especially one whose amplitude varies in proportion to the sine of some variable (such as time).
##### Appears in these related concepts:
voltage
The amount of electrostatic potential between two points in space.