- RLC Circuits: An RLC circuit includes resistors, inductors, and capacitors. These components can be arranged in series or parallel to control the flow of electricity.
- Series Connection: In series RLC circuits, all components share the same current but have different voltages, which are combined vectorially because of their phase differences.
- Parallel Connection: In parallel RLC circuits, all components share the same voltage but have currents that differ and must be vector summed due to phase differences.
- Resonance Phenomenon: Resonance in RLC circuits occurs when the inductive and capacitive reactances balance each other, leading to either minimized or maximized impedance.
- Circuit Analysis: Using phasor diagrams and Kirchhoff’s Laws in analysis helps predict how RLC circuits will respond under various conditions, aiding in design and troubleshooting.
An RLC circuit contains a resistor, an inductor and a capacitor connected to a voltage source. In the ideal model, the resistor dissipates energy while the inductor and capacitor store and return energy. These are passive components. The usual linear analysis also assumes constant R, L and C values within their operating limits.
RLC components can be connected in series, in parallel or in a mixed network. An ideal lossless LC circuit can sustain a natural oscillation, but every practical circuit has loss. The resistor in an unforced RLC circuit dissipates stored energy, so a transient oscillation decays at a rate set by the damping.
Series RLC Circuit
In a series RLC circuit, the resistor, inductor and capacitor form one current path with the source.
The same instantaneous current flows through every element in this series path.
Let VR be the voltage across the resistor R.
Let VL be the voltage across the inductor L.
Let VC be the voltage across the capacitor C.
Let XL be the inductive reactance.
Let XC be the capacitive reactance.
For sinusoidal steady-state analysis, the source voltage is the phasor sum of the three element voltages. Resistor voltage is in phase with current, inductor voltage leads current by 90o and capacitor voltage lags current by 90o. The mnemonic ELI the ICE Man records these phase relationships.
The element voltages have different phase angles, so their magnitudes cannot be added as ordinary scalars. The series-circuit phasor diagram uses current as the reference. The resistor voltage lies on the current axis, while the inductor and capacitor voltages point in opposite quadrature directions.
The Impedance for a Series RLC Circuit
The impedance Z combines circuit resistance R with inductive reactance XL and capacitive reactance XC. If XL > XC, the net impedance is inductive and source current lags source voltage. If XC > XL, the net impedance is capacitive and current leads voltage. If XL = XC, the reactances cancel and the ideal series circuit presents resistance R alone.
Using the element impedances gives 
where 
Substitution gives 
Parallel RLC Circuit
In a parallel RLC Circuit, the resistor, inductor and capacitor are connected across the same two nodes. Each branch has the same voltage, while the source current is the phasor sum of the three branch currents.
The total current drawn from the supply is not equal to mathematical sum of the current flowing in the individual component, but it is equal to its vector sum of all the currents, as the current flowing in resistor, inductor and capacitor are not in the same phase with each other; so they cannot be added arithmetically.
Phasor diagram of parallel RLC circuit, IR is the current flowing in the resistor, R in amps.
IC is the current flowing in the capacitor, C in amps.
IL is the current flowing in the inductor, L in amps.
Is is the supply current in amps.
In the parallel RLC circuit, all the components are connected in parallel; so the voltage across each element is same. Therefore, for drawing phasor diagram, take voltage as reference vector and all the other currents i.e IR, IC, IL are drawn relative to this voltage vector. The current through each element can be found using Kirchhoff’s Current Law, which states that the sum of currents entering a junction or node is equal to the sum of current leaving that node.
As shown above in the equation of impedance, Z of a parallel RLC circuit; each element has reciprocal of impedance (1 / Z) i.e. admittance, Y. So in parallel RLC circuit, it is convenient to use admittance instead of impedance.
Resonance in RLC Circuit
An inductor stores energy in its magnetic field, while a capacitor stores energy in its electric field.
- When a current flows in a inductor, energy is stored in magnetic field.
- When a capacitor is charged, energy is stored in static electric field.
In a natural response, energy can move between the capacitor’s electric field and the inductor’s magnetic field. Resistance converts some of that energy into heat, which damps an unforced oscillation. In a sinusoidally driven circuit, resonance occurs near the frequency at which the inductive and capacitive susceptance or reactance terms cancel, depending on the topology. The circuit can then act as a frequency-selective oscillator or resonator. An RLC circuit does not sustain oscillation by itself unless an active circuit supplies enough energy to offset its losses.
Formula for Resonant Frequency
For an ideal RLC network, the undamped resonant frequency fr is shown below.
At this frequency, ideal series reactances cancel and series impedance reaches its minimum value R. In the ideal three-branch parallel circuit, inductive and capacitive branch currents cancel and input impedance reaches its maximum value R. Coil resistance and other losses limit the practical peak and can shift the measured resonance. The lowest natural frequency in a system with several resonances is its fundamental frequency.
Difference between Series RLC Circuit and Parallel RLC Circuit
| S.NO | RLC SERIES CIRCUIT | RLC PARALLEL CIRCUIT |
| 1 | Resistor, inductor and capacitor are connected in series | Resistor, inductor and capacitor are connected in parallel |
| 2 | Current is same in each element | Current is different in all elements and the total current is equal to vector sum of each branch of current i.e Is2 = IR2 + (IC – IL)2 |
| 3 | Voltage across all the elements is different and the total voltage is equal to the vector sum of voltages across each component i.e Vs2 = VR2 + (VL – VC)2 | Voltage across each element remains the same |
| 4 | For drawing phasor diagram, current is taken as reference vector | For drawing phasor diagram, voltage is taken as reference vector |
| 5 | Voltage across each element is given by : VR= IR, VL = I XL, VC = I XC | Current in each element is given by: IR = V / R , IC = V / XC , IL = V / XL |
| 6 | Its more convenient to use impedance for calculations | Its more convenient to use admittance for calculations |
| 7 | At resonance , when XL = XC, the circuit has minimum impedance | At resonance, when XL = XC, the circuit has maximum impedance |
Equation of RLC Circuit
Consider a RLC circuit with resistor R, inductor L and capacitor C connected in series and driven by a voltage source V. Let Q be capacitor charge and I be loop current. Apply Kirchhoff’s voltage law
In this equation, resistance, inductance, capacitance and source voltage are specified. Current and charge remain unknown. Current is the time rate of charge flow, as shown below.
Differentiating again gives I'(t) = Q”(t).
Differentiating the loop equation with respect to t gives
For a sinusoidal source, take V(0) = 0 and V(t) = Eosinωt.
Differentiating gives V'(t) = ωEocosωt.
Substitute this expression for V'(t).
Assume a sinusoidal particular solution IP(t) = A sin(ωt – ǿ). If IP(t) is a solution, it must satisfy the equation.
Substitute IP(t) and its derivatives.
Expand the cosine term and collect like terms.
Equate the coefficients of sin(ωt – φ) and cos(ωt – φ) on both sides.
The resulting two equations determine φ and A. Dividing them gives
Squaring and adding the two equations gives
Analysis of RLC Circuit Using Laplace Transformation
Step 1: Define the circuit variables, reference directions, source and initial conditions.
Step 2: Apply Kirchhoff’s voltage law to a series loop or Kirchhoff’s current law to a parallel node to obtain a time-domain differential equation.
Step 3: Apply the Laplace transformation, including the stated initial capacitor voltage and inductor current, to convert the equation to the s-domain.
Step 4: Solve the algebraic equation for the required voltage or current.
Step 5: Apply the inverse Laplace transformation and check the result against the initial conditions, final value and physical units.
Applications of RLC Circuit
RLC networks form a low pass filter, high pass filter, band-pass filter or band-stop filter when the output is taken from the appropriate component. Related reactive networks also appear in a voltage multiplier and the frequency-selective section of an oscillator circuit. Common uses include radio tuning, impedance matching and audio crossover networks.
