# RLC Circuit Analysis (Series And Parallel)

Contents

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Key learnings:
• 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 consists of three key components: resistor, inductor, and capacitor, all connected to a voltage supply. These components are passive components, meaning they absorb energy, and linear, indicating a direct relationship between voltage and current.

RLC circuits can be connected in several ways, with series and parallel connections being the most common. Unlike LC circuits, which oscillate indefinitely, the resistor in an RLC circuit causes the oscillations to decay more rapidly.

## Series RLC Circuit

In a series RLC circuit, the resistor, inductor, and capacitor are linked one after another with the voltage supply, creating a continuous path for the current.

Since all these components are connected in series, the current in each element remains the same,

Let VR be the voltage across resistor, R.
VL be the voltage across inductor, L.
VC be the voltage across capacitor, C.
XL be the inductive reactance.
XC be the capacitive reactance.

The total voltage in the RLC circuit is not equal to the algebraic sum of voltages across the resistor, the inductor, and the capacitor; but it is a vector sum because, in the case of the resistor the voltage is in-phase with the current, for inductor the voltage leads the current by 90o and for capacitor, the voltage lags behind the current by 90o (as per ELI the ICE Man).

So, voltages in each component are not in phase with each other; so they cannot be added arithmetically. The figure below shows the phasor diagram of the series RLC circuit. For drawing the phasor diagram for RLC series circuit, the current is taken as reference because, in series circuit the current in each element remains the same and the corresponding voltage vectors for each component are drawn in reference to common current vector.

## The Impedance for a Series RLC Circuit

The impedance Z of a series RLC circuit is defined as opposition to the flow of current due circuit resistance R, inductive reactance, XL and capacitive reactance, XC. If the inductive reactance is greater than the capacitive reactance i.e XL > XC, then the RLC circuit has lagging phase angle and if the capacitive reactance is greater than the inductive reactance i.e XC > XL then, the RLC circuit have leading phase angle and if both inductive and capacitive are same i.e XL = XC then circuit will behave as purely resistive circuit.
We know that
Where,
Substituting the values

## Parallel RLC Circuit

In a parallel RLC Circuit, the resistor, inductor, and capacitor are all connected across the same voltage supply but operate independently, with the voltage constant across each and the total current split among them.

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

In a circuit containing inductor and capacitor, the energy is stored in two different ways.

1. When a current flows in a inductor, energy is stored in magnetic field.
2. When a capacitor is charged, energy is stored in static electric field.

The magnetic field in the inductor is built by the current, which gets provided by the discharging capacitor. Similarly, the capacitor is charged by the current produced by collapsing magnetic field of inductor and this process continues on and on, causing electrical energy to oscillate between the magnetic field and the electric field. In some cases at certain a certain frequency known as the resonant frequency, the inductive reactance of the circuit becomes equal to capacitive reactance which causes the electrical energy to oscillate between the electric field of the capacitor and magnetic field of the inductor. This forms a harmonic oscillator for current. In RLC circuit, the presence of resistor causes these oscillation s to die out over period of time and it is called as the damping effect of resistor.

### Formula for Resonant Frequency

During resonance, at certain frequency called resonant frequency, fr.

When resonance occurs, the inductive reactance of the circuit becomes equal to capacitive reactance, which causes the circuit impedance to be minimum in case of series RLC circuit; but when resistor, inductor and capacitor are connected in parallel, the circuit impedance becomes maximum, so the parallel RLC circuit is sometimes called as anti-resonator. Note that the lowest resonant frequency of a vibrating object is known as its fundamental frequency

## Equation of RLC Circuit

Consider a RLC circuit having resistor R, inductor L, and capacitor C connected in series and are driven by a voltage source V. Let Q be the charge on the capacitor and the current flowing in the circuit is I. Apply Kirchhoff’s voltage law

In this equation; resistance, inductance, capacitance and voltage are known quantities but current and charge are unknown quantities. We know that an current is a rate of electric charge flowing, so it is given by

Differentiating again I'(t) = Q’’ (t)

Differentiating the above equation with respect to ’t’ we get,

Now at time t = 0, V(0) = 0 and at time t = t, V(t) = Eosinωt
Differentiating with respect to ’t’ we get V'(t) = ωEocosωt
Substitute the value of V'(t) in above equation

Let us say that the solution of this equation is IP(t) = Asin(ωt – ǿ) and if IP(t) is a solution of above equation then it must satisfy this equation,

Now substitute the value of IP(t) and differentiate it we get,

Apply the formula of cos (A + B) and combine similar terms we get,

Match the coefficient of sin(ωt – φ) and cos(ωt – φ) on both sides we get,

Now we have two equations and two unknowns i.e φ and A, and by dividing the above two equations we get,

Squaring and adding above equation, we get

## Analysis of RLC Circuit Using Laplace Transformation

Step 1 : Draw a phasor diagram for given circuit.
Step 2 : Use Kirchhoff’s voltage law in RLC series circuit and current law in RLC parallel circuit to form differential equations in the time-domain.
Step 3 : Use Laplace transformation to convert these differential equations from time-domain into the s-domain.
Step 4 : For finding unknown variables, solve these equations.
Step 5 : Apply inverse Laplace transformation to convert back equations from s-domain into time domain.

### Applications of RLC Circuit

It is used as low pass filter, high pass filter, band-pass filter, band-stop filter, voltage multiplier and oscillator circuit. It is used for tuning radio or audio receiver.

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