Circuit Theory 2b - Problems related to RL, LC, RLC Circuits and related Oscillations


DC Circuits with Inductors

Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE Main or Advanced/AIEEE, and anyone else who needs this Tutorial as a reference!


Here's a quick look at what we'll cover in this tutorial :

  • A number of problems with solutions, related to RL, LC,RLC circuits , oscillations.

  • LC and RLC Circuits and the oscillations associated with them. 

LC Circuits 

As you can see it is a combination of C and L elements. Suppose if the capacitor is initially fully charged and the switch is closed, both the current in the circuit and the charge on the capacitor oscillates between maximum and minimum values. And this is why LC circuit is referred by LC oscillations. The electrostatic energy stored in the capacitor is transferred to inductor and transformed into magnetic field energy and vice versa. This oscillation of energy or current takes places infinite times if there are no resistive elements. But in practical as the zero resistance is not possible, the current eventually dies down to zero after certain number of oscillations.

Let us consider the ideal case, when there is no resistance in the circuit. Assume that the initial charge on capacitor is Qmax and the switch is closed just after t=0.

Initially as the capacitor is fully charged the total energy U in the circuit is stored in C. Total energy U = Uc + UL

At t=0 the current in the circuit I=0 and hence energy stored in inductor is zero. After the switch is closed, the rate at which charges leave or enter the capacitor plates is equal to the current in the circuit. As the capacitor begins to discharge after the switch is closed, the energy stored in its electric field decreases. The discharge of the capacitor establishes a current in the circuit, and thus inductor starts storing some energy in form of magnetic field of the inductor. Hence, electric field energy of the capacitor is transferred to the magnetic field of the inductor. 

When the capacitor is fully discharged, there is zero energy stored in it. At this moment, the current reaches its maximum value, and all of the energy is transformed into magnetic field and is stored in the inductor. Now, the current continues in the same direction, decreasing in magnitude, with the inductor getting discharged and eventually capacitor becoming fully charged. But the polarity of capacitor plates is now opposite to the initial polarity. This is followed by another discharge until the circuit returns to its original state of maximum charge Q and the plate polarity 

The oscillations of the LC circuit are analogous to the mechanical oscillations of the spring mass system. Many of those concepts are applicable to LC oscillations. We will show how that the angular frequency of the oscillations depends only on the inductance and capacitance of the circuit. As the Q is varying sinusoidally , the current in the circuit also varies sinusoidally. 
During oscillation of energy, the same energy is being oscillated between both the elements satisfying energy conservation. 

Similarly, we will also cover RLC Circuits.

RLC series circuit

We now consider a practical circuit consisting of an inductor, capacitor and a resistor connected in series. Let us assume that the initial charge on capacitor is Qmax before the switch is closed. Once the switch is closed and a current is set up, the total energy stored in the capacitor and inductor can be found out by previously derived equations. But in this case, the total energy of the circuit is not constant, like it was in LC circuit. This is because of the resistor element in the circuit, which causes energy dissipation across it in for of heat and radiations. As the energy loss across the resistance is given by -I2R.

We explain how to compute the current (as a function of time), the damping factor and the angular frequency of the damped oscillations. We also illustrate the analogies between electrical and mechanical systems..

A quick look at the nature of questions/problems solved in this tutorial:

  • In a given figure, the switch K is kept first at position 1 and steady state condition is reached. At t=0, the switch is moved to position 2. Find the current in both the cases.
  • In a given figure steady state condition is reached with 100V d.c. source. At t=0, switch K is suddenly opened. Find the  expression of the current through the inductor after t=0.5sec.
  • In a given figure, the switch is closed at position 1 at t=0. At t=0.5 msec. the switch is moved to position 2. Find the expression  for the current in both the conditions.
  • In a given figure a d.c. voltage of 100v is applied in the circuit and the switch K is open. the switch k is closed at t=0. Find the complete expression for the current.
  • You will learn to apply KVL and KCL on a variety of circuits, frame differential equations; use basic concepts of differential and integral calculus to obtain a solution.
  • Steady state analysis of circuits containing resistors, inductance and capacitors.

Complete tutorial with solved problems :

MCQ Quiz (Most of these are High School level questions)

MCQ: DC Circuits with R/L/C

Related Tutorials ( Introduction to Electrical Circuits - DC ) :

Circuit Theory 1a - Introduction to Electrical Engineering, DC Circuits, Resistance and Capacitance, Kirchoff Law

 Resistors, Capacitors, problems related to these.   Circuit Theory 1b - More solved problems related to DC Circuits with Resistance and Capacitance

 Capacitors, computing capacitance, RC Circuits, time constant of decay, computing voltage and electrostatic energy across a capacitance  Circuit Theory 2a - Introducing Inductors

 Inductors, inductance, computing self-inductance, flux-linkages, computing energy stored as a magnetic field in a coil,  mutual inductance, dot convention,
introduction to RL Circuits and decay of an inductor. 
 Circuit Theory 2b - Problems related to RL, LC, RLC circuits

 Introducing the concept of oscillations. Solving problems related to RL, LC and RLC circuits using calculus based techniques.  Circuit Theory 3a - Electrical Networks and Network Theorems

 Different kind of network elements: Active and passive, linear and non-linear, lumped and distributed. Voltage and current sources. Superposition theorem, Thevenin (or Helmholtz) theorem and problems based on these.  Circuit Theory 3b - More network theorems, solved problems

 More solved problems and examples related to electrical networks. Star and Delta network transformations, maximum power transfer theorem, Compensation theorem and Tellegen's Theorem and examples related to these.