Solving Homogeneous RLC ODEs

Hello! Let's continue our journey into radioelectronics.

In our last lesson, we successfully derived the second-order ordinary differential equations (ODEs) that govern the behavior of series and parallel RLC circuits. We saw that for both configurations, the governing equation takes the general form of a homogeneous second-order linear ODE with constant coefficients. Now, it's time to solve it.

Introduction

Approximate time to complete: 60 minutes.

Today's lesson addresses the learning outcome: Solving the Homogeneous Second-Order RLC ODE: Overdamped, Critically Damped, and Underdamped Cases. We will uncover the three distinct ways an RLC circuit can respond when disturbed, all of which are contained within the solution to this single differential equation.

Recap from the previous lesson:

We established that the natural response of an RLC circuit (whether it's the current $i(t)$ in a series circuit or the voltage $v(t)$ in a parallel one) is described by:

$$\frac{d^{2}x}{d{t}^2}+2\alpha \frac{dx}{dt}+{\omega }_0^{2}x=0$$

Where:

• ${\omega }_0=\frac{1}{\sqrt{LC}}$ is the undamped resonant frequency.
• $\alpha$ is the neper frequency or damping factor ($\alpha =\frac{R}{2L}$ for series, $\alpha =\frac{1}{2RC}$ for parallel).

The relationship between $\alpha$ and ${\omega }_0$ is the key that unlocks the circuit's behavior. Let's find out how.

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1. The Characteristic Equation: A Bridge to the Solution

To solve this type of ODE, a standard mathematical technique is to assume a solution of the form $x(t)=A{e}^{st}$, where $A$ and $s$ are constants. The logic is that the function and all its derivatives must have the same form (an exponential) to sum to zero.

If we substitute $x(t)=A{e}^{st}$ into our general ODE, we get:

$$s^{2}A{e}^{st}+2\alpha (sA{e}^{st})+{\omega }_0^2(A{e}^{st})=0$$

We can factor out the common term $A{e}^{st}$ (which is never zero), leaving us with a simple quadratic equation:

$$s^2+2\alpha s+{\omega }_0^2=0$$

This is known as the characteristic equation of the differential equation. The roots of this equation, $s_1$ and $s_2$, hold the secret to the circuit's response.

Using the quadratic formula, the roots are:

$$s_{1,2}=-\alpha \pm \sqrt{{\alpha }^2-{\omega }_0^2}$$

The term inside the square root, ${\alpha }^2-{\omega }_0^2$, is the discriminant. Its sign (positive, zero, or negative) determines whether the roots are real and distinct, real and equal, or complex conjugates. This leads to three dramatically different types of circuit behavior.

The following video segment introduces this method, assuming the exponential solution and deriving the characteristic equation and its roots.

Please watch from 10:56 to 16:57. The video uses slightly different coefficients for the series and parallel cases directly, but you can see how they map to our general form with $\alpha$ and ${\omega }_0$.

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2. The Three Natural Responses of an RLC Circuit

The relationship between the damping factor $\alpha$ (representing energy loss from R) and the resonant frequency ${\omega }_0$ (representing energy exchange between L and C) dictates the circuit's response.

To build your intuition, first watch this short video segment that provides excellent graphical illustrations of the three response types.

Focus on the segment from 09:07 to 12:48, which shows what overdamped, underdamped, and critically damped responses look like.

Now, let's connect these visual concepts to the mathematics derived from the characteristic equation.

Case 1: Overdamped Response ($\alpha >{\omega }_0$)

• Condition: The damping is strong compared to the natural resonance.
• Roots: ${\alpha }^2>{\omega }_0^2$, so the discriminant is positive. The roots $s_1$ and $s_2$ are real and distinct (and both negative).
• General Solution: The total solution is a sum of two decaying exponentials. $$x(t)=A_1{e}^{s_{1}t}+A_2{e}^{s_{2}t}$$
• Behavior: The circuit response decays to zero slowly and without any oscillation. Think of a door with a very strong closer; it shuts slowly and smoothly.

Case 2: Critically Damped Response ($\alpha ={\omega }_0$)

• Condition: A special case where damping and resonance are perfectly balanced.
• Roots: ${\alpha }^2={\omega }_0^2$, so the discriminant is zero. There is one real, repeated root: $s_1=s_2=-\alpha$.
• General Solution: From the theory of ODEs, when roots are repeated, the solution takes this form: $$x(t)=(A_1+A_{2}t)e^{-\alpha t}$$
• Behavior: The circuit returns to equilibrium as fast as possible without overshooting or oscillating. This is often the desired response in control systems or shock absorbers.

Case 3: Underdamped Response ($\alpha <{\omega }_0$)

• Condition: The natural tendency to resonate is stronger than the damping. This is the case that allows for oscillation.
• Roots: ${\alpha }^2<{\omega }_0^2$, so the discriminant is negative. The roots are complex conjugates: $$s_{1,2}=-\alpha \pm j\sqrt{{\omega }_0^2-{\alpha }^2}$$ We define a new term, the damped frequency, ${\omega }_d=\sqrt{{\omega }_0^2-{\alpha }^2}$. The roots are then $s_{1,2}=-\alpha \pm j{\omega }_d$.
• General Solution: The solution with complex exponentials can be rewritten using Euler's formula ($e^{j\theta }=\cos \theta +j\sin \theta$) into a more intuitive form: a decaying sinusoid. $$x(t)=e^{-\alpha t}(B_1\cos ({\omega }_{d}t)+B_2\sin ({\omega }_{d}t))$$
• Behavior: The circuit oscillates at the damped frequency ${\omega }_d$, while its amplitude decays exponentially according to $e^{-\alpha t}$. This is like a ringing bell or a plucked guitar string.

The following video segment provides a detailed summary of these three cases and their corresponding mathematical solutions.

Please watch from 16:57 to 26:35. This segment clearly defines the conditions for each damping type and presents the standard form of the solution for each.

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3. Finding the Constants with Initial Conditions

The general solutions we found contain unknown constants ($A_1,A_2$ or $B_1,B_2$). To find their specific values for a given circuit, we need two pieces of information: the initial conditions. For a second-order system, these are:

1. The value of the variable at $t=0^{+}$, i.e., $x(0^{+})$.
2. The value of the variable's first derivative at $t=0^{+}$, i.e., $\frac{dx}{dt}{|}_{t=0^{+}}$.

How to find them:

• $x(0^{+})$ is usually found from the principle that inductor current and capacitor voltage cannot change instantaneously: $i_L(0^{+})=i_L(0^{-})$ and $v_C(0^{+})=v_C(0^{-})$.
• $\frac{dx}{dt}{|}_{t=0^{+}}$ is found by applying KVL/KCL to the circuit at $t=0^{+}$. For example, in a parallel RLC circuit, KCL gives $i_C(0^{+})=-i_R(0^{+})-i_L(0^{+})$. Since $i_C=C\frac{dv}{dt}$, we have $\frac{dv}{dt}{|}_{t=0^{+}}=\frac{i_C(0^{+})}{C}$.

Once you have these two values, you can create a system of two linear equations to solve for the two unknown constants.

The next video segment shows exactly how to set up these equations for all three damping cases.

Watch from 26:35 to 30:48 to see the general procedure for finding the coefficients.

4. Putting It All Together: A Worked Example

Theory is essential, but seeing it applied solidifies understanding. The following video provides a comprehensive, step-by-step numerical example for an underdamped RLC circuit. It covers:

1. Finding initial conditions ($i_L(0^{-})$ and $v_C(0^{-})$).
2. Finding the derivatives at $t=0^{+}$ ($\frac{d{i}_L}{dt}$ and $\frac{d{v}_C}{dt}$).
3. Writing the characteristic equation and finding the eigenvalues ($s_1,s_2$).
4. Identifying the response type (underdamped).
5. Using the initial conditions to solve for the unknown constants in the general solution.

This is a longer segment, but it is an excellent review that ties together everything from this lesson and the previous one. I highly recommend you follow it closely.

Watch the detailed numerical example from 22:08 to 38:44.

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Conclusion

In this lesson, we have solved the fundamental equation governing RLC circuits and uncovered its rich behavior.

Key Takeaways:

• The solution to the RLC circuit's ODE is found by solving its characteristic equation: $s^2+2\alpha s+{\omega }_0^2=0$.
• The nature of the response is determined by comparing the damping factor $\alpha$ to the resonant frequency ${\omega }_0$:
  • Overdamped ($\alpha >{\omega }_0$): A non-oscillatory, slow decay described by $x(t)=A_1{e}^{s_{1}t}+A_2{e}^{s_{2}t}$.
  • Critically Damped ($\alpha ={\omega }_0$): The fastest non-oscillatory decay, described by $x(t)=(A_1+A_{2}t)e^{-\alpha t}$.
  • Underdamped ($\alpha <{\omega }_0$): A decaying oscillation, described by $x(t)=e^{-\alpha t}(B_1\cos ({\omega }_{d}t)+B_2\sin ({\omega }_{d}t))$.
• The specific constants in the solution are found using the two initial conditions: $x(0^{+})$ and $\frac{dx}{dt}{|}_{t=0^{+}}$.

Preview of the Next Module:

You have now mastered the time-domain analysis of first- and second-order circuits built from passive components (R, L, C). This completes a major block of fundamental circuit theory. We are now prepared to enter the world of semiconductors. In the next module, we will start with the physics of the P-N junction, which forms the basis of the diode—the simplest semiconductor device and the gateway to all active electronics like transistors and integrated circuits.