1. For The Circuit Below: A. \( (10 \%) \) Find The Differential Equation That Relates Input And Output.

1. For The Circuit Below: A. \( (10 \%) \) Find The Differential Equation That Relates Input And Output.

Understanding how to derive the differential equation that governs an electrical circuit is fundamental in analyzing and designing systems such as filters, oscillators, and control systems. When given a circuit with resistors, capacitors, and inductors, the goal is to establish a mathematical relationship between the input voltage (or current) and the output voltage (or current). This process involves applying fundamental principles like Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL), along with the constitutive relations of circuit elements.

In this article, we will explore the systematic approach to deriving the differential equation for a given circuit, starting from the basic circuit analysis principles and progressing toward the formulation of the differential equation. Although the actual circuit diagram is not provided here, the methodology remains applicable to a variety of common RLC circuit configurations.

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Understanding the Circuit Components and Their Behavior

Before delving into the derivation, it is essential to understand the behaviors of the basic circuit elements involved, as they form the foundation of the differential equations.

Resistors (R)

• Resistors obey Ohm's law: \( V = R \times I \)
• The voltage across a resistor is proportional to the current flowing through it.
• Resistors do not store energy; they dissipate it as heat.

Capacitors (C)

• Capacitors store energy in the electric field.
• The voltage across a capacitor relates to the charge \( Q \): \( V_C = \frac{Q}{C} \)
• The current through a capacitor is proportional to the rate of change of voltage: \( IC = C \frac{dVC}{dt} \)

Inductors (L)

• Inductors store energy in the magnetic field.
• The voltage across an inductor relates to the current \( IL \): \( VL = L \frac{dI_L}{dt} \)
• The current through an inductor is proportional to the integral of voltage over time.

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Step-by-Step Approach to Deriving the Differential Equation

The process of deriving the differential equation involves several key steps:

1. Identify the Input and Output Variables

• Determine which voltage or current is considered the input and which is the output.
• For example, if the input is an applied voltage \( V{in} \), and the output is the voltage across a capacitor \( V{out} \), these variables form the basis of the analysis.

2. Assign Circuit Variables and Labels

• Label each circuit node, branch, and element.
• Assign variables to unknown quantities, such as node voltages and branch currents.

3. Apply Kirchhoff's Laws

• Use KVL to write equations around loops.
• Use KCL to write equations at nodes where currents meet.

4. Write Constitutive Relations for Each Element

• Express voltages and currents for resistors, capacitors, and inductors according to their behavior.

5. Formulate the Differential Equation

• Combine all the equations to eliminate auxiliary variables.
• Express the relationship explicitly between input and output variables, resulting in a differential equation.

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Example: Deriving the Differential Equation for a Series RLC Circuit

To illustrate the process, consider a classic series RLC circuit with an input voltage \( V{in}(t) \) applied across a resistor \( R \), inductor \( L \), and capacitor \( C \) connected in series. The output voltage \( V{out}(t) \) could be taken across the capacitor.

Step 1: Identify Variables

• Input: \( V_{in}(t) \)


• Output: \( V_{out}(t) \) (voltage across capacitor)

Step 2: Write KVL Equation
Applying Kirchhoff's Voltage Law around the loop:
\[
V{in}(t) = VR(t) + VL(t) + VC(t)
\]

Step 3: Express Element Voltages

• Resistor: \( V_R(t) = R \times I(t) \)


• Inductor: \( V_L(t) = L \frac{dI(t)}{dt} \)


• Capacitor: \( V_C(t) = \frac{Q(t)}{C} \)

Since current \( I(t) \) relates to the charge \( Q(t) \) as \( I(t) = \frac{dQ(t)}{dt} \), and the voltage across the capacitor is \( V{out}(t) = VC(t) = \frac{Q(t)}{C} \).

Step 4: Write Equations in Terms of Output Voltage
Express the current in terms of \( V_{out} \):
\[
Q(t) = C V_{out}(t)
\]
\[
I(t) = \frac{dQ(t)}{dt} = C \frac{dV_{out}(t)}{dt}
\]

Now, rewrite the KVL:
\[
V{in}(t) = R \times I(t) + L \frac{dI(t)}{dt} + V{out}(t)
\]

Substitute \( I(t) \):
\[
V{in}(t) = R C \frac{dV{out}(t)}{dt} + L \frac{d}{dt} \left( C \frac{dV{out}(t)}{dt} \right) + V{out}(t)
\]
\[
V{in}(t) = R C \frac{dV{out}(t)}{dt} + L C \frac{d^2 V{out}(t)}{dt^2} + V{out}(t)
\]

Step 5: Rearrange to Standard Form
Rearranged, the differential equation becomes:
\[
L C \frac{d^2 V{out}(t)}{dt^2} + R C \frac{dV{out}(t)}{dt} + V{out}(t) = V{in}(t)
\]

This is a second-order linear differential equation relating the input voltage \( V{in}(t) \) to the output voltage \( V{out}(t) \).

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General Form of the Differential Equation for RLC Circuits

The example above illustrates a common form of differential equations for RLC circuits. In general, the differential equation relating input and output in an RLC circuit can be expressed as:

\[
an \frac{d^n y(t)}{dt^n} + a{n-1} \frac{d^{n-1} y(t)}{dt^{n-1}} + \dots + a1 \frac{dy(t)}{dt} + a0 y(t) = bm \frac{d^m x(t)}{dt^m} + \dots + b0 x(t)
\]

where:

• \( y(t) \) is the output variable (voltage or current),


• \( x(t) \) is the input variable,


• \( ai \) and \( bj \) are coefficients determined by circuit element values,


• \( n \) and \( m \) are the orders of derivatives, typically dictated by the circuit's configuration.

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Conclusion and Practical Applications

Deriving the differential equation that relates input and output signals in an electrical circuit is a crucial skill in electrical engineering. This process enables engineers to analyze transient responses, design filters, and predict circuit behavior under various input conditions. While the derivation can seem complex initially, following a systematic approach—identifying variables, applying Kirchhoff's laws, expressing element behaviors, and combining equations—makes the process manageable.

In practical applications, once the differential equation is obtained, engineers can solve it using analytical methods or numerical simulations to understand the circuit's response. For example, in communication systems, filters are designed based on these equations to isolate desired frequency components. Similarly, in control systems, understanding the differential relationships helps in designing controllers that ensure stability and desired performance.

Summary of Key Steps:

• Recognize the circuit configuration.


• Identify input and output variables.


• Label all circuit elements and nodes.


• Apply Kirchhoff's Voltage and Current Laws.


• Use element constitutive relations.


• Combine equations to isolate the relationship between input and output.


• Formulate the resulting differential equation.

By mastering these steps, students and practitioners can analyze a wide variety of circuits, predict their behavior, and contribute to innovative electrical and electronic system designs.

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Meta Note: To solve the specific problem given in the original question, a detailed circuit diagram would be necessary. However, the methodology outlined here provides a comprehensive approach applicable to virtually any linear circuit involving resistors, capacitors, and inductors.