Problem 53 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. All numbers are accurate to at least two significant digits. In calculating the current in an electric circuit with an inductance \(L,\) a resistance \(R,\) and a capacitance \(C,\) it is necessary to solve the equation \(L m^{2}+R m+1 / C=0 .\) Solve for \(m\) in the terms of \(L\) \(R,\) and \(C .\) See Fig. 7.7.

Step-by-Step Solution

Answer

The solution for \( m \) is \( m = \frac{-R \pm \sqrt{R^2 - \frac{4L}{C}}}{2L} \).

Step 1: Identify the Type of Equation

The given equation \( L m^{2} + R m + \frac{1}{C} = 0 \) is a quadratic equation in the form of \( ax^2 + bx + c = 0 \). Here, \( a = L \), \( b = R \), and \( c = \frac{1}{C} \).

Step 2: Recall the Quadratic Formula

The quadratic formula to find the roots (solutions) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

Step 3: Substitute the Known Values into the Formula

Substitute \( a = L \), \( b = R \), and \( c = \frac{1}{C} \) into the quadratic formula: \[ m = \frac{-R \pm \sqrt{R^2 - 4L \cdot \frac{1}{C}}}{2L} \].

Step 4: Simplify the Solution Expression

Simplify the expression under the square root by calculating the discriminant \(D = R^2 - \frac{4L}{C} \), and write the solutions in terms of \( R \), \( L \), and \( C \): \[ m = \frac{-R \pm \sqrt{R^2 - \frac{4L}{C}}}{2L} \].

Step 5: Verify the Solution

To ensure accuracy, verify that each step logically follows and leads to the correct solution form. Make sure calculations under the square root are plausible for all given positive values of \( R \), \( L \), and \( C \).

Key Concepts

Electric Circuits

Electric circuits form the backbone of modern electrical engineering. They are pathways through which electric current flows. These pathways are created by connecting electrical components like resistors, inductors, capacitors, and power sources with conductive wires. The ability to solve problems involving electric circuits relies on understanding various properties of these components and their effects on the overall circuit behavior.

An essential aspect to consider in electric circuits is the concept of **series and parallel configurations**:

• **Series circuit:** Components are connected end-to-end, so there is only one path for current flow.
• **Parallel circuit:** Components are connected alongside each other, creating multiple paths for current to flow.

Understanding these configurations helps in analyzing circuit behavior, especially when calculating equivalent resistance, voltage drops, and current distribution.

In complex circuits, components like inductors, resistors, and capacitors interact in ways that may affect the overall properties such as voltage, current, and impedance. Problem-solving in electric circuits often involves using laws such as Ohm's Law and Kirchhoff’s laws to analyze and predict circuit behavior.

Inductance

Inductance is a property of an electrical conductor that measures its opposition to a change in current flow. It is often represented by the symbol **L** and typically measured in henrys (H). Inductors, which are devices designed to store energy in a magnetic field, are characterized by this property.

**Key Points about Inductance**:

• Inductors resist changes in current, which makes them useful for filtering and stabilizing power supplies.
• The inductance of a coil is determined by the number of turns, the area of the coil, and the core material.
• In AC circuits, inductors cause a phase shift between voltage and current.

The impact of inductance in a circuit becomes quite pronounced when dealing with alternating currents (AC). The formula for inductive reactance, which represents the opposition to AC, is given by \( X_L = 2\pi f L \) where \( f \) is the frequency of the AC signal.

In the context of our quadratic equation problem, inductance as part of the circuit determines how quickly the current can change. This can affect the solution to the quadratic equation when finding the circuit's natural oscillation frequencies.

Resistance

Resistance is the opposition to the flow of electric current, making it a fundamental concept in understanding electric circuits. It is represented by the symbol **R** and measured in ohms (\(\Omega\)). Resistors are the components specifically designed to introduce resistance into a circuit.

**Fundamental Aspects of Resistance**:

• The degree of resistance depends on the material, length, and cross-sectional area of the conductor.
• According to Ohm's Law, the relationship between voltage (V), current (I), and resistance (R) is given by \( V = IR \).
• Resistance causes energy dissipation in the form of heat, which is a critical consideration in circuit design.

In circuits, resistors can be used to control current flow, divide voltages, and set biasing levels of transistors.

In our quadratic equation problem, the resistance value affects the damping of the oscillations in the circuit. This is pivotal as it influences the roots of the equation and overall circuit stability and performance.

Capacitance

Capacitance is the ability of a system to store an electric charge. It is denoted by **C** and measured in farads (F). Capacitors are components designed to hold and release energy in a circuit.

**Understanding Capacitance**:

• Capacitors store energy in an electric field created between two conductive plates separated by an insulator.
• They are used to smooth out voltage fluctuations and can act as temporary power sources.
• The amount of charge a capacitor can store depends on its capacitance and the voltage across it, according to \( Q = CV \).

Capacitors have impedance in an AC circuit described by the capacitive reactance \( X_C = \frac{1}{2\pi f C} \), affecting how the circuit responds to different frequencies.

In the quadratic equation provided, the capacitance impacts the frequency response of the circuit, thus altering the solution roots. This relationship is integral to managing how the circuit behaves under varying conditions.