Problem 36 - Elementary Algebra

Question

The resonance frequency \(f\) in an electronic circuit containing inductance \(L\) and capacitance \(C\) in series is given by \(f=\frac{1}{2 \pi \sqrt{L C}}\) (a) Determine the resonance frequency in an electronic circuit if the inductance is 4 and the capacitance is 0.0001 . Use \(\pi=3.14\). (b) Determine the inductance in an electric circuit if the resonance frequency is 7.12 and the capacitance is 0.0001 . Use \(\pi=3.14\).

Step-by-Step Solution

Answer

Answer: The resonance frequency is approximately 3.98, and the inductance is approximately 0.000626.

Step 1: Write down the given values

Inductance (L) is given as 4 and the capacitance (C) is 0.0001.

Step 2: Use the resonance frequency formula

We need to use the formula \(f=\frac{1}{2 \pi \sqrt{L C}}\) and plug in the values for L and C: \(f=\frac{1}{2 \pi \sqrt{4 \times 0.0001}}\).

Step 3: Calculate the resonance frequency

We can now calculate the resonance frequency using the provided value of \(\pi=3.14\). \(f=\frac{1}{2 \times 3.14 \times \sqrt{4 \times 0.0001}} \approx 3.98\). The resonance frequency is approximately 3.98. (b) Determine the inductance in an electric circuit if the resonance frequency is 7.12 and the capacitance is 0.0001. Use \(\pi=3.14\).

Step 1: Write down the given values

Resonance frequency (f) is given as 7.12 and the capacitance (C) is 0.0001.

Step 2: Rearrange the formula for inductance (L)

We first need to rearrange the formula to solve for L: \(L=\frac{1}{(2 \pi f)^2 C}\).

Step 3: Plug in the values for f and C

Now, we can plug in the values for f and C: \(L=\frac{1}{(2 \pi \times 7.12)^2 \times 0.0001}\).

Step 4: Calculate the inductance

We can now calculate the inductance value using the provided value of \(\pi=3.14\). \(L=\frac{1}{(2 \times 3.14 \times 7.12)^2 \times 0.0001} \approx 0.000626\). The inductance is approximately 0.000626.

Key Concepts

Inductance

Inductance is a fundamental concept in electronics, representing the ability of a conductor to store energy in a magnetic field when an electric current flows through it. The unit of inductance is the henry (H). Inductors are the components used to implement inductance in electronic circuits.
Inductors frequently appear in tuning circuits, such as radio receivers, where they can be combined with capacitors to select specific frequencies. The behavior and characteristics of inductors can be influenced by:

• The physical shape and size of the inductor.
• The material of the core used in constructing the inductor, which can impact the magnetic field strength.

In the context of resonance frequency, the value of inductance will directly affect the frequency at which the circuit resonates, following the formula \[ f = \frac{1}{2 \pi \sqrt{L C}} \]This clearly shows that changing the inductance will alter the resonance frequency of the circuit.

Capacitance

Capacitance is the ability of a system to store an electric charge. Capacitors, the components that embody capacitance, store potential energy in an electric field. They are characterized by their capacity value, measured in farads (F). The capacitance of a capacitor can depend on various factors including:

• The surface area of the plates – larger areas provide higher capacitance.
• The distance between the plates – smaller distances increase capacitance.
• The type of dielectric material used between the plates.

When combined with inductance in a circuit, capacitance forms a resonant structure. This means that specific frequencies can be emphasized or reduced based on the values of capacitance and the paired inductance. The relationship between capacitance and resonance frequency can be easily derived from the resonance frequency formula. Increasing capacitance generally results in a lower resonance frequency, while decreasing it achieves the opposite effect.

Electronic Circuit

An electronic circuit is a path or network through which electrons flow, facilitated by a series of electronic components like resistors, capacitors, and inductors. These circuits are the foundation for almost all modern electronics, allowing for functions like signal processing, power amplification, and computation.
Electronic circuits can be divided into different types based on their complexity and applications:

• **Analog circuits**, where signals process continuously variable voltage levels.
• **Digital circuits**, which work with discrete value levels, using logic gates to perform computation.
• **Mixed-signal circuits**, that contain both analog and digital components.

Resonance in an electronic circuit refers to the condition where the inductance and capacitance reactances cancel each other out at a specific frequency, called the resonance frequency. At resonance, the circuit can oscillate at maximum amplitude, making the understanding of the underlying parameters crucial for designing efficient electronic systems.