Problem 1 - Applied Physics

Question

For a circuit with \(R=20 \overline{0} \Omega, L=10.0 \mathrm{mH}\), and \(f=1.25 \mathrm{kHz}\) : (a) find the impedance (in ohms), (b) find the phase angle, and (c) find the current if the voltage is \(45.0 \mathrm{~V}\).

Step-by-Step Solution

Answer

(a) Impedance is 80.02 Ω. (b) Phase angle is 1.325 radians. (c) Current is 0.562 A.

Step 1: Calculate Angular Frequency

First, calculate the angular frequency using the formula \( \omega = 2\pi f \). Substituting the given frequency \( f = 1.25 \text{ kHz} = 1250 \text{ Hz} \), we have:\[ \omega = 2 \times \pi \times 1250 = 2500\pi \text{ rad/s} \]

Step 2: Calculate Inductive Reactance

Calculate the inductive reactance \( X_L \) using the formula \( X_L = \omega L \). Given \( L = 10.0 \text{ mH} = 10.0 \times 10^{-3} \text{ H} \):\[ X_L = 2500\pi \times 10.0 \times 10^{-3} \approx 78.54 \Omega \]

Step 3: Calculate Total Impedance

The total impedance \( Z \) in an RL circuit is given by:\[ Z = \sqrt{R^2 + X_L^2} \]Substitute the values \( R = 20.0 \Omega \) and \( X_L = 78.54 \Omega \):\[ Z = \sqrt{(20.0)^2 + (78.54)^2} \approx 80.02 \Omega \]

Step 4: Calculate Phase Angle

The phase angle \( \phi \) is calculated using \( \tan(\phi) = \frac{X_L}{R} \):\[ \phi = \arctan\left(\frac{78.54}{20.0}\right) \approx 1.325 \text{ radians} \]

Step 5: Calculate Current

Use Ohm's Law to find the current \( I \) with the total voltage \( V = 45.0 \text{ V} \):\[ I = \frac{V}{Z} = \frac{45.0}{80.02} \approx 0.562 \text{ A} \]

Key Concepts

Impedance Calculation

The impedance of a circuit is a key concept, representing how much the circuit resists the flow of electrical current when a voltage is applied. In the simplest terms for an RL circuit, which includes a resistor and an inductor, impedance combines these two elements into a single value.

• The resistance (R) is straightforward, as it's simply the inherent opposition to current in the resistor, measured in ohms (Ω).
• The inductor introduces inductive reactance (X_L), which varies with frequency and also has units of ohms.

To find the overall or total impedance (Z) in an RL circuit, we use the Pythagorean theorem, integrating both the resistance and the inductive reactance:\[Z = \sqrt{R^2 + X_L^2}\]In this exercise, we calculated the impedance to be approximately 80.02 Ω. This step is crucial because impedance affects how the circuit behaves under an AC (alternating current) source, dictating how much current will flow given a certain voltage.

Phase Angle Determination

The phase angle in a circuit indicates the phase difference between the voltage across the entire circuit and the current flowing through it. This can be particularly important in AC circuits where voltages and currents can lag or lead each other.To determine the phase angle (\phi), we rely on the ratio of the inductive reactance to the resistance:\[\tan(\phi) = \frac{X_L}{R}\]From this, the phase angle is the arctangent of the X_L/R ratio. In our problem, we found the phase angle to be approximately 1.325 radians. When converted, this describes how much the current wave is shifting with respect to the voltage wave. A positive phase angle suggests that the current lags the voltage, typical in circuits with inductance. This lag can affect how devices connected to the circuit perform.

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering, relating voltage (V), current (I), and impedance (Z) within a circuit. It can be expressed in the formula:\[I = \frac{V}{Z}\]This simple equation shows how the current is directly proportional to the voltage and inversely proportional to the impedance. In our RL circuit task, once we determined the total impedance, we used Ohm’s Law to find the current flowing through the circuit when a 45.0V source was applied.

• Calculating the current is straightforward with this equation, emphasizing the practical importance of finding the correct impedance.
• Understanding Ohm's Law helps in designing circuits, predicting behavior, and troubleshooting issues when something goes wrong.

In our calculation, we found the current to be approximately 0.562 A. Applying Ohm's Law is especially useful when analyzing AC circuits like this one, as it helps one quickly verify if calculations and measurements are consistent.

Inductive Reactance

Inductive reactance is an important parameter in AC circuits with inductors. It quantifies the opposition of an inductor to changes in current. Unlike resistance, which is constant, inductive reactance depends on the frequency of the alternating current:\[X_L = \omega L\]where \( \omega \) is the angular frequency and \( L \) is the inductance. Inductive reactance increases with both the frequency of the current and the value of the inductance.In our exercise, we were given an inductance of 10.0 mH and calculated the inductive reactance to be around 78.54 Ω using our given frequency.

• This calculation highlights why circuits behave differently at varying frequencies.
• High inductive reactance can significantly reduce current flow, introducing a delay or phase shift between voltage and current, as previously discussed in the phase angle section.

Understanding inductive reactance is vital for designing circuits that perform optimally under various signal conditions.

Angular Frequency Calculation

Angular frequency is a key concept in rotating systems and AC circuits that describes how fast the electrical cycle repeats. It's often denoted as \( \omega \) and is directly related to the frequency (\( f \)) of the signal:\[\omega = 2\pi f\]Angular frequency is expressed in radians per second, embracing the cyclical nature of AC signals and their sinusoidal waveform.In this scenario, we were given a frequency of 1.25 kHz, converted to 1250 Hz, which led to an angular frequency of \(2500\pi\) rad/s.

• This foundational value is used in calculations for other parameters, like inductive reactance.
• Angular frequency helps link the mathematical treatment of the circuit with its physical behavior.

By understanding angular frequency, students can develop insights into how circuit elements like inductors and capacitors affect signal timing and behavior.