Problem 76 - Algebra and Trigonometry with Analytic Geometry

Question

The real part of \(I\) represents the actual current delivered to an electrical appliance in amps. Approximate this current when \(V=163\) cis \(43^{\circ}\) and \(Z=100\) cis \(17^{\circ}\).

Step-by-Step Solution

Answer

The actual current delivered is approximately 1.465 amps.

Step 1: Understand the Problem

To find the actual current, we first need to interpret the problem statement. We are given a complex voltage, \(V = 163 \text{ cis } 43^{\circ}\), and a complex impedance, \(Z = 100 \text{ cis } 17^{\circ}\). The current, \(I\), through the electrical appliance can be found using Ohm's Law, \(I = \frac{V}{Z}\).

Step 2: Apply Ohm's Law

According to Ohm's Law for AC circuits, the current \(I\) is the voltage divided by the impedance. So, we have:\[I = \frac{V}{Z} = \frac{163 \text{ cis } 43^{\circ}}{100 \text{ cis } 17^{\circ}}\]

Step 3: Perform Division of Complex Numbers in Polar Form

To divide two complex numbers in polar form, divide their magnitudes and subtract their angles. Thus:\[|I| = \frac{163}{100} = 1.63\]\[\text{Angle of } I = 43^{\circ} - 17^{\circ} = 26^{\circ}\]Therefore, \(I = 1.63 \text{ cis } 26^{\circ}\).

Step 4: Calculate the Real Part of the Current

The real part of a complex number in polar form \(r \text{ cis } \theta\) is given by \(r \cos \theta\). Thus, the real part of our current is:\[\text{Real Part of } I = 1.63 \cos 26^{\circ}\]Calculating \(\cos 26^{\circ} \approx 0.8988\), we find:\[\text{Real Part of } I \approx 1.63 \times 0.8988 \approx 1.465\]

Step 5: Conclusion

The real part of the current, which represents the actual current delivered to the appliance, is approximately 1.465 amps.

Key Concepts

Complex Numbers

Complex numbers are essential in understanding many concepts in engineering and physics, especially in the context of AC circuits. They consist of a real part and an imaginary part. A complex number can be represented as \[ z = a + bi \]where:

• \(a\) is the real part
• \(b\) is the imaginary part

These numbers are incredibly useful when dealing with quantities that have both magnitude and direction, like current and voltage in electrical circuits. By using complex numbers, we can represent these values succinctly, simplifying calculations. Typically, we use the imaginary unit \(i\), where \(i^2 = -1\). In electrical engineering, we might also see \(j\) used in place of \(i\) to avoid confusion with current, which is often denoted as \(I\). Understanding complex numbers helps us manage the wave-like properties of AC signals effectively.

Polar Form

Polar form is a way of representing complex numbers, especially beneficial for multiplication and division operations, which are common in AC circuit analysis. Instead of the standard form \[ a + bi \]complex numbers can be depicted in polar form as \[ z = r \text{ cis } \theta \]Here,

• \(r\) represents the magnitude or absolute value of the complex number
• \(\theta\) is the angle (in degrees or radians) with the positive real axis

The notation "cis" is shorthand for "\(\cos \theta + i\sin \theta\)" based on Euler's formula. Polar form is advantageous when dividing or multiplying complex numbers since you can simply manipulate the magnitude and add or subtract the angles. This form aligns with natural interpretations of phenomena like rotation and oscillations in AC circuits.

AC Circuits

AC circuits utilize alternating current, where the flow of electric charge periodically reverses direction. This is in contrast to direct current (DC), where the flow of charge is in one direction. The characteristics of AC make it more efficient for long-distance transmission of electricity. In AC circuits, Ohm's Law still applies but in a complex form, allowing us to incorporate phase differences between current and voltage: \[ I = \frac{V}{Z} \]where:

• \(I\) is the current
• \(V\) is the voltage
• \(Z\) is the impedance

The complexity of AC currents is managed using phasor representation, typically employing polar form for quick calculations. The alternating nature introduces concepts like reactance, which plays a crucial role in impedance calculations.

Impedance

Impedance is a fundamental concept in AC circuit analysis. It generalizes resistance to circuits where the current and voltage are not in phase. Unlike resistance, impedance is a complex quantity represented by \[ Z = R + jX \]where:

• \(R\) is the resistance, the real part of impedance
• \(X\) is the reactance, the imaginary part of impedance

Reactance arises from the presence of capacitors and inductors in a circuit, affecting how the circuit responds to frequency. The magnitude of impedance \(|Z|\) affects the amplitude of the current flowing through the circuit. The phase angle \(\theta\) tells us about the phase difference between current and voltage. Calculating and understanding impedance is vital for the analysis and design of AC circuits, ensuring they operate efficiently under various frequencies and loads.