Problem 72 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 69-72: The trigonometric form of complex numbers is often used by electrical engineers to describe the current \(I\), voltage \(V\), and impedance \(Z\) in electrical circuits with alternating current. Impedance is the opposition to the flow of current in a circuit. Most common electrical devices operate on 115 -volt, alternating current. The relationship among these three quantities is \(I=V / Z\). Approximate the unknown quantity, and express the answer in rectangular form to two decimal places. Finding current \(\quad Z=78 \operatorname{cis} 61^{\circ}, \quad V=163 \operatorname{cis} 17^{\circ}\)

Step-by-Step Solution

Answer

The current \( I \) is approximately \( 1.50 - 1.45i \) in rectangular form.

Step 1: Understand the Problem Statement

We need to find the current \( I \) using the relationship \( I = \frac{V}{Z} \), where \( V \) and \( Z \) are provided in trigonometric form as \( 163 \operatorname{cis} 17^{\circ} \) and \( 78 \operatorname{cis} 61^{\circ} \) respectively.

Step 2: Divide Magnitudes

To find \( I \), divide the magnitudes of \( V \) and \( Z \). The magnitude of \( I \) is found by dividing the magnitude of \( V \) by the magnitude of \( Z \):\[|I| = \frac{163}{78}\].

Step 3: Subtract Angles

To determine the angle of \( I \), subtract the angle of \( Z \) from the angle of \( V \):\[\theta_{I} = 17^{\circ} - 61^{\circ} = -44^{\circ}\].

Step 4: Write in Trigonometric Form

Combine the results to write \( I \) in trigonometric form:\[I = \frac{163}{78} \operatorname{cis} (-44^{\circ})\].

Step 5: Convert to Rectangular Form

Convert \( I \) to rectangular form using \( a + bi \):\[I = \frac{163}{78} (\cos(-44^{\circ}) + i \sin(-44^{\circ}))\]. Evaluate \( \cos(-44^{\circ}) \) and \( \sin(-44^{\circ}) \), and compute the expression to obtain the rectangular form.

Step 6: Calculate Cos and Sin

Compute \( \cos(-44^{\circ}) \approx 0.71934 \) and \( \sin(-44^{\circ}) \approx -0.69466 \).

Step 7: Compute Final Rectangular Values

Substitute back to get:\[I \approx 2.089 (0.71934 - 0.69466i) \approx 1.50 - 1.45i\].

Key Concepts

Impedance in Electrical Circuits

Impedance is a fundamental concept when dealing with electrical circuits, especially in the context of alternating current (AC). It can be thought of as the opposition that a circuit presents to the flow of current. Unlike resistance, which only deals with the opposition in DC circuits, impedance takes into account two additional factors:

• Resistance (R): The real part of impedance, which behaves just like pure resistance in DC circuits.
• Reactance (X): The imaginary part, arising from inductors and capacitors, that affects the phase of the current.

Together, these components form a complex number representation of impedance: \[ Z = R + jX \]where \( j \) is the imaginary unit, replacing \( i \) in electrical engineering convention. This complex number form allows engineers to easily calculate and analyze circuit characteristics, such as phase difference and power loss. In tasks involving impedance, trigonometric forms allow for the computation of quantities like voltage and current, simplifying the manipulation of the complex numbers.
When solving problems, we often deal with impedance in its trigonometric form, noted as \( Z = |Z| \operatorname{cis} \theta \), which illustrates the magnitude and phase angle, facilitating complex mathematical operations.

AC Electrical Circuits

Alternating current (AC) electrical circuits are the backbone of modern power systems. In AC circuits, the flow of electric charge periodically reverses direction, which is different from direct current (DC) circuits where the current flows steadily in one direction. This reversing action of AC allows power to be distributed more efficiently over long distances.
The three primary components of AC circuits are:

• Voltage (V): The potential difference that drives current through the circuit, expressed in a sinusoidal form \( V(t) = V_0 \sin(\omega t + \phi) \).
• Current (I): The flow of electric charge that alternates in a sinusoidal manner.
• Impedance (Z): Combines resistance and reactance to govern how the circuit impedes the current flow, affecting both amplitude and phase of the current.

AC circuit analysis often employs complex numbers to describe these quantities because it simplifies the algebra involved in solving circuit equations. The relationship among these quantities in a circuit is given by:\[ I = \frac{V}{Z} \]This equation is key in determining one variable when the others are known, as it integrates the magnitude and phase relationships across AC circuit elements.

Rectangular Form of Complex Numbers

In mathematics and engineering, complex numbers can be expressed in several forms, but the rectangular form is one of the most straightforward. This form expresses a complex number as:\[ a + bi \]where \( a \) is the real part and \( b \) is the imaginary part.
In electrical engineering, converting a complex number from its trigonometric or polar form \( r \operatorname{cis} \theta \) to rectangular form is crucial. This involves using trigonometric identities to expand it as:\[ r(\cos \theta + \sin \theta i) \]This conversion is essential for simplifying calculations involving addition, subtraction, and other operations. For example, when calculating the current \( I \) in a circuit given in the trigonometric form, we first find its magnitude \( r = \frac{V}{Z} \) and angle \( \theta \), then convert it into rectangular form. In our exercise, this approach allows the expression of results in a format that can easily be interpreted or used in further calculations.
Rectangular form offers direct insight into how a complex number will behave when combined with others, making it an indispensable tool in both circuit analysis and broader mathematical fields.