Problem 71 - 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 impedance \(I=8\) cis \(5^{\circ}, \quad V=115\) cis \(45^{\circ}\)

Step-by-Step Solution

Answer

The impedance \( Z \) is approximately \( 11.02 + 9.24i \).

Step 1: Write Down the Given Values

We are given the current \( I = 8 \text{ cis } 5^{\circ} \) and voltage \( V = 115 \text{ cis } 45^{\circ} \). We need to find the impedance \( Z \). We'll first write these values.

Step 2: Use the Relationship I = V/Z to Find Z

From the formula \( I = \frac{V}{Z} \), we can rearrange it to find \( Z \): \( Z = \frac{V}{I} \). Substitute the given values into this equation.

Step 3: Divide the Magnitudes

Divide the magnitudes of \( V \) and \( I \): \( \frac{115}{8} = 14.375 \). This is the magnitude of \( Z \).

Step 4: Subtract the Angles

For the phase angle of \( Z \), subtract the angle of \( I \) from the angle of \( V \): \( 45^{\circ} - 5^{\circ} = 40^{\circ} \). The angle of \( Z \) is \( 40^{\circ} \).

Step 5: Write Z in Trigonometric Form

The impedance \( Z \) in trigonometric form is \( 14.375 \text{ cis } 40^{\circ} \).

Step 6: Convert to Rectangular Form

Convert from trigonometric form to rectangular form using formulas: \( a = r \cos \theta \) and \( b = r \sin \theta \). Here, \( r = 14.375 \) and \( \theta = 40^{\circ} \). First, calculate the real part, \( a = 14.375 \cos 40^{\circ} \approx 11.02 \). Then, calculate the imaginary part, \( b = 14.375 \sin 40^{\circ} \approx 9.24 \). Thus, \( Z \) is approximately \( 11.02 + 9.24i \).

Key Concepts

Trigonometric Form

In electrical engineering, complex numbers are often expressed in what is called the 'trigonometric form.' This form is very useful when dealing with alternating current (AC) circuits. The trigonometric form represents a complex number as \( r \text{ cis } \theta \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the angle (or argument) in degrees or radians.

This form is advantageous because it simplifies the multiplication and division of complex numbers through the use of polar coordinates. When two complex numbers are multiplied, their magnitudes are multiplied and their angles are added. Similarly, when divided, their magnitudes are divided, and the angles are subtracted. This is particularly useful when calculating electrical quantities such as impedance.
Using trigonometric form helps engineers easily manipulate circuit variables like current, voltage, and impedance. It allows for a visual and intuitive understanding of how these quantities relate to each other in an AC circuit.

Rectangular Form

The rectangular form of a complex number expresses it in the format \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. This format is often preferred when graphing complex numbers on a Cartesian plane.

To convert from the trigonometric form to rectangular form, you use the formulas:

• \( a = r \cos \theta \)
• \( b = r \sin \theta \)

Here, \( r \) is the magnitude and \( \theta \) is the angle. This conversion allows you to see how the complex number relates directly in terms of horizontal (real) and vertical (imaginary) components.
In problems involving electrical engineering, once calculations are completed using the trigonometric form, the final step is often to express the result in rectangular form. This conversion provides a clearer numerical understanding of the complex value, which in practical applications, translates into workable engineering parameters.

Alternating Current

Alternating Current (AC) is an essential concept in electrical engineering, representing a type of current where the flow of electric charge periodically reverses direction. This is in contrast to Direct Current (DC), where the flow is unidirectional.

AC is the form of electricity most commonly used in households and businesses. It is efficient in terms of transmission as it can easily be transformed to different voltages. In the context of complex numbers, AC is naturally represented using trigonometric or exponential forms. This is because AC signals are sinusoidal and can be easily described by their amplitude and phase angle, which correspond to the magnitude and angle in complex number terms.
When dealing with AC circuits, complex numbers are employed to analyze the phase difference between current and voltage, as well as the impedance that opposes the flow of AC.

Impedance

Impedance, denoted as \( Z \), plays a critical role in AC circuits. It is a measure of opposition a circuit presents to the flow of alternating current and can combine both resistance and reactance into a single value.

The relationship between current \( I \), voltage \( V \), and impedance is expressed with the formula \( I = \frac{V}{Z} \). Impedance is often represented in complex numbers because it incorporates both magnitude and phase difference with the trigonometric or polar form being particularly useful.
In practical terms, the impedance determines how much voltage is required to drive a given current through the circuit. In AC analysis, calculations of impedance can be greatly simplified using complex notation, further allowing engineers to determine the load's impact on circuit behavior.