Problem 70 - 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 voltage \(\quad I=12 \operatorname{cis} 5^{\circ}, \quad Z=100 \operatorname{cis} 90^{\circ}\)

Step-by-Step Solution

Answer

The voltage \(V\) is approximately \(-104.58 + 1195.43i\) in rectangular form.

Step 1: Recall the relationship

The relationship between current \(I\), voltage \(V\), and impedance \(Z\) is given by the formula \(I = \frac{V}{Z}\). Here, we need to calculate the voltage \(V\). Rearranging the formula, we have \(V = I \cdot Z\).

Step 2: Express components in exponential form

Convert \(I = 12 \operatorname{cis} 5^\circ \) and \(Z = 100 \operatorname{cis} 90^\circ \) to their exponential forms. The cis notation is shorthand for the trigonometric form, specifically \(r \operatorname{cis} \theta = r (\cos \theta + i \sin \theta ) = r e^{i\theta}\). Thus, \(I = 12e^{i \frac{\pi}{36}}\) and \(Z = 100e^{i \frac{\pi}{2}}\).

Step 3: Multiply the complex numbers

To find \(V\), multiply the expressions for \(I\) and \(Z\):\[V = 12e^{i\frac{\pi}{36}} \cdot 100e^{i\frac{\pi}{2}} = 1200e^{i(\frac{\pi}{36} + \frac{\pi}{2})}\] Combine the exponents:\[V = 1200e^{i\frac{\pi}{36} + \frac{\pi}{2}} = 1200e^{i\frac{19\pi}{36}}\]

Step 4: Convert back to rectangular form

Now convert \(V = 1200e^{i\frac{19\pi}{36}}\) back to rectangular form:\[\cos\left(\frac{19\pi}{36}\right) = \cos(95^\circ) \approx -0.08715, \quad \sin\left(\frac{19\pi}{36}\right) = \sin(95^\circ) \approx 0.99619\]Therefore, the rectangular form is:\(V = 1200 (-0.08715 + i \cdot 0.99619)\)Which simplifies to:\(V = -104.58 + 1195.43i\) with each component rounded to two decimal places.

Key Concepts

alternating current

Alternating current, often abbreviated as AC, is a type of electrical current where the flow of electric charge periodically reverses direction. Unlike direct current (DC) which flows in one direction, AC is the form of electricity commonly used in houses and businesses around the world.
The frequency at which the current alternates is measured in hertz (Hz), which indicates the number of cycles per second. In most European countries, this frequency is 50 Hz, while in North America, it is 60 Hz.
AC is particularly advantageous for transmitting power over long distances. This is because the voltage can be easily transformed to higher or lower levels using transformers, minimizing energy losses during transmission.
Electrical devices that use alternating current include:

• Household appliances like refrigerators and washing machines
• Light bulbs and fluorescent lamps
• Personal electronics when connected to the power grid

Understanding AC is crucial for grasping more complex electrical concepts, such as impedance and the use of complex numbers in analyzing circuits.

impedance in electrical circuits

Impedance, denoted as the complex number \(Z\), is a measure of how much an object resists the flow of alternating current. It extends the concept of resistance, which applies to direct current. Impedance includes both resistance (R) and reactance (X), where reactance accounts for the reactance caused by capacitors and inductors in the circuit.
The units of impedance are ohms (\(\Omega\)), similar to resistance, and its complex nature allows it to be expressed as a combination of real and imaginary components: \(Z = R + iX\).
There are two key components of impedance:

• Resistance (R): The real part, which represents the energy dissipated as heat.
• Reactance (X): The imaginary part, which represents the energy stored temporarily in magnetic and electric fields.

Electrical engineers often use the trigonometric form of impedance to better analyze circuits with alternating currents. They can calculate how different components in a circuit resist AC, helping design effective and efficient electronic devices.

rectangular form of complex numbers

The rectangular form of complex numbers is a way of expressing them as a sum of their real and imaginary parts: \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
This form is particularly useful for performing basic arithmetic operations such as addition and subtraction of complex numbers.
In electrical engineering, rectangular form aids in calculating and visualizing components of circuits separately as real and imaginary parts, especially when working with impedance and electronic signals.
For example, when expressing impedance in rectangular form, you can immediately see how much is due to resistance and how much is due to reactance. This visibility is why transforming solutions like voltage and current to this form, as shown in the exercise, is a standard practice in circuit analysis. It helps engineers quickly analyze and troubleshoot components in complex electrical systems.

multiplication of complex numbers in polar form

Multiplying complex numbers becomes much simpler when they are expressed in polar form, often noted as \(r\operatorname{cis} \theta\) or \(re^{i\theta}\). This involves using their magnitudes (\(r\)) and angles (\(\theta\)).
The beauty of polar form is in how it handles multiplication: magnitudes are multiplied while angles are added. This results in an expression \(r_1r_2e^{i(\theta_1 + \theta_2)}\).
This method significantly simplifies calculations involving rotations and scaling, which is why it's preferred in applications such as AC circuit analysis and control systems.
For electrical engineers, multiplying impedance or current values in polar form makes it straightforward to analyze their relationships and effects in a circuit, providing insights into their harmonic sequences and signal behaviors, which are essential in designing efficient electrical systems.
Converting the result back to rectangular form, as showcased in the original exercise, helps to interpret the physical quantities effectively.