Problem 69 - 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 \(I=10 \operatorname{cis} 35^{\circ}, \quad Z=3 \operatorname{cis} 20^{\circ}\)

Step-by-Step Solution

Answer

The voltage in rectangular form is approximately \(17.18 + 24.57i\).

Step 1: Calculate the Magnitude of Voltage

Using the formula \(V = I \times Z\), we calculate the magnitude of the voltage. The magnitude of \(I\) is 10 and the magnitude of \(Z\) is 3. So, the magnitude of \(V\) is \(10 \times 3 = 30\).

Step 2: Calculate the Angle of Voltage

To obtain the angle for voltage \(V\), add the angles of the current \(I\) and impedance \(Z\): \(35^{\circ} + 20^{\circ} = 55^{\circ}\).

Step 3: Write Voltage in Trigonometric Form

Now, express the voltage \(V\) in trigonometric form: \(V = 30 \operatorname{cis} 55^{\circ}\).

Step 4: Convert Voltage to Rectangular Form

Convert \(30 \operatorname{cis} 55^{\circ}\) into rectangular form. Use the formulas: \(a = r \cos \theta\) and \(b = r \sin \theta\), where \(r = 30\) and \(\theta = 55^{\circ}\). Calculate: \(a = 30 \cos 55^{\circ}\) and \(b = 30 \sin 55^{\circ}\). Use a calculator to find: \(a \approx 17.18\) and \(b \approx 24.57\). Thus, the rectangular form is \(17.18 + 24.57i\).

Key Concepts

Alternating Current

Alternating current (AC) is the type of electrical current commonly used in households and electronic devices. In alternating current, the flow of electric charge periodically reverses direction. This is different from direct current (DC) where the flow of electric charge is only in one direction. AC is beneficial because it is easier to transform to different voltages, which is important for power distribution over long distances. AC typically operates at a standard voltage, such as 115 volts in many regions. This makes it compatible with a wide range of devices and appliances. Understanding AC is critical in electrical engineering, as it affects how electrical components and systems are designed and operate in circuits.

Impedance

Impedance, denoted by the symbol \(Z\), measures the opposition that a circuit presents to the passage of alternating current. It combines two types of resistance: ohmic resistance (simple resistance) and reactance (which includes capacitive and inductive effects).

Impedance is a complex number because it contains both a real component (resistance) and an imaginary component (reactance). The total impedance in a circuit affects the current flow and therefore needs to be calculated accurately for safe and efficient circuit design.

Engineers frequently use the formula \(I = V / Z\) to find the current given voltage \(V\) and impedance \(Z\). The unit of impedance is the ohm (\(\Omega\)). Like resistor values, impedance can vary with frequency, so it's especially important in applications where signals change frequency, like in alternating current systems.

Rectangular Form

When working with complex numbers, the rectangular form is one of the two most common ways to express these numbers. In rectangular form, a complex number is written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The letter \(i\) represents the imaginary unit, defined by \(i^2 = -1\).

Expressing complex numbers in rectangular form is often useful in calculations involving addition and subtraction. It gives a straightforward visual representation as points or vectors in a two-dimensional plane, known as the complex plane.

For example, converting from the trigonometric form \(r \text{cis} \theta\) to rectangular form involves calculating \(a = r \cos \theta\) and \(b = r \sin \theta\). This allows you to see the real and imaginary components separately and facilitates further mathematical manipulations or interpretations.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in understanding wave patterns and oscillations in electrical circuits, particularly those involving alternating current. These functions help describe how current and voltage change over time in an AC circuit.

In complex numbers, the trigonometric form \(r \text{cis} \theta\) (where \(\text{cis} \theta = \cos \theta + i \sin \theta\)) is used. This form is handy for multiplying and dividing complex numbers because it allows direct addition and subtraction of angles while multiplying magnitudes.

The usage of such functions is crucial in modeling alternating currents and voltages, helping courses and circuits operate efficiently. By leveraging trigonometric identities and Euler's Formula, engineers simplify the calculations needed to predict how circuits will respond to varying frequencies and phase differences.