Problem 47 - College Algebra with Modeling and Visualization

Question

Electricity \(\quad\) Complex numbers are used in the study of electrical circuits. Impedance \(Z\) (or the opposition to the flow of electricity. voltage \(V\) and current \(I\) can all be represented by complex numbers. They are related by the equation \(Z=\frac{V}{I} .\) Find the value of the missing variable. $$ Z=22-5 i \quad V=27+17 i $$

Step-by-Step Solution

Answer

The current \(I\) is \(1 + i\).

Step 1: Identify the Given Information

We have the impedance as a complex number \(Z = 22 - 5i\) and the voltage \(V = 27 + 17i\). We need to find the current \(I\), using the relation \(Z = \frac{V}{I}\).

Step 2: Rearrange the Formula

Using the relation \(Z = \frac{V}{I}\), rearrange it to solve for \(I\). The rearranged formula becomes \(I = \frac{V}{Z}\).

Step 3: Apply the Formula

Substitute the given values into the rearranged formula: \(I = \frac{27 + 17i}{22 - 5i}\).

Step 4: Multiply by the Conjugate

To divide complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(22 - 5i\) is \(22 + 5i\). Multiply: \(I = \frac{(27 + 17i)(22 + 5i)}{(22 - 5i)(22 + 5i)}\).

Step 5: Simplify the Denominator

Simplify the denominator using the formula \((a-b)(a+b) = a^2 - b^2\). For \((22-5i)(22+5i)\), calculate as follows: \(22^2 - (5i)^2 = 484 - (25)(-1) = 484 + 25 = 509\).

Step 6: Expand the Numerator

Use the distributive property to multiply \((27 + 17i)(22 + 5i)\). Calculate each term: - \(27 \times 22 = 594\), - \(27 \times 5i = 135i\), - \(17i \times 22 = 374i\), - \(17i \times 5i = 85i^2 = -85\).Combine these to get: \(594 + 135i + 374i - 85 = 509 + 509i\).

Step 7: Divide Each Part of the Numerator by the Denominator

Divide each part of the complex numerator by the simplified denominator: \(I = \frac{509}{509} + \frac{509i}{509} = 1 + i\).

Step 8: Verify the Solution

Double-check each calculation to ensure accuracy and verify that the performed operations align with the complex division process.

Key Concepts

Electrical Circuits

Electrical circuits are paths through which electricity flows. In such circuits, complex numbers are incredibly useful for calculations, especially in alternating current (AC) systems. The components of these circuits include resistors, capacitors, and inductors, which oppose the flow of electric current in different ways. This opposition is known as impedance. Impedance combines resistance and reactance.

In an electrical circuit, it's essential to analyze how voltage and current behave. Complex numbers allow engineers to handle phases and magnitudes efficiently, thanks to their ability to represent two-dimensional quantities. Understanding how these components interact is crucial for designing functional and efficient circuits.

Impedance

Impedance ( Z ) is a fundamental concept in electrical circuits. It's the total opposition a component or circuit offers to the flow of alternating current. Unlike resistance, which only deals with the opposition in direct current (DC), impedance considers both resistance and reactance in AC.

• Resistance is the part of impedance that doesn't change with frequency. It resists the flow of electricity regardless of how fast it switches direction.
• Reactance varies with frequency. It comes from capacitors and inductors' storage and release of energy as they create opposing voltages to alternating currents.

Impedance is expressed as a complex number because it accounts for both the magnitude of opposition and the phase difference between current and voltage. By representing impedance with a complex number, we can keep track of these effects easily. The use of complex representation helps in solving electrical circuit problems by transforming them into algebraic equations.

Voltage and Current

Voltage ( V ) and current ( I ) are two vital aspects of any electrical circuit. Voltage resembles the pressure that pushes the flow of electricity, while current is the actual flow. These quantities are interconnected by impedance. In AC circuits, both voltage and current can be represented as complex numbers, accounting for their magnitudes and phase angles.

The relationship between voltage, current, and impedance can be expressed in the equation:\[ Z = \frac{V}{I} \]This equation shows how the three quantities are interrelated. Thus, if two of the values are known, the third can be calculated using complex division.

Complex Division

Complex division is the process used to divide one complex number by another. In the context of electrical circuits, it's essential for finding missing values like current when you know impedance and voltage. The procedure involves multiplying both the numerator and the denominator by the conjugate of the denominator. This turns the division into a safer multiplication problem by eliminating imaginary parts from the denominator.

For example, when dividing \( I = \frac{27 + 17i}{22 - 5i} \), we multiply both parts by \( 22 + 5i \), removing the imaginary component from the denominator. This results in the division being straightforward, as the denominator becomes a real number, simplifying calculations. This process highlights the elegance and utility of complex numbers in solving real-world electrical engineering problems.