Problem 25 - Elementary Algebra

Question

The voltage drops in an AC circuit are \(3.1460+17.175 i\) volts, \(-18.697+3.0820 i\) volts, and \(135.61-20.252 i\) volts. Find the sum of these voltages.

Step-by-Step Solution

Answer

120.059 + 0.005i volts

Step 1: Identify the Given Voltages

The given voltages in the AC circuit are: 1) 3.1460 + 17.175i volts2) -18.697 + 3.0820i volts3) 135.61 - 20.252i volts

Step 2: Break Down the Real and Imaginary Parts

Separate the real and imaginary parts of each voltage: Real parts: 3.1460, -18.697, 135.61 Imaginary parts: 17.175i, 3.0820i, -20.252i

Step 3: Sum the Real Parts

Sum the real parts of the voltages: 3.1460 + (-18.697) + 135.61 = 120.059

Step 4: Sum the Imaginary Parts

Sum the imaginary parts of the voltages: 17.175i + 3.0820i + (-20.252i) = 0.005i

Step 5: Combine the Results

Combine the summed real and imaginary parts to get the total voltage: 120.059 + 0.005i volts

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in AC circuits and many other areas of electrical engineering. A complex number has two components: a real part and an imaginary part. These parts are usually written in the form:
\( z = a + bi \)
Where:

• \( a \) is the real part
• \( bi \) is the imaginary part, and \( i \) is the imaginary unit with the property \( i^2 = -1 \)

Complex numbers allow us to represent quantities that have both magnitude and direction, such as the voltage in an AC circuit. In this exercise, we are given three complex numbers that represent voltage drops: \( 3.1460 + 17.175i \) volts, \( -18.697 + 3.0820i \) volts, and \( 135.61 - 20.252i \) volts.

Real and Imaginary Parts

When dealing with complex numbers, it's often useful to separate them into their real and imaginary parts. This can make operations like addition much simpler. For instance, consider the numbers given in our exercise:

• For \( 3.1460 + 17.175i \), the real part is \( 3.1460 \) and the imaginary part is \( 17.175i \)
• For \( -18.697 + 3.0820i \), the real part is \( -18.697 \) and the imaginary part is \( 3.0820i \)
• For \( 135.61 - 20.252i \), the real part is \( 135.61 \) and the imaginary part is \( -20.252i \)

By writing complex numbers out in this way, we can focus on performing operations on the real and imaginary parts separately. This is particularly useful when adding complex numbers, as we will see in the next section.

Addition of Complex Numbers

Adding complex numbers involves summing their respective real and imaginary parts. Let's walk through the addition process using our example:
1. **Identify Real and Imaginary Parts**

• The real parts are: \( 3.1460 \), \( -18.697 \), \( 135.61 \)
• The imaginary parts are: \( 17.175i \), \( 3.0820i \), \( -20.252i \)

2. **Sum the Real Parts**
Add the real parts together:
\( 3.1460 + (-18.697) + 135.61 = 120.059 \)
3. **Sum the Imaginary Parts**
Now add the imaginary parts:
\( 17.175i + 3.0820i + (-20.252i) = 0.005i \)
4. **Combine Results**
Finally, combine the sums of the real and imaginary parts to get the total voltage:
\( 120.059 + 0.005i \) volts.
By understanding and applying these steps, dealing with complex numbers in AC circuits becomes quite straightforward.