Question

Electrical Circuits The impedance \(Z,\) in ohms, of a circuit element is defined as the ratio of the phasor voltage \(V,\) in volts, across the element to the phasor current \(I\), in amperes, through the element. That is, \(Z=\frac{V}{I} .\) If the voltage across a circuit element is \(18+i\) volts and the current through the element is \(3-4 i\) amperes, determine the impedance.

Short answer

The impedance is \( 2 + 3i \) ohms.

Step-by-step solution

Understand the Impedance Formula

Recall that impedance is defined as the ratio of the phasor voltage to the phasor current: \( Z = \frac{V}{I} \).

Write the Given Voltage and Current as Complex Numbers

The given voltage is \( V = 18 + i \) volts and the current is \( I = 3 - 4i \) amperes.

Perform the Division of Complex Numbers

To find the impedance Z, divide the phasor voltage by the phasor current: \( Z = \frac{18 + i}{3 - 4i} \).

Multiply by the Conjugate of the Denominator

To simplify the division, multiply the numerator and the denominator by the conjugate of the denominator: \( Z = \frac{(18 + i) \times (3 + 4i)}{(3 - 4i) \times (3 + 4i)} \).

Perform the Multiplications

Calculate the numerator: \((18 + i) \times (3 + 4i) = 54 + 72i + 3i + 4(i^2) = 54 + 75i - 4 = 50 + 75i \). Calculate the denominator: \( (3 - 4i) \times (3 + 4i) = 9 - 16(i^2) = 9 + 16 = 25 \).

Write the Result of the Impedance

Finally, divide the simplified numerator by the simplified denominator: \( Z = \frac{50 + 75i}{25} = 2 + 3i \).

Key concepts

complex numbers in electrical engineering

In electrical engineering, complex numbers are essential. They help to represent quantities that have both a magnitude and a direction, such as voltage and current.
The form of a complex number is usually written as: \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
Using complex numbers helps to simplify calculations related to alternating currents (AC) as they naturally embody phase angles.
Voltage and current in AC circuits are often out of phase due to elements like capacitors and inductors. Complex numbers allow engineers to work easily with these phase differences.
So, it is common to encounter terms like 'phasor voltage' and 'phasor current' when dealing with AC circuit analysis.

phasor voltage and current

Phasors are a way to represent sinusoidal functions like AC voltages and currents using complex numbers.
Let's break it down:

• Voltage and Current Phasor: These are complex numbers representing the peak amplitude and phase angle of sinusoidal voltage or current.
• Voltage Phasor: Denoted as \( V \), this represents the AC voltage's magnitude and phase.
• Current Phasor: Denoted as \( I \), this represents the AC current's magnitude and phase.

When you see a phasor like \( V = 18 + i \), it means the real part is 18 volts, and the imaginary part is 1 volt.
Similarly, a current phasor like \( I = 3 - 4i \) has a real part of 3 amperes and an imaginary part of -4 amperes.
These phasor notations make it easier to perform complex mathematical operations, such as addition, subtraction, multiplication, and division.

division of complex numbers

Dividing complex numbers can initially seem tricky, but it's straightforward after a few steps.
To divide complex numbers, you multiply the numerator and the denominator by the conjugate of the denominator.
Given: \[ Z = \frac{V}{I} = \frac{18 + i}{3 - 4i} \].
First, find the conjugate of the denominator. The conjugate of \(3 - 4i\) is \(3 + 4i\).
Next, perform the multiplication:

• Numerator: \( (18 + i) \times (3 + 4i) = 54 + 72i + 3i + 4i^2 = 54 + 75i - 4 = 50 + 75i \)
• Denominator: \( (3 - 4i) \times (3 + 4i) = 9 - 16i^2 = 9 + 16 = 25 \)

Finally, divide the simplified numerator by the simplified denominator: \[ Z = \frac{50 + 75i}{25} = 2 + 3i \]
This gives the impedance as a complex number \(2 + 3i\).Breaking it down step-by-step ensures clarity in dividing complex numbers. This method is crucial for calculating impedance in AC circuits!