Problem 68 - Algebra 2

Question

There are three basic things to be considered in an electrical circuit: the flow of the electrical current \(I,\) the resistance to the flow \(Z\) called impedance, and electromotive force \(E\) , called voltage. These quantities are related in the formula \(E=I \cdot Z .\) The current of a circuit is to be \(35-40 j\) amperes. Electrical engineers use the letter \(j\) to represent the imaginary unit. Find the impedance of the circuit if the voltage is to be \(430-330 j\) volts.

Step-by-Step Solution

Answer

The impedance is \(10 + 2j\) ohms.

Step 1: Identify the Formula

The relationship between current \(I\), impedance \(Z\), and voltage \(E\) is given by the formula \(E = I \cdot Z\). We will use this to find the impedance \(Z\).

Step 2: Substitute Given Values

Insert the known values into the equation. Here, \(E = 430 - 330j\) volts and \(I = 35 - 40j\) amperes.

Step 3: Re-arrange for Impedance

Rearrange the formula to find \(Z\):\[ Z = \frac{E}{I} \]

Step 4: Perform Division

To compute \(Z = \frac{430 - 330j}{35 - 40j}\), multiply the numerator and denominator by the conjugate of the denominator: \[ Z = \frac{(430 - 330j) \cdot (35 + 40j)}{(35 - 40j) \cdot (35 + 40j)} \]

Step 5: Calculate Denominator

Compute the denominator using the difference of squares formula:\[ (35 - 40j) \cdot (35 + 40j) = 35^2 - (40j)^2 = 1225 + 1600 = 2825 \]

Step 6: Calculate Numerator

Compute the product for the numerator:\[ (430 - 330j)(35 + 40j) = (430 \cdot 35 + 430 \cdot 40j - 330j \cdot 35 - 330j \cdot 40j) \] Simplifying, gives us:\[ 15050 + 17200j - 11550j + 13200 = 28250 + 5650j \]

Step 7: Solve for Impedance

Now divide the computed numerator by the denominator:\[ Z = \frac{28250 + 5650j}{2825} \] This simplifies to:\[ Z = 10 + 2j \]

Key Concepts

Impedance

In electrical circuits, impedance is a measure of how much a circuit resists the flow of alternating current. It combines resistance, which is a constant, with reactance, which changes with frequency. Impedance is a crucial part of circuit design because it affects how much current is allowed to pass through the circuit for a given voltage.

Impedance is represented by the symbol \(Z\) and is measured in ohms (\(\Omega\)). The impedance of a circuit is a complex number, often taking the form \(a + bj\), where \(a\) is the real part while \(b\) is the imaginary part, usually associated with reactance. Understanding impedance is key to manipulating how signals are processed in electronic devices.

Imaginary Unit

The imaginary unit, denoted by \(j\) in electrical engineering (or \(i\) in mathematics), is essential for calculating impedance and reactance. It is defined as \(j^2 = -1\). This concept helps engineers to work with alternating current circuits where voltage and current shift out of phase, and components like capacitors and inductors cause reactive effects.

The expression of circuit values using complex numbers (incorporating real and imaginary components) allows engineers to easily perform calculations that involve phase differences and reactance. Using \(j\), engineers can model and analyze circuits that include elements storing and releasing energy, like capacitors and inductors.

Complex Division

When working with complex numbers in circuits, complex division is often necessary to find quantities like impedance. To perform division with complex numbers, you often use the conjugate of the denominator.

For example, to divide \(E=430-330j\) by \(I=35-40j\), multiply numerator and denominator by the conjugate \(35 + 40j\):

• Multiply both parts: \[(430-330j)(35+40j) = 28250 + 5650j\]
• Compute: \[(35-40j)(35+40j) = 2825\]

This process helps in simplifying complex expressions into more manageable forms, making calculations practical.

Electrical Current

Electrical current, represented by \(I\), is the flow of electric charge through a conductor. It is a fundamental aspect of electrical circuits and is measured in amperes (A). In the equation \(E = I \cdot Z\), it illustrates how current is affected by circuit impedance and voltage.

The current in a circuit might be expressed as a complex number, such as \(35 - 40j\). Here, the real part represents the steady-state, or "in-phase" component, and the imaginary part represents the reactive "out-of-phase" component. This view allows engineers to analyze and design alternating current (AC) circuits effectively.

Voltage

Voltage, or electromotive force (\(E\)), is the potential difference that drives current through a circuit. It is measured in volts (V). In the equation \(E = I \cdot Z\), voltage reflects how much energy per charge is available to move through the impedance of the circuit.

Complex voltage, such as \(430-330j\), represents both the magnitude and phase angle that are crucial for AC analysis.

• The real part shows the direct component.
• The imaginary part is associated with phase shift due to reactance.

Understanding voltage in its complex form enables effective manipulation and control of AC circuits.