Problem 61 - Algebra 2 and Trigonometry

Question

Impedance is the resistance to the flow of current in an electric circuit measured in ohms. The impedance, \(Z,\) in a circuit is found by using the formula \(Z=\frac{V}{I}\) where \(V\) is the voltage (measured in volts) and \(I\) is the current (measured in amperes). Find the impedance when \(V=1.8-0.4 i\) volts and \(I=-0.3 i\) amperes.

Step-by-Step Solution

Answer

The impedance is \( \frac{4}{3} + 6i \) ohms.

Step 1: Write down the formula for impedance

The formula for impedance given is \( Z = \frac{V}{I} \), where \( V \) is the voltage in volts and \( I \) is the current in amperes.

Step 2: Substitute the given values into the impedance formula

Substitute \( V = 1.8 - 0.4i \) and \( I = -0.3i \) into the formula: \[ Z = \frac{1.8 - 0.4i}{-0.3i} \].

Step 3: Simplify the complex fraction

Multiply both the numerator and the denominator by the complex conjugate of the denominator, \(-0.3i\) in this case, which is \(0.3i\). This gives: \[ Z = \frac{(1.8 - 0.4i)(0.3i)}{(-0.3i)(0.3i)} \].

Step 4: Multiply in the numerator

Use distributive properties to expand: \((1.8)(0.3i) + (-0.4i)(0.3i) = 0.54i + 0.12 \). The result is: \(0.54i + 0.12\).

Step 5: Simplify the denominator

Simplify the denominator: \((-0.3i)(0.3i) = -0.09(i^2) = 0.09 \), because \(i^2 = -1\).

Step 6: Write the final simplified result

The complex number becomes \( Z = \frac{0.54i + 0.12}{0.09} \). Simplify each term: \( Z = \frac{0.12}{0.09} + \frac{0.54i}{0.09} = \frac{4}{3} + 6i\).

Step 7: Conclusion

Based on the calculated values, the impedance \( Z \) is \( \frac{4}{3} + 6i \) ohms.

Key Concepts

Ohms

Ohms is the unit of measurement for impedance or resistance in an electric circuit.
It indicates how much a circuit resists the flow of electrical current.
When you think about Ohms, imagine trying to push water through a hose.
The higher the resistance, the harder it is for water (or electricity) to move through. This is similar to the resistance that electricity faces in a circuit, except this is measured in ohms.

Ohms are a fundamental part of understanding how circuits work.
When you plug in an appliance, Ohms determine how efficiently electricity is converted to power.
Ohms tell you about the opposition, which is key to ensuring devices work
correctly and don't overload when connected.
For example:

• If a circuit has high ohms, it's hard for the electric current to flow.
• If a circuit has low ohms, the current flows easily.

Complex Numbers

Complex numbers are a combination of real numbers and imaginary numbers.
They take the form of a + bi, where a is the real part and b is the imaginary part.
In our context,
it's often used in electrical engineering to describe electrical quantities that have both magnitude and direction.

Complex numbers make it easier to handle calculations involving electrical circuits,
like impedance, as seen in the given problem.
The imaginary part is indicated with the unit 'i', representing
the square root of -1. This might seem strange, but
it's useful for describing wave-like behaviors of AC signals.

• Real Part: Deals with magnitudes (like voltage level)
• Imaginary Part: Deals with phase differences (how out of sync the wave is)

Electric Circuit

An electric circuit is a path made for electric current to flow.
It usually consists of components like resistors, capacitors, and inductors,
connected in some way to a power source.
Think of it as a pathway created by interconnected wires and components.
Easiest way to understand it? A circle.

Without a complete circle, electricity won't flow.
That's why circuits are closed loops where current can travel.
If there's a break or gap, the current won't reach its destination,
like turning off a switch in the middle of the path. Here's how different parts play a role:

• Power Source: Provides the energy.
• Conductor: Allows current to flow (usually wires).
• Load: Device or resistance that uses the electricity.

Voltage and Current

Voltage and current are key players in any electric circuit.
Voltage ( V ) is like the push that makes electric charges move,
measured in volts. It pushes the electrons through the circuit.
Current ( I ) is the flow of electric charge in the circuit and
is measured in amperes (amps).
Essentially, it's how much charge is being pushed.

In terms of an analogy, voltage is like water pressure in a pipe,
while current is the flow of the water itself.
Without sufficient voltage, current can't flow effectively.
Without current, you can't have electricity being used for work. The relationship in circuits:

• Voltage: Motivates electric charge.
• Current: Shows the movement of charge.
• Impedance: Dictates how they interact (from our problem).