To determine the impedance $Z$ of a circuit element, given by the formula:

$Z = \frac{V}{I}$

we substitute the provided values:

• Voltage $V = 20 + 10i$ volts
• Current $I = 4 - 2i$ amperes

The impedance $Z$ is:

$Z = \frac{20 + 10i}{4 - 2i}$

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator

To simplify the division of complex numbers, multiply both the numerator and denominator by the conjugate of the denominator, $4 + 2i$:

$Z = \frac{(20 + 10i)(4 + 2i)}{(4 - 2i)(4 + 2i)}$

Step 2: Expand the Numerator

Expanding $(20 + 10i)(4 + 2i)$ using the distributive property:

$(20 + 10i)(4 + 2i) = 20 \cdot 4 + 20 \cdot 2i + 10i \cdot 4 + 10i \cdot 2i$ $= 80 + 40i + 40i + 20i^2$

Since $i^2 = -1$, replace $20i^2$ with $-20$:

$= 80 + 40i + 40i - 20 = 60 + 80i$

Step 3: Simplify the Denominator

The denominator is $(4 - 2i)(4 + 2i)$, which is a difference of squares:

$(4 - 2i)(4 + 2i) = 4^2 - (2i)^2 = 16 - 4i^2$

Since $i^2 = -1$, this becomes:

$= 16 + 4 = 20$

Step 4: Write the Impedance in the Form $a + bi$

Now substitute back into the equation:

$Z = \frac{60 + 80i}{20} = 3 + 4i$

Final Answer

The impedance $Z$ is:

$Z = 3 + 4i$

Would you like a further breakdown, or do you have any other questions?

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1. What are complex conjugates, and how are they useful in impedance calculations?
2. How would impedance calculation change with a different complex current or voltage?
3. How does impedance relate to resistance and reactance in AC circuits?
4. What role does the imaginary unit $i$ play in electrical engineering calculations?
5. Can you provide examples of physical systems where impedance is critical?

Tip: When dividing complex numbers, always multiply by the conjugate to simplify to standard form $a + bi$.