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 elements. That​ is, Upper Z equals StartFraction Upper V Over Upper I EndFraction
. If the voltage across a circuit element is 20 plus 10 i volts and the current through the element is 4 minus 2 i ​amperes, determine the impedance.
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Part 1
The impedance is Zequals

enter your response here.
​(Type your answer in the form aplusbi​.)

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?

---

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 \( a + bi \).

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 elements. That​ is, Z = V / I. If the voltage across a circuit element is 20 + 10i volts and the current through the element is 4 - 2i ​amperes, determine the impedance. (Type your answer in the form a + bi.)

Learn how to calculate the impedance Z in an AC circuit using complex numbers for voltage and current. This example uses the formula Z = V / I and covers multiplying by complex conjugates to express impedance in standard form a + bi. Suitable for students in Grades 10-12.

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