Math Problem Statement
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. Question content area bottom Part 1 The impedance is Zequals
enter your response here. (Type your answer in the form aplusbi.)
Solution
To determine the impedance of a circuit element, given by the formula:
we substitute the provided values:
- Voltage volts
- Current amperes
The impedance is:
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, :
Step 2: Expand the Numerator
Expanding using the distributive property:
Since , replace with :
Step 3: Simplify the Denominator
The denominator is , which is a difference of squares:
Since , this becomes:
Step 4: Write the Impedance in the Form
Now substitute back into the equation:
Final Answer
The impedance is:
Would you like a further breakdown, or do you have any other questions?
- What are complex conjugates, and how are they useful in impedance calculations?
- How would impedance calculation change with a different complex current or voltage?
- How does impedance relate to resistance and reactance in AC circuits?
- What role does the imaginary unit play in electrical engineering calculations?
- 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 .
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Algebra
Impedance in Electrical Circuits
Formulas
Impedance formula Z = V / I
Multiplication by complex conjugate
Theorems
Complex Conjugate Theorem
Suitable Grade Level
Grades 10-12
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