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 ZZ of a circuit element, given by the formula:

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

we substitute the provided values:

  • Voltage V=20+10iV = 20 + 10i volts
  • Current I=42iI = 4 - 2i amperes

The impedance ZZ is:

Z=20+10i42iZ = \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+2i4 + 2i:

Z=(20+10i)(4+2i)(42i)(4+2i)Z = \frac{(20 + 10i)(4 + 2i)}{(4 - 2i)(4 + 2i)}

Step 2: Expand the Numerator

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

(20+10i)(4+2i)=204+202i+10i4+10i2i(20 + 10i)(4 + 2i) = 20 \cdot 4 + 20 \cdot 2i + 10i \cdot 4 + 10i \cdot 2i =80+40i+40i+20i2= 80 + 40i + 40i + 20i^2

Since i2=1i^2 = -1, replace 20i220i^2 with 20-20:

=80+40i+40i20=60+80i= 80 + 40i + 40i - 20 = 60 + 80i

Step 3: Simplify the Denominator

The denominator is (42i)(4+2i)(4 - 2i)(4 + 2i), which is a difference of squares:

(42i)(4+2i)=42(2i)2=164i2(4 - 2i)(4 + 2i) = 4^2 - (2i)^2 = 16 - 4i^2

Since i2=1i^2 = -1, this becomes:

=16+4=20= 16 + 4 = 20

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

Now substitute back into the equation:

Z=60+80i20=3+4iZ = \frac{60 + 80i}{20} = 3 + 4i

Final Answer

The impedance ZZ is:

Z=3+4iZ = 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 ii 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+bia + bi.

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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