Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. The current in a circuit with voltage \(E\), resistance \(R\), capacitive reactance \(X_{c^{\prime}}\) and inductive resistance \(X_{L}\) is $$I=\frac{E}{R+\left(X_{L}-X_{c}\right) i}$$ Find \(I\) if \(E=12\left(\cos 25^{\circ}+i \sin 25^{\circ}\right), R=3, X_{L}=4\) and \(X_{c}=6 .\) Give the answer in rectangular form.

Step-by-Step Solution

Answer

The current \(I\) is approximately \(2.097 + 2.844i\).

Step 1: Simplify the Denominator

First, calculate the expression in the denominator of the given formula for the current \(I\). This expression is \(R + (X_L - X_c) i\), where \(R = 3\), \(X_L = 4\), and \(X_c = 6\). Substitute these values into the expression:\[R + (X_L - X_c) i = 3 + (4 - 6)i = 3 - 2i.\]

Step 2: Convert Voltage to Rectangular Form

The voltage \(E\) is given in polar form as \(12(\cos 25^{\circ} + i \sin 25^{\circ})\). Convert it to rectangular form:\[E = 12(\cos 25^{\circ} + i \sin 25^{\circ}) = 12(0.9063 + i \times 0.4226) = 10.8756 + i \cdot 5.0712\]

Step 3: Calculate the Magnitude of Denominator for Rationalization

To simplify the division by a complex number, calculate the magnitude \(|3 - 2i| = \sqrt{3^2 + (-2)^2} = \sqrt{13}\).

Step 4: Rationalize the Denominator

Multiply numerator and denominator by the conjugate of the denominator to simplify the fraction. The conjugate of \(3 - 2i\) is \(3 + 2i\):\[I = \frac{(10.8756 + 5.0712i)(3 + 2i)}{(3 - 2i)(3 + 2i)} = \frac{37.4028 + 21.7506i + 15.2136i + 10.1424i^2}{9 + 4}\]Since \(i^2 = -1\), the expression simplifies to:\[= \frac{37.4028 + 36.9642i - 10.1424}{13} = \frac{27.2604 + 36.9642i}{13}\]

Step 5: Divide Each Component by 13

Divide the real and imaginary components of the numerator by 13 to obtain the current \(I\) in rectangular form:\[I = \frac{27.2604}{13} + i \frac{36.9642}{13} \\approx 2.097 + i \cdot 2.844\]

Step 6: Write Final Answer in Rectangular Form

Thus, the current in the circuit, in rectangular form, is approximately:\[ I \approx 2.097 + 2.844i \]

Key Concepts

Rectangular Form

Complex numbers in rectangular form are expressed as the sum of a real part and an imaginary part. The general form is given as \(a + bi\), where \(a\) is the real part and \(bi\) represents the imaginary part. In this format, it is straightforward to visualize points or vectors on a two-dimensional plane known as the complex plane. Here:

• The horizontal axis (x-axis) represents the real component.
• The vertical axis (y-axis) represents the imaginary component.

For example, the complex number \(3 - 2i\) consists of a real part \(3\) and an imaginary part \(-2\). This translates to a point on the complex plane located at coordinates (3, -2).

The rectangular form is particularly useful for performing arithmetic operations such as addition and subtraction directly, as it's similar to operating with vector components. In the given problem, the current \(I\) is calculated and eventually expressed in rectangular form to provide clarity for further calculations or interpretations.

Polar Form

Polar form is an alternative way to express complex numbers. Instead of using a real and imaginary component, it uses a magnitude and an angle. This compact representation is handy for multiplication and division operations.

A complex number in polar form is shown as \(r(\cos\theta + i\sin\theta)\) or sometimes as \(r\text{cis}\theta\). Here, \(r\) stands for the modulus or magnitude, and \(\theta\) represents the angle or argument.

• The magnitude \(r\) is the distance from the origin to the point \((a, b)\) on the complex plane.
• The angle \(\theta\) measures the direction of the vector representing the complex number from the positive x-axis.

To convert from polar to rectangular form, as shown in the problem, use the formulas: \[a = r \cdot \cos\theta\] \[b = r \cdot \sin\theta\] where \(a + bi\) is the rectangular form. This approach was used for voltage \(E\), transforming it from the polar form given to rectangular form, making the division operation and subsequent calculations more manageable.

Rationalization

Rationalization is a process applied to simplify the division of complex numbers. When dividing two complex numbers, it's desirable to have a real number in the denominator. To achieve this, multiply both the numerator and the denominator by the conjugate of the denominator.

For a complex denominator \(a + bi\), its conjugate is \(a - bi\). Multiplying by the conjugate eliminates the imaginary part in the denominator, as:\[(a+bi)(a-bi) = a^2 - (bi)^2 = a^2 + b^2 \]This results in a real number since \(i^2 = -1\).

• In the example, the expression \(3 - 2i\) was rationalized by multiplying by its conjugate \(3 + 2i\).

This step is crucial for simplifying the fraction of complex numbers to a form where components can be separately addressed and is a standard arithmetic adjustment in solving complex-number-based problems.