Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Alternating Current The alternating current in amps in an electric inductor is \(I=\frac{E}{Z}\) where \(E\) is the voltage and \(Z=R+X_{L} i\) is the impedance. If \(E=8\left(\cos 20^{\circ}+i \sin 20^{\circ}\right), R=6,\) and \(X_{L}=3\) find the current. Give the answer in rectangular form.

Step-by-Step Solution

Answer

The current is approximately \( 0.8199 - 0.1364i \).

Step 1: Express Impedance in Rectangular Form

The impedance \( Z \) is given by \( Z = R + X_L i \). Substituting the values \( R = 6 \) and \( X_L = 3 \), we have: \[ Z = 6 + 3i \]

Step 2: Substitute Values into Current Equation

Now, substitute the voltage \( E = 8(\cos 20^{\circ} + i \sin 20^{\circ}) \) and the impedance \( Z = 6 + 3i \) into the current equation \( I = \frac{E}{Z} \): \[ I = \frac{8(\cos 20^{\circ} + i \sin 20^{\circ})}{6 + 3i} \]

Step 3: Simplify Voltage to Rectangular Form

First, calculate \( \cos 20^{\circ} \approx 0.9397 \) and \( \sin 20^{\circ} \approx 0.3420 \). Thus, the voltage \( E \) in rectangular form is:\[ E = 8(0.9397 + 0.3420i) = 7.5176 + 2.736i \]

Step 4: Compute the Current in Rectangular Form

To find \( I = \frac{E}{Z} \), multiply the numerator and the denominator by the complex conjugate of the denominator, \( 6 - 3i \): \[I = \frac{(7.5176 + 2.736i)(6 - 3i)}{(6 + 3i)(6 - 3i)}\]This simplifies the denominator to \( 6^2 + 3^2 = 36 + 9 = 45 \).

Step 5: Calculate Numerator by Expanding

Expand the numerator: \[(7.5176 + 2.736i)(6 - 3i) = (7.5176 \times 6) + (7.5176 \times (-3i)) + (2.736i \times 6) + (2.736i \times (-3i))\]Simplify each term:1. \( 7.5176 \times 6 = 45.1056 \)2. \( 7.5176 \times (-3i) = -22.5528i \)3. \( 2.736i \times 6 = 16.416i \)4. \( 2.736i \times (-3i)= -8.208 \) (since \( i^2 = -1 \))Combine the terms: \[ 45.1056 + (-8.208) + (-22.5528i) + 16.416i = 36.8976 - 6.1368i \]

Step 6: Final Division to Get Current Value

Divide the resulting complex number by the real number we simplified in Step 4:\[I = \frac{36.8976 - 6.1368i}{45}\] Thus, \[ I = 0.8199 - 0.1364i \]

Step 7: Conclusion

The current \( I \) in rectangular form is \( 0.8199 - 0.1364i \).

Key Concepts

Complex Numbers

Complex numbers are fundamental in understanding alternating current systems, especially when dealing with quantities like voltage and impedance. A complex number typically has two components: a real part and an imaginary part. The general form of a complex number is given by:

• Real part (\(a\)): This is a standard real number.
• Imaginary part (\(b\)): This is accompanied by \(i\), where \(i\) is the imaginary unit defined as \(i^2 = -1\).

Written together, a complex number appears as \(a + bi\).

In the context of AC circuits, these components are essential in representing sinusoidal quantities, enhancing the analysis of such systems. The beauty of complex numbers lies in their ability to simplify calculations that involve angles and magnitudes. This makes them perfect for electrical engineering applications.

To compute with complex numbers, operations like addition, subtraction, multiplication, and division are straightforward and involve simple arithmetic for real and imaginary parts respectively. This intricacy is crucial for handling AC circuit parameters.

Impedance

Impedance, denoted as \(Z\), is a critical concept in AC circuits. It extends the idea of resistance to accommodate both resistive and reactive elements in a circuit. Impedance is a complex quantity and can be expressed as:\[Z = R + X_L i\]
Here:

• \(R\) represents the resistance, a measure of how much a component opposes the flow of electrical current.
• \(X_L\) represents the inductive reactance, reflecting the behavior of inductors in the circuit with AC sources.

Impedance combines these two factors (resistance and reactance) to describe how an AC circuit will behave under different conditions.

In practical terms, impedance affects the amplitude and phase of the current relative to the voltage. This is crucial when designing circuits with specific performance requirements. By using the rectangular form of complex numbers, impedance can be easily manipulated to calculate current, voltage, or power in AC circuits.

Rectangular Form

Rectangular form is one format to express complex numbers and is particularly handy for computations involving addition and subtraction. This form is crucial when using complex numbers to represent electrical properties like impedance and current in AC circuits.

The rectangular form is represented as: \[a + bi\], where \(a\) is the real part, and \(b\) is the imaginary part.

Using rectangular form is beneficial in AC analysis as it allows for straightforward computational procedures, especially when evaluating expressions like impedance, which combines both resistive and reactive components. When solving for the current or voltage in a circuit, as in the original problem, expressing all quantities in rectangular form ensures clarity and consistency.

Moreover, converting between rectangular and polar forms is often necessary, depending on the requirements of the analysis or the preferences of the engineer. Polar form might be more intuitive for understanding phase shifts, whereas rectangular form excels in algebraic operations.