Question

In a sinusoidal steady-state AC circuit, the voltage across a passive element is given by Ohm's law: where \(\mathbf{V}\) is the voltage across the element, \(\mathbf{I}\) is the current through the element, and \(Z\) is the impedance of the clement. Note that all three of these values are complex, and that these complex numbers are usually specified in the form of a magnitude at a specific phase angle expressed in degrees. For example, the voltage might be \(\mathbf{V}=120 \angle 30^{\circ} \mathrm{V}\). Write a program that reads the voltage across an element and the impedance of the element, and calculates the resulting current flow. The input values should be given as magnitudes and angles expressed in degrees, and the resulting answer should be in the same form. Use the function to_complex from Exercise \(6.3\) to convert the numbers to rectangular for the actual computation of the current, and the function to polar from Exercise \(6.2\) to convert the answer into polar form for display (see Figure \(6.13\) ),

Short answer

To calculate the current in a sinusoidal steady-state AC circuit given the voltage across an element and its impedance: 1. Define two functions, `to_complex` and `to_polar`, to convert between polar and rectangular forms of complex numbers. 2. Read the input values for voltage and impedance as magnitudes and angles in degrees. 3. Convert the voltage and impedance to rectangular forms using the `to_complex` function. 4. Calculate the current by dividing the voltage by the impedance in rectangular form. 5. Convert the rectangular form of the current back to polar form using the `to_polar` function. 6. Display the current magnitude and angle in degrees.

Step-by-step solution

1. Define the to_complex function from Exercise 6.3

This function takes the magnitude and angle (in degrees) as its inputs and returns the complex number in rectangular form. ```python def to_complex(magnitude, angle_degrees): angle_radians = math.radians(angle_degrees) real = magnitude * math.cos(angle_radians) imaginary = magnitude * math.sin(angle_radians) return complex(real, imaginary) ```

2. Define the to_polar function from Exercise 6.2

This function takes a complex number as its input and returns its magnitude and angle in degrees as a tuple. ```python def to_polar(complex_number): magnitude = abs(complex_number) angle_degrees = math.degrees(cmath.phase(complex_number)) return magnitude, angle_degrees ```

3. Read Input

Prompt the user to input the voltage and impedance in the form of magnitude and angle (in degrees). ```python V_mag, V_angle = map(float, input("Enter the voltage magnitude and angle (in degrees): ").split()) Z_mag, Z_angle = map(float, input("Enter the impedance magnitude and angle (in degrees): ").split()) ```

4. Calculate Current

Convert the voltage and impedance into rectangular form using the to_complex function and then use Ohm's law formula to calculate the current. ```python V_rect = to_complex(V_mag, V_angle) Z_rect = to_complex(Z_mag, Z_angle) I_rect = V_rect / Z_rect ```

5. Display the Result

Convert the rectangular form of the current into polar form using the to_polar function and then display the result. ```python I_mag, I_angle = to_polar(I_rect) print(f"The current magnitude is: {I_mag:.2f}") print(f"The current angle (in degrees) is: {I_angle:.2f}") ``` Now let's put everything together and write the complete program. ```python import math import cmath def to_complex(magnitude, angle_degrees): angle_radians = math.radians(angle_degrees) real = magnitude * math.cos(angle_radians) imaginary = magnitude * math.sin(angle_radians) return complex(real, imaginary) def to_polar(complex_number): magnitude = abs(complex_number) angle_degrees = math.degrees(cmath.phase(complex_number)) return magnitude, angle_degrees V_mag, V_angle = map(float, input("Enter the voltage magnitude and angle (in degrees): ").split()) Z_mag, Z_angle = map(float, input("Enter the impedance magnitude and angle (in degrees): ").split()) V_rect = to_complex(V_mag, V_angle) Z_rect = to_complex(Z_mag, Z_angle) I_rect = V_rect / Z_rect I_mag, I_angle = to_polar(I_rect) print(f"The current magnitude is: {I_mag:.2f}") print(f"The current angle (in degrees) is: {I_angle:.2f}") ```

Key concepts

Ohm's Law in AC Circuits

When we talk about Ohm's Law in the context of alternating current (AC) circuits, it's important to note that the voltage, current, and impedance are not simple real numbers as in direct current (DC) circuits, but complex numbers. This is because AC parameters have both a magnitude and a phase. The Ohm's Law formula for AC circuits is typically written as \( \mathbf{V} = \mathbf{I} \times Z \), where \( \mathbf{V} \) represents the voltage, \( \mathbf{I} \) the current, and \( Z \) the complex impedance.

Understanding this law is essential for analyzing AC circuits and determining how voltage and current will change over time. Engineers use the real part of these complex numbers to represent the active power, which does actual work, and the imaginary part to represent reactive power, which stores and releases energy in the circuit.

Complex Impedance

Complex impedance \( Z \) in AC circuits is the total opposition a circuit presents to alternating current. Unlike resistance, which is only an aspect of Ohm's Law in DC, impedance accounts for both the resistance \( (R) \) and the reactance \( (X) \) which includes the capacitive and inductive properties of the circuit. This is often expressed in rectangular form as \( R + jX \) where \( j \) represents the imaginary unit.

Impedance takes into account the fact that components in AC circuits can delay the time at which the peak current and voltage occur due to the storage of energy in the electric and magnetic fields, respectively. Complex impedance therefore provides a comprehensive measure that’s essential for predicting and analyzing the circuit's behavior under varying frequencies.

Conversion between Polar and Rectangular Form

In circuit analysis, it's often necessary to convert complex numbers between polar form, which specify a magnitude and an angle, and rectangular form, which break the number down into real and imaginary parts. The polar form is useful for understanding the physical relationship between voltage, current, and impedance (for example, their phase difference), while the rectangular form is convenient for performing actual calculations, such as addition and division.

Mathematically, the conversion from polar to rectangular is done using cosine and sine for the real and imaginary parts, respectively, and from rectangular to polar using the Pythagorean theorem for magnitude and arctan function (or phasor calculations) to find the angle. These conversions are frequently used in engineering tasks and can be easily implemented in a programming environment like MATLAB.

MATLAB Programming for Engineers

MATLAB is a programming language and environment widely used by engineers for technical computing. It's especially well-suited for numerical computation and modeling, which are vital to electrical engineering tasks like AC circuit analysis. In MATLAB or similar computational tools, engineers can create scripts and functions to automate complex calculations, perform signal processing, and visualize data. This includes easily manipulating complex numbers and converting between different forms, as illustrated in the above exercise.

By learning MATLAB programming, engineers gain a powerful tool for designing, analyzing, and testing electrical systems. It offers functions and predefined commands for complex mathematics, making it much easier to implement and solve equations that contain complex numbers, such as those used in AC circuit analysis.