Complex Numbers in AC Circuit

In AC circuits, the properties of current and voltage are characterized by their changes over time. Knowledge of complex numbers is extremely important to understand the effects of these changes on electrical circuits precisely. This article explains the specific methods of calculating complex numbers and provides a detailed visual explanation. Doing so deepens the understanding of AC circuit characteristics, laying the foundation for more efficient and precise circuit design.

What is a Complex number?

AC circuits refer to circuits where the current and voltage change over time. A basic understanding of complex numbers related to this is crucial for accurately capturing the characteristics of AC circuits. This article comprehensively explains the introduction and calculation of complex numbers, their representation methods, and basic concepts.

Definition of Complex Numbers

Complex numbers are generally a combination of real numbers and imaginary numbers. The typical formula for a complex number is as follows:

\( Z = A + jB \)

In this formula, \(A\) is the real part, \(B\) is the imaginary part, and \(j\) is the unit of imaginary numbers.
It is also common to convert complex numbers into polar coordinate form, which can be expressed with the following equation:

\( Z = re^{j\theta} \)

Algebraic Operations on Complex Numbers

Complex numbers consist of real and imaginary parts and are represented on the complex plane. Understanding the operations of addition, subtraction, multiplication, and division of complex numbers enables us to understand the properties and applications of AC circuits. Here, we explain these operations using the complex numbers \(Z_1\) and \(Z_2\):

\( Z_1 = A_1 + jB_1 \)
\( Z_2 = A_2 + jB_2 \)

1. Addition of Two Complex Numbers:
   
   Complex number addition and subtraction involve calculating the real and imaginary parts separately. The sum of \(Z_1\) and \(Z_2\) is calculated by adding the real and imaginary components of each:

   \( Z_1 + Z_2 = (A_1 + A_2) + j(B_1 + B_2) \)

2. Subtraction of Two Complex Numbers:
   In subtraction, each real and imaginary part is subtracted:
   \( Z_1-Z_2=\left(A_1-A_2\right)+j\left(B_1-B_2\right) \)
3. Multiplication of Two Complex Numbers:
   Complex number multiplication involves multiplying the real components together, the imaginary components together, and using the property that the square of the imaginary unit is negative:
   \( Z_1 \cdot Z_2 = (A_1A_2 – B_1B_2) + j(A_1B_2 + A_2B_1) \)
4. Division of Two Complex Numbers:
   Complex number division is performed by multiplying the numerator and the denominator by the conjugate of the denominator. The quotient of \(Z_1\) divided by \(Z_2\) is expressed as:
   \(\displaystyle \frac{Z_1}{Z_2} = \displaystyle \frac{A_1 A_2 + B_1 B_2}{A_2^2 + B_2^2} + j \displaystyle \frac{A_2 B_1 – A_1 B_2}{A_2^2 + B_2^2}\)

Complex Number Representation of Impedance

The impedance of a circuit is represented as a complex number. Resistances, inductances, and capacitances are each expressed as complex numbers and operations involving these impedances utilize complex number arithmetic.

As an application in the analysis of alternating current (AC) circuits, calculations are performed using complex number operations to optimize the circuit impedance (impedance matching), thereby maximizing energy transmission. This enhancement in signal transmission and power delivery between circuits improves efficiency.

Moreover, complex number analysis provides tools like the Nyquist plot, which visualises a system’s stability and frequency response. This plot represents the transfer function on a complex plane, aiding in the understanding of the system’s behavior.

Furthermore, complex number analysis offers methods to optimize circuit bandwidth and signal-to-noise ratio, following Bode’s law, thereby expecting an improvement in circuit performance.

Complex Numbers in Math

Understanding complex numbers mathematically aids in the analysis and design of AC circuits. Below, we discuss mathematical aspects such as Euler’s formula, trigonometric functions, differentiation, integration, and absolute values.

Euler’s Formula

Euler’s formula plays a central role in representing complex numbers. This formula, linking exponential and trigonometric functions, concisely represents complex numbers in both polar and Cartesian coordinates.

1. Definition of the Complex Exponential Function
   The complex exponential function is defined using the base of natural logarithms, \(e\) as follows:

   \( e^{j\theta} = \cos\theta + j\sin\theta \)

   Here, \(j\) is the imaginary unit.

2. Derivation of Euler’s Formula
   Let a complex number \(Z\) be \( Z = a + jb \) Using the complex exponential function to express it, we get:

   \( e^{z} = e^{a + jb} = e^{a} e^{jb} \)

   Substituting the definition of the complex exponential function into this expression, we have:

   \(e^z = e^a (\cos b + j \sin b)\)

   Simplifying this, we derive Euler’s formula:

   \( e^{j\theta} = \cos\theta + j\sin\theta \)

This formula is essential in electrical engineering, particularly in simplifying the calculations involving alternating currents and voltages.

Applications of Euler’s Formula

When applied to the polar coordinate form of complex numbers, Euler’s formula provides a method for expressing complex numbers in polar coordinates. This form particularly utilizes the concepts of trigonometric functions and rotation, representing a complex number \(Z\) as \({re}^{j\theta}\) and consists of the following elements:

1. Polar Coordinate Form of Complex Numbers
   The polar coordinate form of a complex number \(Z\) is expressed as:

   \( Z = re^{j\theta} \)

   Here, r represents the complex number’s magnitude (absolute value), and \theta represents its phase angle. This derivation uses Euler’s formula.

2. Derivation Using Euler’s Formula
   By using Euler’s formula \(e^{j\theta}=cos\theta+isin\theta\), the polar coordinate form is derived as:

   \( Z = re^{j\theta} = r(\cos\theta + j\sin\theta) \)

   This form explicitly displays the properties of trigonometric functions, decomposing complex numbers into angles and magnitudes on the complex plane.

3. Significance of the Polar Coordinate Form
   • A) Magnitude \(r\):
     Represents the distance from the origin of the complex number, indicating its absolute value.
   • B) Phase Angle \(\theta\):
     The angle formed counterclockwise by the complex number from the x-axis, continuously represented in radians.
4. Graphical Representation
   The complex number \(Z={re}^{j\theta}\) points to a location on the complex plane that is \(r\) units from the origin and rotated by an angle \(θ\) from the x-axis. This concept is derived from the circular motion associated with trigonometric functions.
5. Outstanding Features of Euler’s Formula
   Euler’s formula links trigonometric functions with exponential functions, simplifying operations and analysis of complex numbers. The polar coordinate form is particularly convenient for computations like multiplication and division of complex numbers, represented simply as the addition and subtraction of angles.
6. Multiplication and Division of Complex Numbers Using Polar Coordinates
   Using polar coordinates simplifies the multiplication and division of complex numbers. When complex numbers are represented in polar form, their multiplication and division become more straightforward than in Cartesian form. This is because the operations involve the complex numbers’ moduli and angles (arguments), leading to more straightforward calculations.

   Multiplication・・
   \( Z_1 \cdot Z_2 = r_1 e^{j\theta_1} \cdot r_2 e^{j\theta_2} = r_1 r_2 e^{j(\theta_1 + \theta_2)} \)

   Division・・
   \( \displaystyle \frac{Z_1}{Z_2} = \displaystyle \frac{r_1 e^{j\theta_1}}{r_2 e^{j\theta_2}} = \displaystyle \frac{r_1}{r_2} e^{j(\theta_1 – \theta_2)} \)

Trigonometric Functions

Trigonometric functions are frequently used to analyze AC circuits and sine waves. In particular, sine and cosine functions are commonly featured in the expression of sine waves. By understanding the basic properties and characteristics of the graphs of trigonometric functions, you can grasp the characteristics of waveforms in AC circuits.
Sine Function:

\(y=sin\theta\)

• Periodicity: The sine function is periodic with a period of 2\(\pi\) or 360 degrees.
• Amplitude: The amplitude is 1, oscillating between -1 and 1.
• Symmetry: It is an odd function, with \(sin\left(-\theta\right)=-sin\left(\theta x\right)\).
• Maximum and Minimum Values: The maximum value is 1, and the minimum is -1.
• Phase Shift: \( sin\left(-\theta\right)=-sin\left(\theta x\right), \), a phase shift of \(\pi\) represents a phase inversion.

1. Cosine Function:

   \(y=cos\theta\)

   

   • Periodicity: The cosine function is also periodic with a 2\(\pi\) or 360 degrees period.
   • Amplitude: The amplitude is 1, oscillating between -1 and 1.
   • Symmetry: It is an even function, with \( cos(-\theta x)=cos(\theta x).\).
   • Maximum and Minimum Values: The maximum value is 1, and the minimum is -1.
   • Phase Shift: \(cos(\theta x+\pi)=-cos(\theta x)\), a phase shift of \(\pi\) represents a phase inversion.
2. Tangent Function

   \(y = \tan \theta = \displaystyle \frac{\sin \theta}{\cos \theta}\)

   

   • ・Periodicity: The tangent function is periodic with a period of \(\pi\) or 180 degrees.
   • ・Points of Divergence: The tangent function diverges at points where the cosine function is zero.

Differentiation and Integration

Differentiation and integration are mathematical techniques essential for capturing changes in current and voltage. It is crucial to consider changes over time in the analysis of AC circuits. Understand the concepts of differentiation and integration to grasp the temporal characteristics of signals in AC circuits.

1. Definition of Differentiation
   Differentiation is an operation that represents the rate of change of a function, showing the instantaneous change at a point. When you differentiate the function \(f(x)\) concerning \(x\), you obtain the derivative \(f'(x)\). The derivative represents the slope of the original function and is defined as follows:

   \(f'(x) = \displaystyle \frac{\lim}{t \to 0} \displaystyle \frac{f(x+t) – f(x)}{t}\)

2. Definition of Integration

   Integration is an operation for calculating the area or accumulation of a function, achieved by summing up infinitesimal intervals. Integrating the function \(f(x)\) concerning \(x\) yields either an indefinite integral \(\int f(x)\ dx\) dx or a definite integral \( \int_a^b f(x) \, dx \). The indefinite integral finds a primitive function, and the definite integral represents the area over the interval \([a,b]\):

   \(\int f(x) \, dx = F(x) + C\)

   where \(F(x)\) is a primitive function of \(f(x)\), and \(C\) is the constant of integration.

3. Temporal Characteristics of Signals in AC Circuits

   Representation of AC Signals:
   In AC circuits, signals that change over time are commonly used. AC signals are typically represented by sinusoidal waves, as follows:

   \((t) = V_{\text{max}} sin(\omega t + \theta)\)

   \(V_{\text{max}}\) is the amplitude, \(t\) is the angular frequency, \(t\) is time, and \(θ\) is the initial phase.

   ・Differentiation of the Signal
    Differentiating the AC signal \(V_t\) provides the derivative:

   \(\displaystyle \frac{dV(t)}{dt} = V_{\text{max}} \omega \cos(\omega t + \theta)\)

   This represents the signal’s instantaneous rate of change, where the amplitude is multiplied by the angular frequency.

   ・Integration of the Signal
    Integrating the AC signal \(V(t)\) results in:

   \(\int V(t) \, dt = -\displaystyle \frac{V_{\text{max}}}{\omega} \cos(\omega t + \theta) + C\)

   Here, the negative sign is applied to the cosine wave integral divided by the angular frequency. The integration constant C is maintained throughout the process.

   ・Integration of the Signal

   • A) Differentiation: Signal differentiation indicates the rate of change and instantaneous motion and is used to understand circuits’ responses and characteristics.
   • B) Integration: Signal integration represents the accumulation or area of the signal, relating to the storage of charge and energy.

Absolute Value

The absolute value of a complex number represents its magnitude, which indicates the amplitude or strength of a signal in AC circuits. We will study the methods for calculating absolute values and their applications to precisely understand the behavior of complex numbers. This understanding allows us to predict the behavior of circuits mathematically.

1. Form of Complex Numbers

A complex number \(z\) is generally expressed in the form \(z=a+bj\)
Here, \(a\) represents the real part, and \(b\) the imaginary part. We will explore how to derive the absolute value from this form.

2. Calculation of Absolute Value

The following equation determines the absolute value (modulus) of a complex number \(z=a+bj\)

\(|z| = \sqrt{a^2 + b^2}\)

This represents the distance from the origin to the point on the complex plane, applying the Pythagorean theorem.

3. Meaning of Complex Numbers on the Complex Plane

Plotting the complex number \(z=a+bj\) on the complex plane, it appears as the point (\(a,b\)) in a rectangular coordinate system. The distance between this point and the origin represents the absolute value, indicating amplitude or signal strength.

4. Applications to Amplitude and Signal Strength

a. Definition of Amplitude
In AC circuits, complex numbers are usually converted into polar form, represented by amplitude and phase angle. Here, \(r\) denotes the amplitude, which coincides with the absolute value.

\(Z = re^{j\theta}\)
\(r=|Z|\)

b. Concept of Signal Strength
Amplitude indicates signal strength, meaning a larger amplitude corresponds to a stronger signal, and a smaller one to a weaker signal. This is especially true in AC circuits, as the amplitude of voltage or current represents the amount of energy transferred.

5. Simple Example

For instance, the absolute value of the complex number \(z=3+4j\) is calculated as follows:

\(|3 + 4j| = \sqrt{3^2 + 4^2} = 5\)

This represents the distance from the origin to the point (3,4). Amplitude and signal strength can be similarly calculated.

Sine Waves and AC Circuits

In AC circuits, sine waves are commonly used. The AC voltage is expressed in the following form:

\(V(t) = V_{\text{max}} \sin(\omega t + \theta)\)

where \(V_{\text{max}}\) is the amplitude, \(ω\) is the angular frequency, \(t\) is time, and \(θ\) is the initial phase.

• Amplitude: The maximum height of the waveform.
• The amplitude of the voltage represents the amount of power transmitted.
• Period: The time it takes to complete one cycle.
• Angular frequency: Represented as \(\omega=2\pi f\), where \(f\) is the frequency.
• Initial phase: The phase angle at time zero.

Phase Difference and Phase Shift

Phase difference and phase shift are crucial concepts for describing the temporal relationship of waveforms in AC circuits. The following section provides a detailed explanation of these concepts.

Phase

Phase is an indicator of how far the waveform of voltage or current has progressed over time. Phase is usually expressed in radians or degrees and represented as a constant angle in periodic waveforms. The phase of a sine wave starts at a position of 0 degrees or 0 radians and increases as time progresses.

In AC circuits, the phase of voltage and current is represented by an angle, typically expressed as follows:

\(V(t) = V_{\text{max}} \sin(\omega t + \theta)\)

Here, \(V_(t)\) is the voltage at time \(t\) , \(V_{\text{max}}\) is the amplitude (maximum value), \(\omega\) is the angular frequency (multiplied by 2\(\pi\), and \(\theta\) is the phase (phase angle).

Phase Difference

The phase difference refers to the difference in phase between two waveforms. It is typically expressed as an angle per cycle. For instance, if two waveforms start at the same position, the phase difference is 0 degrees or 0 radians. If the phase difference is 90 degrees, one waveform is ahead by a quarter of a cycle ahead of the other.

Phase Shift

Phase shift refers to the phenomenon where the waveform shifts in time. This can occur due to components such as resistance (\(R\)), inductance (\(L\)), and capacitance (\(C\)) in an AC circuit or during signal processing. When a phase shift occurs, the peaks and troughs of the waveform shift in time, indicating how much the signal in the circuit is delayed or advanced.

There are two ways to express phase shift mathematically: in the time domain and in complex number form.

\(a\). Representation in the time domain

\(V_1(t) = V_{\text{max}} \sin(\omega t + \theta_1)\)
\(V_2(t) = V_{\text{max}} \sin(\omega t + \theta_2)\)

Here, \(v_1(t)\) and \(v_2(t)\) are the two waveforms, and \(\theta1\) and \(\theta2\) are their respective phases. The phase shift is represented by \(\theta_2-\theta_1\).

\(b\). Complex Number Form

The phase shift can be represented as the product of complex numbers in complex number form. Given two complex numbers \(Z1\) and \(Z2\) with phases \(\theta_1\) and \(\theta_2\), respectively:

\(Z_1 = V_{\text{max}} e^{j \theta_1}\)
\(Z_2 = V_{\text{max}} e^{j \theta_2}\)

The phase shift can be obtained by multiplying these complex numbers:

\(Z_{\text{total}} = Z_1 \cdot Z_2 = V_{\text{max}}^2 e^{j(\theta_1 + \theta_2)}\)

where \(j\) is the imaginary unit.

Positive Phase Shift

A positive phase shift indicates the waveform is delayed on the time axis. This is usually considered as the waveform shifting to the right. In an AC circuit with only inductance (\(L\)), the AC \(I_L\) flowing through the inductance (\(L\)) leads the voltage \(V_L\) across the inductance by \( \displaystyle \frac{\pi}{2}\)(90 degrees).

Negative Phase Shift

A negative phase shift indicates the waveform is advanced on the time axis. This is usually considered as the waveform shifting to the left. In an AC circuit with only capacitance (\(C\)), the \(AC I_C\) flowing through the capacitance (\(C\)) lags behind the voltage \(V_C\) across the capacitance by \( \displaystyle \frac{\pi}{2}\)(90 degrees).

Complex Number Analysis in AC Circuits

In AC circuits, impedance, coils, capacitance, and resistors are vital components. Particularly in RLC circuits, which combine resistance (\(R\)), inductance (\(L\)), and capacitance (\(C\)), complex number analysis using complex impedance helps understand the behavior at specific frequencies.

Complex Number Analysis in Series AC Circuits

A series circuit is one where the resistor \(R\), inductance \(L\), and capacitance \(C\) are connected in series. When analyzing a series circuit using complex impedance, each element’s complex impedance is calculated, and the total impedance is determined.

1. Complex Impedance of Resistor in Series AC Circuits

The complex impedance \(Z_R\) of a resistor is represented by the real part, which is the value of the resistance \(R\), and the imaginary part is zero.

\(Z_R=R+(j×0)=R\)

2. Complex Impedance of Inductance in Series AC Circuits

The complex impedance \(Z_L\) of an inductance is represented with an imaginary component expressed by \(j\).

\(Z_L=j\omega L\)

Here, \(j\) is the imaginary unit, \( \omega\) is the angular frequency, and \(L\) is the inductance.

3. Complex Impedance of Capacitance in Series AC Circuits

The complex impedance \(Z_C\) of a capacitance also has an imaginary component expressed by \(j\)

\(Z_C = \displaystyle \frac{1}{j \omega C}\)

Here, \(j\) is the imaginary unit, \( \omega\) is the angular frequency, and \(C\) is the capacitance.

4. Calculation of Total Impedance in Series AC Circuits

In a series circuit, the total impedance is the sum of the complex impedances of each element.

\(Z_{\mathrm{total}}=Z_R+Z_L+Z_C=R+j\omega\ L-\displaystyle \frac{j}{\omega C}\)

The magnitude of the total impedance is given by:

\(|Z_{\text{total}}| = \sqrt{R^2 + \left(\omega L – \displaystyle \frac{1}{\omega C}\right)^2}\)

5. Calculation of Steady-State Current in Series AC Circuits

Using the total impedance, the steady-state current I can be determined using Ohm’s law.

\(I = \displaystyle \frac{V}{Z_{\text{total}}}\)

Here, V is the voltage applied to the circuit.

6. Calculation of Voltages across Each Element in Series AC Circuits

he voltages across each element are calculated using Ohm’s law:

\(V_R=I×Z_R\)
\(V_L=I×Z_L\)
\(V_C=I×Z_C\)

7. Calculation of AC Power for Each Element in Series AC Circuits

The power for each element is calculated as the product of the voltage drop and the current.

\(P_R=V_R×I\)
\(P_L=V_L×I\)
\(P_C=V_C×I\)

Complex Number Analysis in Parallel AC Circuits

In a parallel circuit, the resistor R, inductance L, and capacitance C are connected in parallel. The circuit’s behavior can be analyzed by calculating the complex impedance of each element and determining the total impedance.

1. Complex Impedance of Resistor in Parallel AC Circuits

The complex impedance \(Z_R\) of a resistor is represented by the real part, which is the value of the resistance \(R\), and the imaginary part is zero.

\(Z_R=R+(j×0)=R\)

2. Complex Impedance of Inductance in Parallel AC Circuits

The complex impedance \(Z_L\) of an inductance is represented with an imaginary component expressed by \(j\).

\(Z_L=j\omega L\)

Here, \(j\) is the imaginary unit, \(\omega\) is the angular frequency, and \(L\) is the inductance.

3. Complex Impedance of Capacitance in Parallel AC Circuits

The complex impedance \(Z_C\) of a capacitance also has an imaginary component expressed by \(j\).

\(Z_C = \displaystyle \frac{1}{j \omega C}\)

Here, \(j\) is the imaginary unit, \(\omega\) is the angular frequency, and \(C\) is the capacitance.

4. Calculation of Total Impedance in Parallel AC Circuits

In a parallel circuit, the total impedance is calculated by taking the reciprocal of the sum of the reciprocals of each element’s impedance.

\(\displaystyle \frac{1}{Z_{\text{total}}} = \displaystyle \frac{1}{Z_R} + \displaystyle \frac{1}{Z_L} + \displaystyle \frac{1}{Z_C} = \displaystyle \frac{1}{R} + \displaystyle \frac{1}{j \omega L} + j \omega C\)

\(Z_{\text{total}} = \displaystyle \frac{1}{\displaystyle \frac{1}{R} + j\omega L + j \omega C} = \displaystyle \frac{\omega^2 L^2 R}{R^2(1 – \omega^2 LC) + \omega^2 L^2} + j \displaystyle \frac{\omega LR^2(1 – \omega^2 LC)}{R^2(1 – \omega^2 LC) + \omega^2 L^2}\)

The magnitude of the total impedance is given by:

\(|Z_{\text{total}}| = \sqrt{\left(\displaystyle \frac{\omega^2 L^2 R}{R^2 (1 – \omega^2 LC) + \omega^2 L^2}\right)^2 + \left(\displaystyle \frac{\omega L R^2 (1 – \omega^2 LC)}{R^2 (1 – \omega^2 LC) + \omega^2 L^2}\right)^2}\)

\(|Z_{\text{total}}| = \displaystyle \frac{\omega R L}{\sqrt{R^2 (1 – \omega^2 L C)^2 + \omega^2 L^2}}\)

5. Calculation of Steady-State Current in Parallel AC Circuits

Using the total impedance, Ohm’s law determines the steady-state current I.

\(I=V･Z_total\)

Here, V is the voltage applied to the circuit.

6. Calculation of Voltages across Each Element in Parallel AC Circuits

The voltages across each element are calculated using Ohm’s law:

\(V_R=I×Z_R\)
\(V_L=I×Z_L\)
\(V_C=I×Z_C\)

7. Calculation of Power for Each Element in Parallel AC Circuits

The power for each element is calculated as the product of the voltage drop and the current.

\(P_R=V_R×I\)
\(P_L=V_L×I\)
\(P_C=V_C×I\)

Practical Applications

Importance of Practical Applications

1. Performance Improvement
   By practically applying complex number analysis, it is possible to enhance the performance of electronic circuits and systems. Efficient energy conversion and signal processing can improve the system’s response and reliability.
2. Design Efficiency
   Complex number analysis provides mathematical methods that improve the efficiency of the design phase. Designing based on accurate calculations reduces trial and error, thus shortening the design time.
3. Reliability Enhancement
   Practical applications allow for a theoretical and mathematical understanding of the operation of electrical circuits and systems. This can lead to improved system reliability and more efficient troubleshooting.
4. Multidisciplinary Use
   Complex number analysis is used not only in electrical engineering but also in control engineering, communication engineering, and acoustical engineering. Therefore, it can be applied to problem-solving across different fields.

Examples of Practical Applications

1. Impedance Matching
   Impedance matching is a technique that adjusts the impedance between the transmitting and receiving sides in signal and power transmission to achieve maximum energy transfer. This minimizes signal loss and improves system efficiency.
2. Filter Design
   In practical electronic circuits, filters are used to control the frequency characteristics of signals. Using complex number analysis, it is possible to optimize characteristics such as bandpass and attenuation, allowing for the extraction of desired signals.
3. Power Analysis
   In practical power circuits and power transmission systems, power, current, and voltage analysis is crucial. Complex number analysis enables detailed analysis of these parameters, achieving efficient energy conversion.
4. Signal Processing
   Complex number analysis is frequently used in signal processing in areas such as audio processing and communication engineering. It is useful for extracting specific frequency components and adjusting the phase and amplitude of signals.
5. Control Systems
   Complex number analysis is applied to evaluate system stability and response speed in control systems. It helps understand the behavior of control targets and controllers, enabling optimal control design.

Example: RC Filter Design

Finally, we will explain how to apply the knowledge gained so far by presenting specific examples.

An RC filter is a circuit that uses a resistor (\(R\)) and a capacitor (\(C\)) to allow certain frequency components to pass through. Here, we will design the behavior of an RC filter at a specific cutoff frequency using complex number analysis.
By analyzing the frequency characteristics of RC filters using complex numbers, we can predict how these filters will behave over a range of frequencies. For a low-pass filter, it passes low frequencies and blocks high frequencies, making it useful for applications where high-frequency noise needs to be removed. For a high-pass filter, it passes high frequencies and blocks low frequencies, making it suitable for applications where low-frequency noise or interference needs to be filtered out.

Understanding these characteristics is crucial for designing filters that meet specific requirements in various electronic and signal processing applications. Adjusting the cutoff frequency and component values a

Low-Pass Filter Design

A low-pass filter allows low-frequency signals to pass through while attenuating high-frequency signals.

1. Setting Specifications

Set the cutoff frequency \(f_c.\)
Determine the capacitance \(C\).

2. Complex Number Analysis

The impedance of the capacitor \(Z_C\) is given by:
\(Z_C = \displaystyle \frac{1}{j\omega C}\)
For a low-pass filter, the impedance of the resistor \(Z_R\) is \(R\) . The total impedance \(Z_{\text{total}}\) is:
\(Z_{\text{total}} = Z_R + Z_C = R + \displaystyle \frac{1}{j\omega C}\)

3. Calculation of Transfer Function

The transfer function \(H(\omega)\) is derived as a frequency-dependent form.
The transfer function is determined by the following equation.

\(H(\omega) = \displaystyle \frac{V_{\text{out}}}{V_{\text{in}}} = \displaystyle \frac{Z_C}{Z_{\text{total}}}\)

\(H_{LP}(\omega) = \displaystyle \frac{V_{\text{out}}}{V_{\text{in}}} = \displaystyle \frac{Z_C}{Z_{\text{total}}} = \displaystyle \frac{\displaystyle \frac{1}{j\omega C}}{R + \displaystyle \frac{1}{j\omega C}} = \displaystyle \frac{1}{1 + j\omega RC}\)

• ・At low frequencies \(\left(\omega\rightarrow0\right)\):
  The term \(j\omega RC\) becomes very small, approaching zero.
  Therefore, \(H_{LP}\left(\omega\right)\approx1\).
  This means the output signal is almost equal to the input signal, indicating that low frequencies pass through with little to no attenuation.
• ・At high frequencies \(\left(\omega\rightarrow\infty\right)\):
  The term \(j\omega RC\) becomes very large.
  Therefore,\(H_{LP}\left(\omega\right)\approx0.F\)
  This means the output signal is significantly attenuated, indicating that high frequencies are blocked.

4. Design of Cutoff Frequency

Analyze the transfer function at the cutoff frequency \(f_C\) .
Determine the frequency where \(∣H(\omega)∣\) becomes 0.707 (-3dB), and designate it as \(f_C\) .
Also, the cutoff frequency is determined as follows:
\(f_c = \displaystyle \frac{1}{2\pi RC}\)

High-Pass Filter Design

A high-pass filter allows high-frequency signals to pass through while attenuating low-frequency signals.
The transfer function \(H_{HP}\left(\omega\right)\) for a high-pass filter is given by:

\(H_{HP}(\omega) = \displaystyle \frac{V_{\text{out}}}{V_{\text{in}}} = \displaystyle \frac{Z_R}{Z_{\text{total}}} = \displaystyle \frac{R}{R + \displaystyle \frac{1}{j\omega C}} = \displaystyle \frac{j\omega RC}{j\omega RC + 1}\)

• ・At low frequencies \(\left(\omega\rightarrow0\right)\):
  The term \(j\omega RC\) becomes very small, approaching zero.
  Therefore, \(H_{LP}\left(\omega\right)\approx0\).
  This means the output signal is significantly attenuated, indicating that low frequencies are blocked.
• ・At high frequencies \(\left(\omega\rightarrow\infty\right)\):
  The term \(j\omega RC\) becomes very large.
  Therefore,\(H_{LP}\left(\omega\right)\approx1\).
  This means the output signal is almost equal to the input signal, indicating that high frequencies pass through with little to no attenuation.

Explanation

By using complex number analysis, the characteristics of an RC filter can be mathematically modeled. Designing the cutoff frequency is a crucial step to ensure that the filter’s frequency response has the desired properties. By appropriately adjusting the design variables, which are the cutoff frequency and capacitance, the performance of the filter can be customized.