Sinusoids, Harmonics, and Electrical Power Systems Flashcards

Fundamental Mathematical Conventions and Rounding

• Rounding to Decimal Points: Professional engineering and mathematical calculations require standardized rounding procedures. When rounding to 2 decimal points:

  • A value such as $0.028$ becomes $0.03$.

  • Any value falling within the numerical range of $0.025$ to $0.035$ is rounded to $0.03$.

  • Values below this threshold, such as $0.024$, round down to $0.02$.

• Numerical Formatting: Some computational input systems require numbers to be entered without thousands-separators. For instance, the value ten thousand should be written as $10000$ rather than $10,000$.

Introduction to Sinusoids and Signal Applications

• Sinusoids in Engineering: Sinusoidal signals are the foundational building blocks for many modern technologies and power distribution systems.

• Microphone: This device serves as a transducer that converts physical sound waves (acoustic energy) into time-varying electrical sinusoidal signals.

• Speaker: Functioning as the inverse of a microphone, a speaker converts electrical sinusoids back into physical sound energy.

• Motor: Motors utilize sinusoidal currents to generate the electromagnetic fields necessary to produce mechanical rotation.

• Generator: Generators perform electromechanical conversion, turning mechanical energy (such as rotation) into sinusoidal electrical energy.

• Electricity Consumption: In residential environments, power is distributed and consumed in the form of Alternating Current ($AC$) sinusoids, where voltage and current vary periodically over time.

• Energy of Time-Varying Signals: Engineers must track how power and energy fluctuate over specific durations rather than looking at a static value, which is essential for billing and system stability.

Mathematical Representation of Sinusoidal Signals

• General Expression: The standard mathematical form for a time-varying sinusoidal signal is represented as:

  • $v(t) = V_p \sin(\omega t + \phi)$

• Key Parameters:

  • Amplitude ($A$ or $V_p$): This is also known as the Peak Voltage. It represents the maximum vertical displacement from the zero-reference line to the crest of the wave.

  • Angular Frequency ($\omega$): This is measured in units of $\text{rad/s}$. It is mathematically related to frequency by the formula: $\omega = 2\pi f$

  • Phase ($\phi$): This represents the initial angle of the sinusoid at the starting time $t = 0$, indicating how much the wave is shifted horizontally.

  • Frequency ($f$): Measured in Hertz ($\text{Hz}$), it is the number of cycles per second. It is defined as the reciprocal of the period: $f = \frac{1}{T}$

  • Time Period ($T$): The duration of time required for the signal to complete one full cycle.

• Peak-to-Peak Voltage ($V_{pp}$): This represents the total vertical distance from the negative peak to the positive peak. It is calculated as: $V_{pp} = 2 \times V_p$

• Periodic Nature: A signal is considered periodic if it repeats its values after a regular interval $T$. This is defined by the mathematical equality: $v(t + kT) = v(t)$, where $k$ is an integer.

Power and Energy in Time-Varying Signals

• Instantaneous Power ($p(t)$): This is the power at any specific moment in time, defined as the product of instantaneous voltage and instantaneous current:

  • $p(t) = v(t) \times i(t)$

• Purely Resistive Load ($R$): In a circuit with resistance $R$, the power can be expressed as:

  • $p(t) = \frac{v(t)^2}{R}$

  • $p(t) = i(t)^2 R$

• Continuous AC Signals: For a signal where $v(t) = V_p \sin(\omega t)$ and $i(t) = I_p \sin(\omega t)$, and where current is derived from voltage via Ohm's Law ($I_p = \frac{V_p}{R}$), the instantaneous power is:

  • $p(t) = \frac{V_p^2}{R} \sin^2(\omega t)$

• Energy ($E$): Energy is calculated as the integral of power over a duration of time:

  • $E = \int p(t)\,dt$

• Energy in One Cycle ($E$): To find energy over one period ($T$):

  • $E = \int_0^T \frac{V_p^2}{R} \sin^2(w t)\,dt$

  • Using the trigonometric identity $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$, the integral becomes:

  • $E = \frac{V_p^2}{R} \int_0^T \frac{1 - \cos(2\omega t)}{2}\,dt$

  • Completing the integration results in: $E = \frac{V_p^2 T}{2R}$

• Average Power ($P_{avg}$): The average power over one cycle is the total energy divided by the period:

  • $P_{avg} = \frac{E}{T} = \frac{V_p^2}{2R}$

Root Mean Square (RMS) Values

• DC Equivalent: The Root Mean Square value represents the equivalent $DC$ (Direct Current) voltage that would produce the same amount of heat or power dissipation in a resistor as the subject $AC$ signal.

  • Setting $DC$ power equal to average $AC$ power:

  • $P_{dc} = \frac{V_{dc}^2}{R}$

  • $P_{dc} = P_{avg}$

  • $V_{dc}^2 = \frac{V_p^2}{2}$

• RMS Definition: For a sinusoidal signal, the $V_{rms}$ is defined as:

  • $V_{rms} = \frac{V_p}{\sqrt{2}} \approx 0.707 V_p$

Fundamentals of Harmonics

• Wave Relationships: The velocity ($v$) of a wave is related to its wavelength ($\lambda$) and frequency ($f$) by the equation:

  • $v = \lambda f$

• Vibration Modes on a String: For a string of length $L$, different modes of vibration (harmonics) occur:

  • 1st Harmonic (Fundamental Mode): The wavelength is exactly twice the length of the string ($L = \frac{\lambda_1}{2}$), meaning $\lambda_1 = 2L$. The fundamental frequency is $f_0 = \frac{v}{2L}$.

  • 2nd Harmonic: The wavelength equals the string length ($L = \lambda_2$). The frequency is $f_2 = \frac{v}{L}$, which is equal to $2f_0$.

  • 3rd Harmonic: The string length is three times the half-wavelength ($L = \frac{3\lambda_3}{2}$), resulting in $\lambda_3 = \frac{2L}{3}$. The frequency is $f_3 = \frac{3v}{2L}$, which is equal to $3f_0$.

• General Harmonic Rule: The frequency of the $n$-th harmonic is a strictly integer multiple of the fundamental frequency:

  • $f_n = n \times f_0$

• Signal Representation: Any complex waveform can be mathematically decomposed into or represented as a combination of various sine waves (harmonics) at different frequencies and amplitudes.

Electricity Consumption and Duty Cycle

• Energy Calculation: The total energy consumed by electrical appliances is calculated using the formula:

  • $\text{Energy} = \text{Power} \times \text{Time} \times \text{Number of Devices} \times \text{Duty Cycle}$

• Duty Cycle: This is the percentage of time that a device is actively in its "ON" state within a given period:

  • $\text{Duty Cycle}\% = \frac{T_{ON}}{T_{ON} + T_{OFF}} \times 100\%$

• Application Examples: Appliances like Air Conditioners ($AC$) cycle on and off to maintain a specific ambient temperature. For example, the compressor may alternate between cycles for a target of $24^{\circ} C$ or $28^{\circ} C$.

  • Setting the thermostat to a higher temperature (e.g., $26^{\circ} C$ vs $24^{\circ} C$) reduces the overall duty cycle, thereby decreasing the consumed energy and cost.

Electromechanical Conversion and Induction

• Conversion Devices:

  • Motor: Responsible for converting electrical energy into mechanical energy.

  • Generator: Responsible for converting mechanical energy into electrical energy.

• Physics of Induction:

  • Angular Velocity ($\omega_0$): Defined by the rate of change of displacement over time $\omega_0 = \frac{\theta(t)}{t}$, leading to $\theta(t) = \omega_0 t$.

  • Magnetic Flux ($\Phi$): The flux through a rotating coil is given by $\Phi = BA \cos(\omega_0 t)$, where $B$ is magnetic field strength and $A$ is the area of the coil.

  • Electromotive Force (EMF, $e$): Based on Faraday's Law, the induced EMF is the negative rate of change of flux: $e = -\frac{d\Phi}{dt} = BA\omega_0 \sin(\omega_0 t)$.

Advanced Circuit Elements and Potential Difference

• Component Modeling:

  • Inductor: The voltage across an inductor is proportional to the time rate of change of current: $v(t) = L \frac{di}{dt}$.

  • Capacitor: The current through a capacitor is proportional to the time rate of change of voltage: $i(t) = C \frac{dv}{dt}$. Alternatively, the voltage is the integral of the current: $v(t) = \frac{1}{C} \int i(t)\,dt$.

• Potential Difference ($V_{AB}$): The difference in electrical potential between two points $A$ and $B$ is defined as $V_{AB} = V_A - V_B$.

  • Calculations are performed by traversing a path from $A$ to $B$ and summing voltage gains and drops according to Kirchhoff's Voltage Law ($KVL$).

Questions & Discussion

• Sinusoid Example 1: Given an input current $i(t) = 50 \sin(50t + \frac{\pi}{3})\,A$.

  1. Peak Value ($I_p$): Identified as $50\,A$.

  2. Angular Frequency ($\omega$): Identified as $50\,rad/s$.

  3. Frequency ($f$): Calculated as $\frac{50}{2\pi}\,Hz$.

  4. Phase ($\phi$): Identified as $\frac{\pi}{3}$.

• Sinusoid Example 2: Given an input voltage $v(t) = 10 \sin(\pi t)$.

  1. Angular Frequency ($\omega$): Identified as $\pi\,rad/s$.

  2. Time Period ($T$): Calculated as $\frac{2\pi}{\pi} = 2\,s$.

  3. Frequency ($f$): Calculated as $\frac{1}{2} = 0.5\,Hz$.

• Harmonics Problem: If the 3rd harmonic frequency ($f_3$) is $150\,Hz$, find the fundamental frequency ($f_0$).

  • Calculation: $f_3 = 3 \times f_0$

  • $150 = 3 \times f_0$

  • $f_0 = 50\,Hz$

• Electricity Cost Calculation: A $1.5\,kW$ heater is used for $20$ minutes daily for $30$ days. Cost is $8$ currency units per $\text{kWh}$.

  • Units Conversion: $20\,minutes = \frac{1}{3}\,hours$.

  • Daily Energy: $1.5\,kW \times \frac{1}{3}\,h = 0.5\,kWh$.

  • Monthly Energy: $0.5\,kWh/day \times 30\,days = 15\,kWh$ (or $15$ units).

  • Total Cost: $15\,units \times 8 = 120$.

• Inductor Calculation: Calculate inductance $L$ if voltage $v = 60\,mV$ and current changes from $0.4\,A$ to $1\,A$ in $10\,ms$.

  • Formula: $v = L \frac{\Delta i}{\Delta t}$

  • $60 \times 10^{-3} = L \times \frac{1 - 0.4}{10 \times 10^{-3}}$

  • $60 \times 10^{-3} = L \times \frac{0.6}{10^{-2}}$

  • $60 \times 10^{-3} = L \times 60$

  • $L = 1\,mH$

• Potential Difference Example: Find $V_{AB}$ for a circuit loop with sources of $3\,V$, $12\,V$, and $2\,V$.

  • Equation: $V_A + 3 - 12 - 2 - V_B = 0$

  • Rearrangement: $V_A - V_B = 12 + 2 - 3$

  • Result: $V_{AB} = 11\,V$