⚡Electrical Circuits and Systems I Unit 9 – Sinusoidal Steady-State Analysis

Key Concepts and Definitions

• Sinusoidal steady-state refers to the condition when a circuit is driven by a sinusoidal source and the response is also sinusoidal with the same frequency
• Phasors are complex numbers that represent sinusoidal quantities, simplifying the analysis of AC circuits
  • Phasors capture the amplitude and phase information of sinusoidal signals
• Impedance is the opposition to the flow of alternating current in a circuit, consisting of resistance and reactance
  • Resistance is the real part of impedance and represents the opposition to current flow
  • Reactance is the imaginary part of impedance and represents the opposition to changes in current or voltage
• Admittance is the reciprocal of impedance and represents the ease with which current flows in a circuit
• Complex power is the product of voltage and current phasors, consisting of real power (active) and reactive power
  • Real power is the average power consumed by the circuit
  • Reactive power is the power that oscillates between the source and the load

Fundamental Principles of Sinusoidal Steady-State

• In sinusoidal steady-state, all voltages and currents in the circuit are sinusoidal with the same frequency
• The amplitude and phase of the sinusoidal quantities remain constant over time
• Kirchhoff's laws (KVL and KCL) still apply in the phasor domain, allowing for the analysis of AC circuits using phasors
• The principle of superposition holds for linear circuits in sinusoidal steady-state, enabling the analysis of circuits with multiple sources
• The frequency of the sinusoidal source determines the behavior of the circuit elements (resistors, inductors, and capacitors)
  • Resistors have a constant impedance equal to their resistance
  • Inductors have an impedance that increases with frequency ($X_L = j\omega L$)
  • Capacitors have an impedance that decreases with frequency ($X_C = \frac{1}{j\omega C}$)

Complex Numbers and Phasors

• Complex numbers consist of a real part and an imaginary part ($a + jb$), where $j = \sqrt{-1}$
• Phasors are complex numbers that represent sinusoidal quantities, with the real part corresponding to the cosine component and the imaginary part corresponding to the sine component
• Phasor notation simplifies the analysis of AC circuits by eliminating the need to work with time-varying sinusoidal functions
• Phasors can be expressed in rectangular form ($a + jb$) or polar form ($A\angle\theta$)
  • In rectangular form, $a$ is the real part, and $b$ is the imaginary part
  • In polar form, $A$ is the magnitude, and $\theta$ is the phase angle
• Phasor arithmetic (addition, subtraction, multiplication, and division) follows the rules of complex number arithmetic

Impedance and Admittance

• Impedance ($Z$) is the ratio of the voltage phasor to the current phasor in a circuit element or a complete circuit
  • Impedance is a complex quantity, with the real part representing resistance and the imaginary part representing reactance
  • The unit of impedance is the ohm ($\Omega$)
• Admittance ($Y$) is the reciprocal of impedance and represents the ease with which current flows in a circuit
  • Admittance is also a complex quantity, with the real part representing conductance and the imaginary part representing susceptance
  • The unit of admittance is the siemens ($S$)
• The impedance of circuit elements in series is the sum of their individual impedances ($Z_{total} = Z_1 + Z_2 + \ldots$)
• The admittance of circuit elements in parallel is the sum of their individual admittances ($Y_{total} = Y_1 + Y_2 + \ldots$)

Circuit Analysis Techniques

• Nodal analysis involves applying Kirchhoff's current law (KCL) at each node in the circuit and solving for the node voltages using phasors
  • The node voltages are referenced to a common ground node
  • Once the node voltages are known, the branch currents can be calculated using Ohm's law
• Mesh analysis involves applying Kirchhoff's voltage law (KVL) to each mesh in the circuit and solving for the mesh currents using phasors
  • The mesh currents are the currents circulating in each independent loop of the circuit
  • Once the mesh currents are known, the branch voltages can be calculated using Ohm's law
• Thévenin's theorem allows for the simplification of a complex circuit into an equivalent circuit consisting of a single voltage source and a series impedance
  • The Thévenin equivalent voltage is the open-circuit voltage at the terminals of interest
  • The Thévenin equivalent impedance is the impedance seen from the terminals of interest when all sources are turned off
• Norton's theorem is the dual of Thévenin's theorem and allows for the simplification of a complex circuit into an equivalent circuit consisting of a single current source and a parallel impedance
  • The Norton equivalent current is the short-circuit current at the terminals of interest
  • The Norton equivalent impedance is the same as the Thévenin equivalent impedance

Power in AC Circuits

• Instantaneous power is the product of the instantaneous voltage and current in a circuit
• Average power (real power) is the time average of the instantaneous power over one period of the sinusoidal waveform
  • Real power represents the power consumed by the circuit and is measured in watts ($W$)
  • Real power is calculated as $P = \frac{1}{2}VI\cos\theta$, where $V$ and $I$ are the rms values of voltage and current, and $\theta$ is the phase angle between them
• Reactive power is the power that oscillates between the source and the load without being consumed
  • Reactive power is measured in volt-amperes reactive ($VAR$)
  • Reactive power is calculated as $Q = \frac{1}{2}VI\sin\theta$
• Complex power is the sum of real power and reactive power ($S = P + jQ$)
  • Complex power is measured in volt-amperes ($VA$)
  • The magnitude of complex power is called apparent power ($|S| = \sqrt{P^2 + Q^2}$)
• Power factor is the ratio of real power to apparent power ($\cos\theta = \frac{P}{|S|}$) and represents the efficiency of power utilization in the circuit

Frequency Response

• Frequency response describes how a circuit behaves over a range of frequencies
• The frequency response of a circuit can be characterized by its magnitude and phase response
  • The magnitude response shows how the amplitude of the output signal varies with frequency
  • The phase response shows how the phase of the output signal varies with frequency
• Bode plots are used to graphically represent the frequency response of a circuit
  • A Bode magnitude plot shows the magnitude response in decibels ($dB$) versus frequency on a logarithmic scale
  • A Bode phase plot shows the phase response in degrees versus frequency on a logarithmic scale
• Resonance occurs when the inductive and capacitive reactances in a circuit are equal in magnitude, resulting in a purely resistive impedance
  • At resonance, the impedance of the circuit is minimized, and the current is maximized
  • The resonant frequency is given by $f_0 = \frac{1}{2\pi\sqrt{LC}}$ for an LC circuit
• Filters are circuits designed to pass or attenuate signals based on their frequency
  • Low-pass filters allow low-frequency signals to pass while attenuating high-frequency signals
  • High-pass filters allow high-frequency signals to pass while attenuating low-frequency signals
  • Band-pass filters allow a specific range of frequencies to pass while attenuating frequencies outside that range
  • Band-stop filters attenuate a specific range of frequencies while allowing frequencies outside that range to pass

Practical Applications and Examples

• Power systems use sinusoidal steady-state analysis to study the generation, transmission, and distribution of electrical energy
  • The frequency of the power system is typically 50 Hz or 60 Hz
  • Power factor correction is used to improve the efficiency of power transmission by reducing reactive power
• Audio and video systems use sinusoidal steady-state analysis to design filters, equalizers, and amplifiers
  • Crossover networks in loudspeakers use filters to divide the audio signal into different frequency ranges for the woofer, midrange, and tweeter
  • Graphic equalizers use a series of band-pass filters to adjust the amplitude of specific frequency bands in an audio signal
• Communication systems use sinusoidal steady-state analysis to design filters, mixers, and modulators
  • Band-pass filters are used to select the desired frequency band in a receiver
  • Mixers are used to shift the frequency of a signal by multiplying it with a local oscillator signal
• Control systems use sinusoidal steady-state analysis to study the frequency response of the system and design compensators
  • The frequency response of a control system determines its stability and performance
  • Lead and lag compensators are used to modify the frequency response of the system to achieve the desired performance
• Biomedical engineering uses sinusoidal steady-state analysis to study the electrical activity of the body
  • Electroencephalography (EEG) measures the electrical activity of the brain using sinusoidal steady-state analysis
  • Electrocardiography (ECG) measures the electrical activity of the heart using sinusoidal steady-state analysis