AC Network Analysis: Impedance and Techniques

• AC network problems involve analyzing circuits with alternating current (AC) sources and components like resistors, inductors, and capacitors.

Impedance

• Impedance (Z) stands as the AC equivalent of resistance.
• Impedance is measured in ohms (Ω).
• It is the total opposition to current flow in an AC circuit.
• Impedance is a complex quantity.
• It comprises a real part (resistance, R) and an imaginary part (reactance, X).
• Z = R + jX, where j is the imaginary unit (√-1).
• Reactance (X) signifies opposition to current flow due to inductors and capacitors.
• Inductive reactance (XL) is proportional to frequency.
• XL = 2πfL, where f is frequency and L is inductance.
• Capacitive reactance (XC) is inversely proportional to frequency.
• XC = 1/(2πfC), where C is capacitance.
• The magnitude of impedance is |Z| = √(R² + X²).
• The phase angle (θ) between voltage and current is given by θ = arctan(X/R).

AC Circuit Analysis Techniques

• Many DC circuit analysis techniques extend to AC circuits.
• This can be achieved by using impedance instead of resistance.
• Ohm's Law: V = IZ, where V and I are phasor representations of voltage and current.
• Kirchhoff's Laws: KVL and KCL apply to AC circuits.
• KVL and KCL involve using phasor voltages and currents.
• Series Impedances: Total impedance equals the sum of individual impedances: Ztotal = Z1 + Z2 + Z3 + ...
• Parallel Impedances: The reciprocal of total impedance is the sum of reciprocals of individual impedances: 1/Ztotal = 1/Z1 + 1/Z2 + 1/Z3 + ...
• Voltage Divider Rule: In a series circuit, the voltage across an impedance is proportional to its impedance: V1 = Vtotal * (Z1 / Ztotal).
• Current Divider Rule: In a parallel circuit, the current through an impedance is inversely proportional to its impedance: I1 = Itotal * (Ztotal / Z1).
• Ztotal represents the equivalent impedance of the parallel combination.
• Nodal Analysis: Define node voltages and apply KCL at each node.
• Currents can be expressed in terms of node voltages and impedances.
• Mesh Analysis: Define mesh currents and apply KVL around each mesh.
• Voltages can be expressed in terms of mesh currents and impedances.
• Superposition Theorem: In a linear circuit with multiple independent sources, the total response is the sum of the responses due to each source acting alone.
• Thevenin's Theorem: Any linear circuit can be replaced by an equivalent circuit containing a voltage source (Vth) in series with an impedance (Zth).
• Norton's Theorem: Any linear circuit can be replaced by an equivalent circuit containing a current source (In) in parallel with an impedance (Zn).

Phasors

• Phasors represent sinusoidal voltages and currents as complex numbers.
• A sinusoidal voltage v(t) = Vm cos(ωt + φ) can be represented as a phasor V = Vm∠φ.
• Vm is the amplitude
• ω is the angular frequency
• φ is the phase angle
• Phasors simplify AC circuit analysis
• This is done by converting differential equations into algebraic equations.
• Mathematical operations like addition, subtraction, multiplication, and division can be performed on phasors.
• Differentiation in the time domain corresponds to multiplication by jω in the phasor domain.
• Integration in the time domain corresponds to division by jω in the phasor domain.

AC Power

• Instantaneous power (p(t)) is the product of instantaneous voltage and current; p(t) = v(t) * i(t).
• Average power (P) represents the average value of instantaneous power over one cycle.
• Average power is measured in watts (W).
• For a sinusoidal voltage and current, P = Vrms * Irms * cos(θ).
• Vrms and Irms stand for the RMS values of voltage and current.
• θ represents the phase angle between them.
• Reactive power (Q) is the power oscillating between the source and reactive components (inductors and capacitors).
• Reactive power is measured in Volt-Ampere Reactive (VAR).
• Q = Vrms * Irms * sin(θ).
• Apparent power (S) is the product of RMS voltage and RMS current.
• Apparent power is measured in Volt-Amperes (VA).
• S = Vrms * Irms.
• Average power, reactive power, and apparent power are related through the power triangle: S² = P² + Q².
• Power factor (PF) is the ratio of average power to apparent power: PF = P/S = cos(θ).
• A power factor of 1 means voltage and current are in phase in a purely resistive circuit.
• A power factor of 0 indicates voltage and current are 90 degrees out of phase in a purely reactive circuit.
• Power factor correction adjusts the power factor closer to 1.
• This can be done by adding capacitors or inductors to the circuit, improving efficiency.

Resonance

• Resonance arises in AC circuits when inductive and capacitive reactance are equal (XL = XC).
• At resonance, the impedance is purely resistive.
• Additionally, the current is maximum for a given voltage.
• Series Resonance: XL = XC, thus 2πfL = 1/(2πfC).
• The resonant frequency is f = 1/(2π√(LC)).
• Impedance in a series resonant circuit is minimal at resonance.
• Parallel Resonance: XL = XC.
• The resonant frequency is f = 1/(2π√(LC)).
• Impedance in a parallel resonant circuit is maximal at resonance.
• Quality factor (Q) measures the sharpness of resonance.
• For a series resonant circuit, Q = (1/R)√(L/C).
• For a parallel resonant circuit, Q = R√(C/L).
• Bandwidth (BW) spans the frequencies around resonance.
• Current or voltage in the bandwidth is at least 70.7% of its maximum.
• BW = f/Q, where f is the resonant frequency.

AC Circuit Theorems and Simplifications

• Source Transformation: A voltage source in series with an impedance can be transformed into a current source in parallel with the same impedance, and vice versa.
• Delta-Wye (Pi-T) Transformation: A delta (Δ) network of impedances can be transformed into an equivalent wye (Y) network, and vice versa, simplifying circuit analysis.
• Superposition: In a linear network containing multiple independent sources, the voltage or current for any element can be found by algebraically summing the contribution of each independent source acting alone.
• Thevenin's and Norton's Theorems: These theorems simplify complex circuits into simpler equivalents.
• These equivalents allow easier analysis of a specific part of the circuit.
• Thevenin's Theorem replaces a circuit with a voltage source (Vth) in series with an impedance (Zth).
• Norton's Theorem replaces a circuit with a current source (In) in parallel with an impedance (Zn).

Explore AC network problems involving resistors, inductors, and capacitors. Learn about impedance (Z), the AC equivalent of resistance, and how to calculate it using resistance (R) and reactance (X). Discover AC circuit analysis techniques.