In a parallel RC circuit, the total admittance Y is given by:
In a parallel RC circuit, admittances add directly. Y = G + jBC where G = 1/R and BC = ωC.
The Q-factor (quality factor) of a resonant circuit is defined as:
Q-factor represents selectivity and bandwidth characteristics. For series RLC: Q = ω₀L/R = 1/(ω₀RC) = (1/R)√(L/C).
In Thevenin's theorem, the Thevenin equivalent voltage (VTh) is found by:
Thevenin voltage is the open-circuit voltage measured across the load terminals after removing the load.
A capacitor and inductor are in series with resonance occurring at 100 Hz. If L = 25 mH, the capacitance C is approximately:
At resonance: f = 1/(2π√LC). C = 1/(4π²f²L) = 1/(4π²×10000×0.025) ≈ 101.3 µF.
A bridge circuit is balanced when:
For a balanced AC bridge, the product of opposite arm impedances must be equal: Z1×Z3 = Z2×Z4.
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The bandwidth of a resonant circuit is related to Q-factor by:
Bandwidth is inversely proportional to Q-factor. Higher Q means narrower bandwidth. BW = f₀/Q.
In a delta-wye (Δ-Y) conversion, if the delta resistances are RA, RB, RC, the wye resistance R1 (between node 1 and star point) is:
In Δ-Y conversion, each Y-resistor equals the product of adjacent Δ-resistors divided by the sum of all Δ-resistors.
In a series RLC circuit at resonance, the impedance is:
At resonance, XL = XC, so they cancel. Z = R (minimum), and current is maximum.
The equivalent resistance of two resistors in parallel is always:
For parallel resistors: Req = (R1×R2)/(R1+R2), which is always less than the smaller individual resistor.
In Norton's theorem, the Norton equivalent current (IN) is found by:
Norton current is the short-circuit current at the load terminals. IN = VTh/RTh.
The voltage regulation of a source is defined as:
Voltage regulation measures the percentage change in output voltage from no-load to full-load conditions.
A resistor of 10 Ω is connected in series with an inductor of 0.1 H across a 50 Hz AC supply. What is the impedance of the circuit?
Impedance Z = √(R² + (ωL)²) where ω = 2πf = 2π(50) = 314.16 rad/s. XL = ωL = 314.16 × 0.1 = 31.416 Ω. Z = √(10² + 31.416²) = √(100 + 987.96) = √1087.96 = 32.98 Ω. Correction: XL = 2π × 50 × 0.1 = 31.416 Ω, Z = √(100 + 987.96) ≈ 12.63 Ω is for series RL with proper calculation.
What is the power factor of a circuit with impedance Z = 8 + j6 Ω?
Z = 8 + j6 Ω, |Z| = √(8² + 6²) = √(64 + 36) = 10 Ω. Power factor = cos(φ) = R/|Z| = 108 = 0.8 (lagging)
In a purely capacitive circuit with V = 100V (RMS) and C = 50 μF at 50 Hz, what is the capacitive reactance?
Capacitive reactance XC = 1/(ωC) = 1/(2πfC) = 1/(2π × 50 × 50×10⁻⁶) = 1/(0.0157) = 636.6 Ω
What happens to the impedance of a series RLC circuit when frequency approaches resonant frequency?
At resonance, XL = XC, so Z = √(R² + (XL - XC)²) = R (minimum). This occurs at f₀ = 1/(2π√LC)
The Q-factor of a resonant circuit is defined as:
Quality factor Q = f₀/BW = ω₀L/R = 1/(ω₀RC), representing the sharpness of resonance curve
A 50 Ω transmission line has a velocity factor of 0.67. What is the wavelength at 1 GHz?
Velocity of propagation v = c × vf = 3×10⁸ × 0.67 = 2.01×10⁸ m/s. Wavelength λ = v/f = 2.01×10⁸/10⁹ = 0.201 m
The instantaneous power in an AC circuit is given by:
Instantaneous power is the product of instantaneous voltage and current: p(t) = v(t)i(t). Average power is VI cos(φ)
A circuit has resistance R = 8 Ω and reactance X = 6 Ω (inductive). If the apparent power is 100 VA, what is the real power?
S = 100 VA, Z = √(8² + 6²) = 10 Ω, Power factor = R/Z = 108 = 0.8, Real power P = S × pf = 100 × 0.8 = 80 W
In maximum power transfer theorem, the load resistance should be:
Maximum power transfer occurs when RL = RTh (conjugate of source impedance), delivering Pmax = VTh²/(4RTh)
