GATE Electrical Engineering Electric circuits Syllabus

How are the Thevenin and Norton equivalents related by source transformation?

$R_N = R_{Th}$ and $V_{Th} = I_N R_{Th}$, equivalently $I_N = \dfrac{V_{Th}}{R_{Th}}$. Norton current equals the short-circuit current and Thevenin voltage equals the open-circuit voltage.

State the superposition theorem and its key limitation.

In a linear circuit with multiple independent sources, the response (voltage or current) in any element equals the algebraic sum of the responses caused by each independent source acting alone (others deactivated). Limitation: it applies only to linear quantities; it cannot be used directly to compute power, which is a nonlinear ($I^{2}R$) quantity.

When deactivating sources for superposition, how are dependent sources handled?

Dependent (controlled) sources are never deactivated; they remain active in every step of the superposition analysis. Only independent sources are turned off one at a time.

State the Maximum Power Transfer theorem for a DC resistive network.

Maximum power is delivered to a load resistance $R_L$ when it equals the Thevenin resistance of the source network: $R_L = R_{Th}$.

What is the maximum power delivered to the load under matched conditions in a DC circuit?

$P_{max} = \dfrac{V_{Th}^{2}}{4 R_{Th}}$, achieved when $R_L = R_{Th}$. At this point the efficiency is only $50\%$.

For an AC source with Thevenin impedance $Z_{Th} = R_{Th} + jX_{Th}$, what load impedance gives maximum power transfer?

The load must be the complex conjugate of the source impedance: $Z_L = Z_{Th}^{*} = R_{Th} - jX_{Th}$. The reactances cancel and the resistances match.

If only the magnitude of an AC load impedance can be varied (load angle fixed) or the load is purely resistive, what is the maximum power transfer condition?

When the load is purely resistive, maximum power transfer occurs at $R_L = |Z_{Th}| = \sqrt{R_{Th}^{2} + X_{Th}^{2}}$.

Define the impedance of an inductor and a capacitor in the sinusoidal steady state (phasor domain).

Inductor: $Z_L = j\omega L$. Capacitor: $Z_C = \dfrac{1}{j\omega C} = -\dfrac{j}{\omega C}$. Resistor: $Z_R = R$.

Relate the RMS value of a sinusoid to its peak value.

For a sinusoid of amplitude $V_m$, the RMS value is $V_{rms} = \dfrac{V_m}{\sqrt{2}} \approx 0.707\,V_m$. The average value over a full cycle is zero; the average over a half-cycle is $\dfrac{2V_m}{\pi}$.

What is admittance, and how do its components relate to impedance?

Admittance $Y = \dfrac{1}{Z} = G + jB$, where $G$ is conductance and $B$ is susceptance, measured in siemens (S). For $Z = R + jX$, $G = \dfrac{R}{R^{2}+X^{2}}$ and $B = -\dfrac{X}{R^{2}+X^{2}}$.

What is the resonant frequency of a series RLC circuit?

$\omega_0 = \dfrac{1}{\sqrt{LC}}$ (rad/s), or $f_0 = \dfrac{1}{2\pi\sqrt{LC}}$ (Hz). At resonance $X_L = X_C$, the net reactance is zero, and the impedance is purely resistive and minimum ($Z = R$).

What characterizes a series RLC circuit at resonance (current, impedance, power factor)?

Impedance is minimum and purely resistive ($Z = R$), current is maximum ($I = V/R$), the circuit appears purely resistive with unity power factor, and $X_L = X_C$ so the inductor and capacitor voltages are equal and opposite.

Define the quality factor $Q$ of a series RLC resonant circuit.

$Q = \dfrac{\omega_0 L}{R} = \dfrac{1}{\omega_0 C R} = \dfrac{1}{R}\sqrt{\dfrac{L}{C}}$. It is the ratio of reactive power to average power (or $2\pi$ times energy stored per energy dissipated per cycle) at resonance.

How are bandwidth, resonant frequency, and quality factor related in a resonant circuit?

Bandwidth $BW = \omega_2 - \omega_1 = \dfrac{\omega_0}{Q}$ (where $\omega_1,\omega_2$ are the half-power frequencies). For a series RLC circuit $BW = \dfrac{R}{L}$. Higher $Q$ gives a narrower, more selective bandwidth.

What characterizes a parallel RLC (tank) circuit at resonance?

At resonance the impedance is maximum and purely resistive, the total current drawn from the source is minimum, and the circuit shows unity power factor. The resonant frequency is again $\omega_0 = \dfrac{1}{\sqrt{LC}}$ for an ideal parallel RLC.

How are the half-power (cutoff) frequencies $\omega_1$ and $\omega_2$ related to the resonant frequency?

The resonant frequency is the geometric mean of the half-power frequencies: $\omega_0 = \sqrt{\omega_1 \omega_2}$.

In a balanced three-phase system, what is the phase relationship between the three phase voltages?

The three phase voltages have equal magnitude and are displaced by $120^{\circ}$ from one another. For positive (ABC) sequence: $V_a = V\angle 0^{\circ}$, $V_b = V\angle{-120^{\circ}}$, $V_c = V\angle{+120^{\circ}}$, and they sum to zero.

In a balanced star (Y) connection, how are line and phase voltages and currents related?

Line voltage leads phase voltage and $V_L = \sqrt{3}\,V_{ph}$ (with a $30^{\circ}$ phase shift); line current equals phase current, $I_L = I_{ph}$.

In a balanced delta ($\Delta$) connection, how are line and phase voltages and currents related?

Line voltage equals phase voltage, $V_L = V_{ph}$; line current is $I_L = \sqrt{3}\,I_{ph}$ (line current lags phase current by $30^{\circ}$).

Write the total real power consumed by a balanced three-phase load in terms of line quantities.

$P = \sqrt{3}\,V_L I_L \cos\phi$, where $\phi$ is the per-phase impedance angle (the power-factor angle between phase voltage and phase current). This formula holds for both star and delta loads.

Write the total reactive power and apparent power for a balanced three-phase load in terms of line quantities.

Reactive power $Q = \sqrt{3}\,V_L I_L \sin\phi$ and apparent power $S = \sqrt{3}\,V_L I_L$, with $S = \sqrt{P^{2} + Q^{2}}$.

How is total three-phase power measured by the two-wattmeter method, and how is the power factor obtained?

Total power $P = W_1 + W_2$. The power-factor angle is $\tan\phi = \sqrt{3}\,\dfrac{W_1 - W_2}{W_1 + W_2}$. Both wattmeters read equal positive values at unity p.f.; one reads zero at p.f. $= 0.5$.

State the star-to-delta and delta-to-star transformation formulas for resistors.

$\Delta\to Y$: $R_a = \dfrac{R_{ab}R_{ca}}{R_{ab}+R_{bc}+R_{ca}}$ (each star arm = product of two adjacent delta arms over the sum of all three). $Y\to\Delta$: $R_{ab} = \dfrac{R_a R_b + R_b R_c + R_c R_a}{R_c}$ (sum of pairwise products divided by the opposite star arm).

For a balanced (symmetric) network, how do the star-delta resistance values relate?

If all three star resistances are equal to $R_Y$ and all three delta resistances equal $R_\Delta$, then $R_\Delta = 3R_Y$, equivalently $R_Y = \dfrac{R_\Delta}{3}$.

Define complex power $S$ in an AC circuit and its components.

$S = VI^{*} = P + jQ$, where $P = VI\cos\phi$ is real (active) power in watts (W), $Q = VI\sin\phi$ is reactive power in volt-amperes reactive (VAR), and $|S| = VI$ is apparent power in volt-amperes (VA). Here $I^{*}$ is the conjugate of the phasor current.

Define power factor and distinguish leading from lagging power factor.

Power factor $= \cos\phi = \dfrac{P}{S}$, where $\phi$ is the angle by which current lags or leads voltage. Lagging p.f. occurs in inductive loads (current lags voltage, $Q > 0$); leading p.f. occurs in capacitive loads (current leads voltage, $Q < 0$).

What is the power triangle, and how are its sides related?

The power triangle is a right triangle with real power $P$ (horizontal), reactive power $Q$ (vertical), and apparent power $S$ (hypotenuse): $S = \sqrt{P^{2}+Q^{2}}$, $P = S\cos\phi$, $Q = S\sin\phi$, and $\tan\phi = \dfrac{Q}{P}$.

Why is reactive power important and what is the sign convention for inductive and capacitive elements?

Reactive power $Q$ represents energy that oscillates between source and reactive elements without net dissipation. By convention inductors absorb positive reactive power ($Q > 0$) and capacitors supply reactive power / absorb negative ($Q < 0$). Capacitors are used to supply $Q$ for power-factor correction of inductive loads.

What is the average power absorbed by a purely reactive element (ideal inductor or capacitor) in steady state?

Zero. A purely reactive element has $\phi = \pm 90^{\circ}$, so $P = VI\cos(\pm 90^{\circ}) = 0$; it stores and returns energy each cycle but dissipates none.

Compare ideal voltage and current sources in terms of internal resistance and source transformation.

An ideal voltage source has zero internal (series) resistance; an ideal current source has infinite internal (parallel) resistance. A practical voltage source ($V_s$ in series with $R_s$) and a practical current source ($I_s$ in parallel with $R_s$) are interchangeable via source transformation with $V_s = I_s R_s$. Ideal sources cannot be transformed because of their zero/infinite resistance.

Give the average power delivered to a general AC load in terms of RMS phasors and impedance angle.

$P = V_{rms} I_{rms}\cos\phi = I_{rms}^{2} R = \dfrac{V_{R,rms}^{2}}{R}$, where $\phi$ is the angle of the load impedance $Z = |Z|\angle\phi$ and $R = \operatorname{Re}\{Z\}$ is the resistive part. Only the resistive part consumes average power.