Questions: Alternating Current

Q1. Define the rms value of an alternating current and derive its relation to the peak value.

Answer (English): The rms value is the square root of the mean of the current squared over one complete cycle. For i = i_m sin omega t, the mean of sin squared omega t over a cycle is one half, so the mean of i squared is i_m squared/2 and the rms value is i_m divided by the square root of two, about 0.707 i_m. The rms value is the effective value: a resistor carrying a steady current equal to I_rms dissipates the same average power as it does with the alternating current, which is why rms values are used in all power formulas.

Hinglish: RMS value current ke square ke mean ka square root hoti hai. i = im sin omega t ke liye sin^2 ka mean ek cycle par 1/2 hota hai, isliye rms = im/root 2, yaani 0.707 im. Yahi effective value hai: same rms current wali steady current se resistor mein wahi average power nikalta hai, isliye power formulas mein rms use hoti hai.

Q2. Write the expression for the current in a pure inductor and state its phase relation.

Answer (English): For a pure inductor the voltage is v = v_m sin omega t and the circuit equation is v = L di/dt. Integrating gives i = (v_m / omega L) sin (omega t minus pi by 2), so the current lags the voltage by a quarter of a cycle. The amplitude of the current is v_m/X_L with X_L = omega L. The average power over a cycle is zero, because when the voltage is maximum the current is zero and vice versa, so energy is stored in the magnetic field in one quarter cycle and returned in the next.

Hinglish: Pure inductor mein v = L di/dt hota hai, integrate karne par i = (vm/omega L) sin(omega t - pi/2) milta hai, isliye current voltage se 90 degree lag karta hai. Current ka amplitude vm/XL hota hai jahan XL = omega L. Average power zero hoti hai kyunki voltage maximum ho to current zero hota hai, aur energy magnetic field mein store hokar wapas milti hai.

Q3. Why is the average power zero in a pure capacitive circuit?

Answer (English): In a pure capacitor the current leads the voltage by pi by 2, so the instantaneous power, which is the product of voltage and current, is positive during one quarter cycle and negative during the next, and its average over a full cycle is zero. The positive part is the energy being stored in the electric field of the capacitor while it charges, and the negative part is the same energy being returned to the source while it discharges. Since no energy is dissipated, the average power is zero and the current is called wattless.

Hinglish: Pure capacitor mein current voltage se 90 degree lead karta hai, isliye instantaneous power ek quarter mein positive aur doosre mein negative hoti hai, aur poore cycle ka average zero. Charging ke waqt energy electric field mein store hoti hai aur discharging mein wapas source ko mil jati hai. Koi energy dissipate nahi hoti, isliye average power zero aur current ko wattless kehte hain.

Q4. Derive the impedance of a series LCR circuit using a phasor diagram.

Answer (English): The same current flows through R, L and C, so take the current phasor along the reference direction. The voltage across R is in phase with the current, the voltage across L leads by 90 degrees and the voltage across C lags by 90 degrees. The two reactive voltages are opposite, so their resultant is (X_L minus X_C) times the current, perpendicular to the current. Adding that to the resistive voltage by the Pythagoras rule gives the resultant voltage amplitude as the current times the square root of R squared plus (X_L minus X_C) squared. Dividing by the current gives Z equal to that square root, and the phase angle satisfies tan phi = (X_L minus X_C)/R.

Hinglish: Series LCR mein current sabme same hota hai, isliye current ko reference lo. R ka voltage in phase, L ka 90 degree lead aur C ka 90 degree lag. Dono reactive voltages opposite hain, isliye net (XL - XC) current ke perpendicular hota hai. Pythagoras se resultant voltage = current x root(R^2 + (XL - XC)^2), aur current se divide karke Z milta hai. Phase ke liye tan phi = (XL - XC)/R.

Q5. What is resonance in a series LCR circuit and what is the resonant frequency?

Answer (English): Resonance occurs when the inductive and capacitive reactances are equal, so their effects cancel and the impedance falls to its minimum value R. The current then reaches its maximum value v_m/R, the voltage and current are in phase, and the circuit behaves as a pure resistance. The condition X_L = X_C gives omega_0 L = 1/(omega_0 C), so the resonant angular frequency is omega_0 = 1 divided by the square root of L C, and the ordinary frequency is that divided by 2 pi. The resonant frequency depends only on L and C, while the sharpness of the resonance peak depends on R through the quality factor Q = omega_0 L / R.

Hinglish: Resonance tab hota hai jab XL = XC ho, dono cancel ho jayein aur impedance minimum R ban jaye. Phir current maximum vm/R ho jata hai, voltage aur current in phase hote hain aur circuit pure resistance jaisa behave karta hai. XL = XC se omega0 = 1/root(LC) milta hai. Resonant frequency sirf L aur C par depend karti hai, par peak ki sharpness R se aati hai through Q = omega0 L / R.

Q6. What is the power factor and why is it important in power transmission?

Answer (English): The power factor of an AC circuit equals cos phi, with phi the angle by which the current differs in phase from the voltage; multiplying V_rms I_rms by it gives the average power. It equals one for a pure resistor, zero for a pure inductor or capacitor, and lies between zero and one for a mixed circuit. It matters in transmission because a low power factor means that a larger current is needed to deliver the same useful power, and the cable losses are proportional to the square of the current. Correcting the power factor with a capacitor bank, for example, reduces the current and the loss.

Hinglish: Power factor cos phi hota hai, jahan phi voltage aur current ka phase angle hai; average power Vrms Irms cos phi hoti hai. Pure resistor mein 1, pure L ya C mein 0, aur mixed circuit mein 0 aur 1 ke beech. Transmission mein low power factor matlab same useful power ke liye zyada current aur cable loss current ke square ke proportional, isliye capacitor bank se power factor improve karte hain.

Q7. What are LC oscillations and what is their frequency?

Answer (English): When a charged capacitor is connected across an inductor, the capacitor discharges through the inductor and the current builds up a magnetic field; then the magnetic field collapses and recharges the capacitor with the opposite polarity. The charge and current oscillate at the angular frequency one divided by the square root of L C, with the energy swapping between the electric field of the capacitor and the magnetic field of the inductor. In an ideal circuit the oscillation would continue forever; in a real circuit the resistance removes energy as heat and the oscillation dies away.

Hinglish: Charged capacitor ko inductor se jodne par capacitor discharge hota hai, current magnetic field banata hai, phir field collapse hokar capacitor ko ulti polarity se charge karta hai. Charge aur current omega = 1/root(LC) par oscillate karte hain aur energy electric aur magnetic field ke beech ghoomti hai. Ideal circuit mein oscillation chalti rehti hai, real circuit mein resistance energy heat mein badal deti hai.

Q8. Explain the working of a transformer and the relation between its voltages and currents.

Answer (English): A transformer has two coils wound on a common soft iron core. The alternating current in the primary produces an alternating magnetic flux in the core, and this shared flux passes through the secondary and induces an emf there. Since the flux through every turn is identical, each turn develops the same emf, so the voltage ratio matches the turns ratio. For an ideal transformer the power is the same on both sides, so the currents are in the inverse ratio. A step-up transformer raises the voltage and lowers the current, which is why power is transmitted at high voltage.

Hinglish: Transformer mein do coils common soft iron core par wound hote hain. Primary ka alternating current core mein flux banata hai jo secondary ko link karke emf induce karta hai. Har turn par same flux link hota hai, isliye Vs/Vp = Ns/Np. Ideal transformer mein power same rehti hai, isliye Is/Ip = Np/Ns. Step-up transformer voltage badhata hai aur current ghatata hai, isliye transmission high voltage par hota hai.

Q9. List the energy losses in a real transformer.

Answer (English): Four losses matter. The copper loss is the I squared R heat in the resistance of the primary and secondary windings. The hysteresis loss is the energy spent in magnetising and demagnetising the iron core once per cycle. The eddy current loss is the heat from currents induced in the bulk of the core, which is reduced by making the core from thin insulated laminations. The flux leakage loss is the part of the primary flux that does not link the secondary, which reduces the voltage ratio below the turns ratio. A good transformer reduces these so that its efficiency can be above 90 per cent.

Hinglish: Chaar loss hain. Copper loss windings ke resistance mein I^2R heat. Hysteresis loss core ko har cycle magnetise aur demagnetise karne mein. Eddy current loss core mein induce hui currents se heat, jo thin insulated laminations se kam hoti hai. Flux leakage woh flux jo secondary ko link nahi karta. Achhe transformer ki efficiency 90 percent se zyada hoti hai.

Q10. How does the impedance of a series LCR circuit change as the frequency is increased from a low value?

Answer (English): At low frequency the capacitive reactance 1/(omega C) is large and dominates, so the impedance is large and the current leads the voltage. As the frequency rises, X_C falls and X_L rises until they become equal at the resonant frequency, where the impedance falls to its minimum value R and the current is maximum with zero phase difference. As the frequency rises further, the inductive reactance omega L dominates, the impedance rises again and the current lags the voltage. The impedance frequency graph is therefore U shaped, and the current graph is a peak at the resonant frequency whose width is set by the resistance.

Hinglish: Low frequency par XC bada hota hai isliye impedance bada aur current lead karta hai. Frequency badhne par XC ghatta hai aur XL badhta hai; jab woh barabar ho jate hain tab resonance aata hai, impedance minimum R aur current maximum hota hai. Aage frequency badhne par XL dominate karta hai, impedance phir badhta hai aur current lag karta hai. Isliye impedance graph U shaped hota hai aur current ka peak resonant frequency par aata hai, jiski chaudai R se tay hoti hai.

Quiz below: five questions on rms values, reactance, LCR resonance, power and the transformer. Write the phase relation with every circuit answer.