Attempt any THREE of the following: a) With neat diagram, explain the phasor representation of sinusoidal quantity. b) An AC circuit consists of two branches in parallel. Branch I:... Attempt any THREE of the following: a) With neat diagram, explain the phasor representation of sinusoidal quantity. b) An AC circuit consists of two branches in parallel. Branch I: R = 10Ω and L = 0.1H in series; Branch II: C = 50μF. If the circuit is supplied from 200V, 50Hz supply, calculate: i) Branch impedances ii) Branch Currents iii) Circuit Power factor iv) Power consumed by Ckt c) With the help of neat phasor diagram, derive the relationship between line and phase values of voltage in balanced star connection. d) State the equivalent delta connection for star connection of three resistances R1, R2 and R3, with proper equations. e) Explain the concept of 'duality' in electric circuit with one example. Read more

Steps to Solve

1. Calculate Branch Impedance for Branch I

   Given:

   • Resistance, $R = 10 \Omega$
   • Inductance, $L = 0.1 , H$
   • Angular frequency, $\omega = 2\pi f = 2\pi \times 50 , Hz = 314.16 , rad/s$

   The impedance for Branch I (R-L series) is given by:

   $$ Z_1 = R + j\omega L = 10 + j(314.16 \times 0.1) = 10 + j31.416 , \Omega $$

   Calculate the magnitude of $Z_1$:

   $$ |Z_1| = \sqrt{R^2 + (j\omega L)^2} = \sqrt{10^2 + 31.416^2} = \sqrt{100 + 987.699} \approx 34.47 , \Omega $$

2. Calculate Branch Impedance for Branch II

   Given:

   • Capacitance, $C = 50 , \mu F = 50 \times 10^{-6} , F$

   The impedance for Branch II (C):

   $$ Z_2 = \frac{1}{j\omega C} = \frac{1}{j(314.16 \times 50 \times 10^{-6})} = \frac{1}{j0.0157} = -j63.66 , \Omega $$

3. Calculate Branch Currents

   The voltage across both branches is the same, $V = 200 , V$.

   For Branch I:

   $$ I_1 = \frac{V}{Z_1} = \frac{200}{10 + j31.416} $$

   To compute, multiply the numerator and the denominator by the complex conjugate:

   $$ I_1 = \frac{200(10 - j31.416)}{(10 + j31.416)(10 - j31.416)} $$

   The denominator simplifies to:

   $$ = 10^2 + 31.416^2 = 100 + 987.699 \approx 1087.699 $$

   Now calculating:

   $$ I_1 \approx \frac{2000 - j6283.2}{1087.699} \approx 1.84 - j5.78 , A $$

   For Branch II:

   $$ I_2 = \frac{V}{Z_2} = \frac{200}{-j63.66} = -j3.14 , A $$

4. Calculate Circuit Power Factor

   The power factor (PF) for Branch I (leading or lagging) can be computed as:

   $$ \text{PF} = \cos(\theta) $$ where $\theta = \tan^{-1}\left(\frac{Imaginary}{Real}\right) = \tan^{-1}\left(\frac{31.416}{10}\right)$

   The total current is $I_{total} = I_1 + I_2$.

5. Calculate Power Consumed by Circuit

   The power consumed can be calculated using:

   $$ P = VI\cos(\theta) $$

   Total power can be calculated considering both branches.