Question

- An AC series circuit consists of a resistor with resistance of \( 90 \Omega \), a coil with inductance of \( 1.3 H \) and a capacitor with capacitance of \( 10 \mu F \). The circuit is connected to an AC voltage source with amplitude of \( 100 V \) and frequency of \( 50 Hz \). Write an equation for instantaneous values of voltage and current in the circuit, if the initial phase of the current is zero. Use a phasor diagram to answer the above question.

Answer 1

To find the instantaneous values of voltage and current, as well as interpret the situation using a phasor diagram, we'll solve step-by-step:

Step 1: Calculate angular frequency

The angular frequency $$\omega$$ for the AC circuit is given as: $$\omega = 2\pi f$$ Substituting $$f = 50 \, \text{Hz}$$: $$\omega = 2\pi \times 50 = 314 \, \text{rad/s}$$

Step 2: Calculate impedances

• Inductive reactance $$X_L$$: $$X_L = \omega L = 314 \times 1.3 = 408.2 \, \Omega$$

• Capacitive reactance $$X_C$$: $$X_C = \frac{1}{\omega C} = \frac{1}{314 \times 10 \times 10^{-6}} = 318.47 \, \Omega$$

• Net reactance $$X$$: $$X = X_L - X_C = 408.2 - 318.47 = 89.73 \, \Omega

• Step 3: Impedance of the circuit

  The total impedance $$Z$$ is calculated as: $$Z = \sqrt{R^2 + X^2}$$ $$Z = \sqrt{90^2 + 89.73^2} \approx 127.1 \, \Omega$$

  Step 4: Current amplitude

  The amplitude of current $$I_0$$ can be found using Ohm's Law: $$I_0 = \frac{V_0}{Z}$$ $$I_0 = \frac{100}{127.1} \approx 0.787 \, \text{A}$$

• Step 5: Instantaneous current equation

  With the initial phase of current set to zero, the instantaneous current $$i(t)$$ is: $$i(t) = I_0 \sin(\omega t)$$ $$i(t) = 0.787 \sin(314t) \, \text{A}$$

  Step 6: Instantaneous voltage equation

  The voltage $$v(t)$$ is given by: $$v(t) = V_0 \sin(\omega t + \phi)$$

  Here, the phase angle $$\phi$$ is determined by: $$\tan \phi = \frac{X}{R} = \frac{89.73}{90} \approx 0.996$$ $$\phi = \tan^{-1}(0.996) \approx 44.8^\circ$$

  Thus, the instantaneous voltage becomes: $$v(t) = 100 \sin(314t + 44.8^\circ) \, \text{V}$$

• Step 7: Phasor diagram

  The phasor diagram illustrates:

• Voltage across the resistor in phase with current.

• Voltage across the inductor leading the current by $$90^\circ$$.

• Let me know if you'd like to go further, such as sketching phasors conceptually or exploring resonance in AC circuits!

• Voltage across the capacitor lagging the current by $$90^\circ$$. The resultant voltage, considering phase differences, aligns with the applied voltage.