Circuits with Resistance and Inductance - Examples

• Introduction to circuits with resistance and inductance
• Definition of resistance and inductance
• Overview of circuit components
• Introduction to the concept of electrical current
• Series and parallel circuits
• Calculation of total resistance in series and parallel circuits
• Calculation of total inductance in series and parallel circuits
• Application of Ohm’s law in circuits with resistance and inductance
• Examples of circuits with resistance and inductance
• Solving problems involving circuits with resistance and inductance

11. Example 1: Series Circuit with Resistance and Inductance

• A circuit with a resistor and an inductor connected in series
• Given values: Resistance (R) = 10 ohms, Inductance (L) = 0.5 H
• Initial current = 2 A
• Find the total impedance of the circuit
• Calculate the current in the circuit using Ohm’s law
• Using the equation for total impedance in a series circuit, calculate the total impedance

12. Example 2: Parallel Circuit with Resistance and Inductance

• A circuit with a resistor and an inductor connected in parallel
• Given values: Resistance (R) = 15 ohms, Inductance (L) = 1 H
• Initial current = 3 A
• Find the total impedance of the circuit
• Calculate the current in the circuit using Ohm’s law
• Using the equation for total impedance in a parallel circuit, calculate the total impedance

13. Example 3: Inductive Reactance

• A circuit with an inductor and an AC power supply
• Given values: Frequency (f) = 50 Hz, Inductance (L) = 0.2 H
• Determine the inductive reactance using the formula XL = 2πfL
• Calculate the impedance of the circuit using the formula Z = √(R^2 + XL^2)
• Find the phase angle using the formula θ = arctan(XL/R)
• Calculate the current in the circuit using Ohm’s law

14. Example 4: Resonant Frequency

• A circuit with a capacitor and an inductor in parallel
• Given values: Capacitance (C) = 10 μF, Inductance (L) = 0.5 H
• Calculate the resonant frequency using the formula fr = 1 / (2π√(LC))
• Find the reactance of the inductor at the resonant frequency using the formula XL = 2πfL
• Calculate the current in the circuit using Ohm’s law

15. Example 5: Series RLC Circuit

• A circuit with a resistor, capacitor, and inductor connected in series
• Given values: Resistance (R) = 5 ohms, Capacitance (C) = 50 μF, Inductance (L) = 0.1 H
• Calculate the total impedance using the formula Z = √(R^2 + (XL - XC)^2)
• Find the resonant frequency using the formula fr = 1 / (2π√(LC))
• Calculate the current in the circuit at the resonant frequency

16. Example 6: Power in an Inductive Circuit

• A circuit with an inductor and a power source
• Given values: Voltage (V) = 10 V, Inductance (L) = 2 H, Frequency (f) = 60 Hz
• Calculate the inductive reactance using the formula XL = 2πfL
• Calculate the power factor using the formula PF = cos(θ)
• Find the phase angle using the formula θ = arctan(XL/R)
• Calculate the power in the circuit using the formula P = V * I * PF

17. Example 7: Energy Stored in an Inductor

• A circuit with an inductor and current flowing through it
• Given values: Inductance (L) = 2 H, Current (I) = 5 A
• Calculate the energy stored in the inductor using the formula E = 0.5 * L * I^2
• Determine the change in energy stored when the current is doubled

18. Example 8: RL Circuit Time Constant

• A circuit with a resistor and an inductor in series
• Given values: Resistance (R) = 5 ohms, Inductance (L) = 0.2 H
• Calculate the time constant using the formula τ = L / R
• Determine the time it takes for the current to reach 63.2% of its final value

19. Example 9: LCR Circuit

• A circuit with a resistor, capacitor, and inductor connected in series
• Given values: Resistance (R) = 10 ohms, Capacitance (C) = 20 μF, Inductance (L) = 0.5 H
• Calculate the resonant frequency using the formula fr = 1 / (2π√(LC))
• Determine the bandwidth using the formula BW = fr / Q, where Q is the quality factor
• Find the half-power frequencies using the formula f1 = fr - (BW / 2) and f2 = fr + (BW / 2)

20. Example 10: AC Circuit Analysis with Resistance and Inductance

• A circuit with a resistor and an inductor in series
• Given values: Voltage (V) = 12 V, Resistance (R) = 8 ohms, Inductance (L) = 0.3 H
• Calculate the inductive reactance using the formula XL = 2πfL, with frequency (f) as a variable
• Determine the phase angle using the formula θ = arctan(XL/R)
• Calculate the current in the circuit using Ohm’s law and the phase angle

21. Example 11: RL Circuit Time Response

• A circuit with a resistor and an inductor in series
• Given values: Resistance (R) = 10 ohms, Inductance (L) = 0.5 H
• Initial current = 0 A
• Determine the time constant using the formula τ = L / R
• Calculate the current at different time intervals using the formula I(t) = I0 * (1 - e^(-t/τ))
• Plot the current-time graph for the circuit

22. Example 12: LC Circuit

• A circuit with a capacitor and an inductor in series
• Given values: Capacitance (C) = 20 μF, Inductance (L) = 0.2 H
• Initial voltage across the capacitor = 10 V
• Calculate the resonant frequency using the formula fr = 1 / (2π√(LC))
• Determine the total energy stored in the circuit using the formula E = 0.5 * C * V^2
• Calculate the maximum charge on the capacitor using the formula Q = C * V

23. Example 13: Power in an Inductive Circuit (AC)

• A circuit with an inductor and an AC power supply
• Given values: Voltage (V) = 20 V, Inductance (L) = 1 H, Frequency (f) = 50 Hz
• Calculate the inductive reactance using the formula XL = 2πfL
• Determine the phase angle using the formula θ = arctan(XL/R)
• Find the power factor using the formula PF = cos(θ)
• Calculate the power in the circuit using the formula P = V * I * PF

24. Example 14: AC Circuit Analysis with Resistance and Inductance (Phasor Diagram)

• A circuit with a resistor and an inductor in series
• Given values: Voltage (V) = 10 V, Resistance (R) = 5 ohms, Inductance (L) = 2 H
• Calculate the inductive reactance using the formula XL = 2πfL, with frequency (f) as a variable
• Determine the impedance of the circuit using the formula Z = √(R^2 + XL^2)
• Draw the phasor diagram for the circuit, showing voltage, current, and impedance

25. Example 15: Power Factor Correction

• A circuit with a power factor of 0.8
• Given values: Apparent power (S) = 1000 VA, Power factor (PF) = 0.8
• Determine the real power using the formula P = S * PF
• Find the reactive power using the formula Q = S * sin(θ)
• Calculate the capacitor value required for power factor correction using the formula C = Q / (2πfV^2)

26. Example 16: Resonance in an AC Series Circuit

• A circuit with a resistor, capacitor, and inductor connected in series
• Given values: Resistance (R) = 10 ohms, Capacitance (C) = 50 μF, Inductance (L) = 0.1 H
• Calculate the resonant frequency using the formula fr = 1 / (2π√(LC))
• Determine the impedance of the circuit at resonance using the formula Z = R
• Calculate the current in the circuit at resonance using Ohm’s law

27. Example 17: Quality Factor of an LCR Circuit

• A circuit with a resistor, capacitor, and inductor connected in series
• Given values: Resistance (R) = 5 ohms, Capacitance (C) = 20 μF, Inductance (L) = 0.2 H
• Calculate the resonant frequency using the formula fr = 1 / (2π√(LC))
• Determine the bandwidth using the formula BW = fr / Q
• Calculate the quality factor using the formula Q = fr / BW

28. Example 18: Power Loss in a Transformer

• A transformer with an efficiency of 95%
• Given values: Input power (Pin) = 400 W, Output power (Pout) = ?
• Determine the output power using the formula Pout = Pin * Efficiency
• Calculate the power loss in the transformer using the formula Power loss = Pin - Pout

29. Example 19: Induced EMF in a Coil

• A coil with a changing magnetic field
• Given values: Number of turns (N) = 200, Magnetic field (B) = 0.5 T, Area (A) = 0.1 m^2, Time (t) = 0.5 s
• Calculate the induced EMF using the formula EMF = -N * dΦ/dt
• Determine the change in magnetic flux using the formula ΔΦ = B * A
• Calculate the induced EMF in the coil

30. Example 20: Self-Inductance of a Solenoid

• A solenoid with a current of 3 A
• Given values: Number of turns per unit length (n) = 1000 turns/m, Length (l) = 0.2 m, Current (I) = 3 A
• Calculate the magnetic field inside the solenoid using the formula B = μ0 * n * I
• Determine the magnetic flux through each turn using the formula Φ = B * A
• Calculate the self-inductance of the solenoid using the formula L = N * Φ / I