Lecture - 4 Inductance and capacitance equivalent circuits

1. Lecture - 4 Inductance and capacitance equivalent circuits 

2. Outline • Introduction. • The inductor. • The inductor v-i equation. • Power and energy in an inductor. • Series and parallel inductors. • The capacitor • The capacitor v-i equation. • Power and energy in a capacitor. • Series and parallel capacitors.

3. Introduction • Resistor: a passive element which dissipates energy only. • Two important passive linear circuit elements: • Capacitor • Inductor • Inductors and capacitors are passive elements; they can store and release energy, but they cannot generate or dissipate energy. • The instantaneous power at the terminals of an inductor or capacitor can be positive or negative, depending on whether energy is being delivered to or extracted from the element.

4. The inductor i • Inductance is the circuit parameter used to describe an inductor. • Inductance is symbolized by the letter L, is measured in henrys (H). • It is represented graphically as a coiled wire. • An inductor consists of a coil of conducting wire. • An inductor is a passive element designed to store energyin the magnetic field. + v L -

5. The inductor v-i equation • When the current through an inductor is a constant, then the voltage across the inductor is zero, same as a short circuit. • No sudden change of the current through an inductor is possible except an infinite voltage across the inductor is applied which is not possible. • Switching inductive circuits is an important engineering problem, because arcing and voltage surges must be controlled to prevent equipment damage.

6. Power and Energy in an inductor • The power and energy relationships for an inductor can be derived directly from the current and voltage relationships.

7. Series and Parallel Inductors • The equivalent inductance of series connected inductors is the sum of the individual inductances: • The equivalent inductance of parallel connected inductors is the reciprocal of the sum of the reciprocals of the individual inductors: • The inductor in various connection has the same effect as the resistor. Hence, the Y-Δ transformation of inductors can be similarly derived.

8. Example 1 The current through a 0.1 H inductor is i(t) = 10te-5t A. Find the voltage across the inductor and the energy stored in it. -------------------------------------------

9. Example 2 • Assume that the initial energy stored in the inductors of the figure is zero. Find the equivalent inductance with respect to the terminals a,b. -------------------------------------------- 30||20 = 12H 80||(8 + 12) = 16H 60||(14 + 16) = 20H 15||(20 + 10) = 20H Lab= 5 + 10 = 15H

10. The Capacitor • The circuit parameter of capacitance is represented by the letter C. • It is measured in farads (F) and it is symbolized graphically by two short parallel conductive plates. • A capacitor is a passive element designed to store energyin the electric field.

11. The capacitor v-i equation • If v is a constant voltage, then i=0; a constant voltage across a capacitor creates no current through the capacitor, the capacitor in this case is the same as an open circuit. • Voltage cannot change instantaneously across the terminals of a capacitor because such a change would produce infinite current, a physical impossibility. • Every practical capacitor has a maximum limit on its operating voltage in order not to break down the insulation of the dielectric.

12. Power and Energy in a capacitor • The power and energy relationships for an inductor can be derived directly from the current and voltage relationships.

13. Series and Parallel Capacitors • The equivalent inductance of series connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances: • The equivalent inductance of parallel connected capacitors is the sum of the individual capacitors: • These results enable us to look the capacitor in this way: 1/C has the equivalent effect as the resistance. The equivalent capacitor of capacitors connected in parallel or series can be obtained via this point of view, so is the Y-△ connection and its transformation.

14. Example 3 The voltage at the terminals of the 0.6 µF capacitor shown in the figure is 0 for t < 0 and 40e-15,000tsin30,000tV for t ≥ 0. Find: • i(0); • the power delivered to the capacitor at t = π /80 ms; • the energy stored in the capacitor at t = π /80 ms.

15. Example 3

16. Example 4 • Find the equivalent capacitance with respect to the terminals a,b for the circuit shown in the figure.

17. Example 4

18. Summary • Inductance is a linear circuit parameter that relates the voltage induced by a time-varying magnetic field to the current producing the field. • Capacitance is a linear circuit parameter that relates the current induced by a time-varying electric field to the voltage producing the field. • Inductors and capacitors are passive elements; they can store and release energy, but they cannot generate or dissipate energy. • The instantaneous power at the terminals of an inductor or capacitor can be positive or negative, depending on whether energy is being delivered to or extracted from the element.

19. Summary • An inductor: • does not permit an instantaneous change in its terminal current. • does permit an instantaneous change in its terminal voltage. • behaves as a short circuit in the presence of a constant terminal current. • A capacitor: • does not permit an instantaneous change in its terminal voltage. • does permit an instantaneous change in its terminal current. • behaves as an open circuit in the presence of a constant terminal voltage.

20. Summary • Series connected: • Parallel connected: