## Capacitors and Inductors

Objectives: We want to be able to
• use the element equations of the capacitor and inductor to calculate the element voltage from the element current or visa versa
• analyze a switched dc circuit containing capacitors and inductors.
• replace series or parallel combinations of capacitors by an equivalent capacitor.
• replace series or parallel combinations of inductors by an equivalent inductor.
• determine the energy stored by a capacitors or an inductor.
• represent a first or second order circuit by a differential equation.

Reading: Sections 2.9, 7.2 thru 7.11, 8.2 and 9.2.

1. Section 7.2 describes capacitors, giving the equations that relate the capacitor current and voltage. In contrast to circuit elements introduced previously, the constitutive equations of a capacitor involve integration and differentiation.
1. Section 7.2 contains several examples in which the constitutive equations of the capacitor are used to calculate the capacitor voltage from the capacitor current and visa versa.
2. Section 7.11 shows how we can check that these calculations have been done correctly.
3. The capacitor current and voltage can be complicated functions of time. Section 7.10 uses MATLAB to plot these functions so that we can see what they look like.
2. Capacitors store energy. Section 7.3 shows that we can determine how much energy is stored from the capacitor voltage.
3. Section 7.4 shows how to determine the equivalent capacitance of series or parallel capacitors.
4. Section 7.5 describes inductors, giving the equations that relate the inductor current and voltage. Similar to a capacitor, the constitutive equations of a inductor involve integration and differentiation. Section 7.5 contains several examples in which the constitutive equations of the inductor are used to calculate the inductor volatage from the inductor currrent and visa versa.
5. Inductors store energy. Section 7.6 shows that we can determine how much energy is stored from the inductor current.
6. Section 7.7 shows how to detemine the equivalent inductance of series or parallel inductors.
7. Circuits containing capacitor and inductors are represented by differential equations. Generally, the order of the differential equation is equal to the number of capacitors in the circuit plus the number of inductors.
1. Circuits containing either a single capacitor or a single inductor are "first order circuits". Section 8.2 shows how to represent thses circuits by a differential equation.
2. Capacitors and inductors are called "reactive elements". Circuits containing two reactive elements are "second order circuits". Section 9.2 shows how to represent these circuits by a differential equation.
8. We are frequently interested in the reponse of circuits to abrupt changes, such as the opening or closing of a switch. Section 2.9 describes switches.

Lecture Notes:

1. Some observations regarding capacitors and inductors.
2. The constitutive equations of the capacitor or inductor are used to calculate the voltage from the current and visa versa.
1. In these examples the capacitor current and voltage are represented analytically.
2. In these examples the capacitor current and voltage are represented graphically.
3. Series and parallel capacitors can be replaced by equivalent capacitors.
4. Series and parallel inductors can be replaced by equivalent inductors.
5. Having included capacitors and inductors in our circuits, we need to distinguish between several types of circuit.
6. Circuits without capacitors and inductors are represented by algebraic eqautions, but we nee to represent circuits with capacitors and inductors by differential equations.

Handouts: Exercises and solutions.

### On-line Exercises:

1. Capacitors and Inductors