We study electric circuits as an application of second order linear differential equations.

### The Circuit

In this section we consider the *RLC circuit*, shown schematically in the figure
below.

As we’ll see, the circuit is an electrical analog of a spring-mass system with damping.

Nothing happens while the switch is open (dashed line). When the switch is closed
(solid line) we say that the *circuit is closed*. Differences in electrical potential in a
closed circuit cause current to flow in the circuit. The battery or generator, shown in
the figure, creates a difference in electrical potential between its two terminals,
which we’ve marked arbitrarily as positive and negative. (We could just as well
interchange the markings.) We’ll say that if the potential at the positive terminal is
greater than the potential at the negative terminal, if the potential at the
positive terminal is less than the potential at the negative terminal, and
if the potential is the same at the two terminals. We call the *impressed
voltage*.

At any time , the same current flows in all points of the circuit. We denote current by . We say that if the direction of flow is around the circuit from the positive terminal of the battery or generator back to the negative terminal, as indicated by the arrows in the figure, if the flow is in the opposite direction, and if no current flows at time .

Differences in potential occur at the resistor, induction coil, and capacitor. Note that
the two sides of each of these components are also identified as positive and negative.
The *voltage drop across* each component is defined to be the potential on the
positive side of the component minus the potential on the negative side.
This terminology is somewhat misleading, since “drop” suggests a decrease
even though changes in potential are signed quantities and therefore may be
increases. Nevertheless, we’ll go along with tradition and call them voltage
drops. The voltage drop across the resistor, as shown in the figure, is given by

*resistance*of the resistor. The voltage drop across the induction coil is given by where is a positive constant, the

*inductance*of the coil.

A capacitor stores electrical charge , which is related to the current in the circuit by the equation

where is the charge on the capacitor at . The voltage drop across a capacitor is given by where is a positive constant, the*capacitance*of the capacitor.

The following table names the units for the quantities that we’ve discussed. The units are defined so that

Symbol | Name | Unit |

Impressed Voltage | volt | |

Current | ampere | |

Charge | coulomb | |

Resistance | ohm | |

Inductance | henry | |

Capacitance | farad | |

According to Kirchoff’s Law, the sum of the voltage drops in a closed circuit equals the impressed voltage. Therefore, from (eq:6.3.1), (eq:6.3.2), and (eq:6.3.4),

This equation contains two unknowns, the current in the circuit and the charge on the capacitor. However, (eq:6.3.3) implies that , so (eq:6.3.5) can be converted into the second order equation in . To find the current flowing in an circuit, we solve (eq:6.3.6) for and then differentiate the solution to obtain .In Trench 6.1 and 6.2 we encountered the equation

in connection with spring-mass systems. Except for notation this equation is the same as (eq:6.3.6). The correspondence between electrical and mechanical quantities connected with (eq:6.3.6) and (eq:6.3.7) is shown in the table below.

Electrical | Mechanical | ||

charge | displacement | ||

curent | velocity | ||

impressed voltage | external force | ||

inductance | Mass | ||

resistance | damping | ||

1/capacitance | spring constant | ||

The equivalence between (eq:6.3.6) and (eq:6.3.7) is an example of how mathematics unifies fundamental similarities in diverse physical phenomena. Since we’ve already studied the properties of solutions of (eq:6.3.7) in In Trench 6.1 and 6.2, we can obtain results concerning solutions of (eq:6.3.6) by simply changing notation, according to the table.

#### Free Oscillations

We say that an circuit is in *free oscillation* if for , so that (eq:6.3.6) becomes

Case 1. The oscillation is *underdamped* if . In this case, and in (eq:6.3.9) are complex
conjugates, which we write as
where
The general solution of (eq:6.3.8) is
which we can write as

Actual circuits are usually underdamped, so the case we’ve just considered is the most important. However, for completeness we’ll consider the other two possibilities.

Case 2. The oscillation is *overdamped* if . In this case, the zeros and of the
characteristic polynomial are real, with (see (eq:6.3.9)), and the general solution of (eq:6.3.8) is

Case 3. The oscillation is *critically damped* if . In this case, and the general solution
of (eq:6.3.8) is

If , the exponentials in (eq:6.3.10), (eq:6.3.11), and (eq:6.3.12) are negative, so the solution of any homogeneous
initial value problem
approaches zero exponentially as . Thus, all such solutions are *transient*, in the sense
defined In Trench 6.2 in the discussion of forced vibrations of a spring-mass system
with damping.

The characteristic equation of (eq:6.3.13) is which has complex zeros . Therefore the general solution of (eq:6.3.13) is

Differentiating this and collecting like terms yields To find the solution of the initial value problem (eq:6.3.14), we set in (eq:6.3.15) and (eq:6.3.16) to obtain therefore, and , so is the solution of (eq:6.3.14). Differentiating this yields#### Forced Oscillations With Damping

An initial value problem for (eq:6.3.6) has the form

where is the initial charge on the capacitor and is the initial current in the circuit. We’ve already seen that if then all solutions of (eq:6.3.17) are transient. If , we know that the solution of (eq:6.3.17) has the form , where satisfies the complementary equation, and approaches zero exponentially as for any initial conditions , while depends only on and is independent of the initial conditions. As in the case of forced oscillations of a spring-mass system with damping, we call the*steady state*charge on the capacitor of the circuit. Since and also tends to zero exponentially as , we say that is the

*transient*current and is the

*steady state*current. In most applications we’re interested only in the steady state charge and current.

Electrical | Mechanical | ||

charge | displacement | ||

curent | velocity | ||

impressed voltage | external force | ||

inductance | Mass | ||

resistance | damping | ||

1/capacitance | spring constant | ||

yields the steady state charge where Therefore the steady state current in the circuit is

### Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)