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RL and RC Circuits

Review of Capacitance and Inductance

Capacitance

A capacitor consists of two metal electrodes which can be given equal and opposite charges.

If the electrodes have charges Q and Q, then there is an electric field between them whichoriginates on Q and terminates on Q. There is also a potential difference between the

electrodes which is proportional to Q. The capacitance of the configuration is defined as

The capacitance is a measure of the capacity of the electrodes to hold charge for a givenpotential difference.

The symbol of capacitance is shown below:

Combinations of Capacitors

Capacitors in parallel have the same electrical potential as the source. Thus, the charge onthe equivalent capacitor is the sum of the charges on C1 and C2.

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For capacitors connected in series, the potential of the source is the sum of the potentialsacross C1 and C2. This is a consequence of conservation of energy.

or

Inductance

Inductance is the property of a conductor by which a change in current in the conductor"induces" (creates) a voltage (electromotive force) in both the conductor itself (self-

inductance) and in any nearby conductors (mutual inductance).

Symbol for inductance:

Inductance is measured in Henry (H).

The relationship between the self-inductance L of an electrical circuit (in henries), voltage v,and current i is

Combinations of Inductors

Inductors connected in series are added normally

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Example:

If inductors are connected in parallel, they add in reciprocal

Example:

***

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First Order Circuits

RC Circuita circuit comprising a resistor and a capacitor

RL Circuita circuit comprising a resistor and inductor

Capacitors and inductors are storage elements

Two ways to excite a first-order circuit:

Through initial conditions of the storage elements (source-free circuits)

Through the independent sources

Source-free circuits we assume that energy is initially stored in the capacitive or

inductive element.

Source-free circuits may have dependent sources, but do not have independent sources.

The Source-Free RC Circuit

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The natural respond depends on the nature of the circuit alone, with no external sources. In

fact, the circuit has a response only because of the energy initially stored in the capacitor.

Voltage response of the RC circuit:

(unit in seconds)

In terms of the time constant,

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The Source-Free RL Circuit

Consider the series connection of a resistor and an inductor as shown in the circuit below. Our

goal is to determine the circuit response, which we will assume to be the current i(t)through

the inductor. We select the inductor current as the response in order to take advantage of the

idea that the inductor has an initial current I0, or

()

As shown on the graph below, the natural response of the RL circuit is an exponential decay of

the initial current.

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The time constant for the RL circuit is:

Thus, the current in terms of the time constant is:

When a circuit has a single inductor and several resistors and dependent sources, the Theveninequivalent can be found at the terminals of the inductor to form a simple RL circuit. Also,, one

can use Thevenins theorem when several inductors can be combined to form a single

equivalent inductor.

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