In this lecture the following are introduced:
• Resistive-Capacitative transients and RC time constant
• Resistive-Inductive transients and RL time constant

From previous semester - Recap One.

Capacitors

A capacitor is a device that stores electric charge. Any conductor that is formed into an extended surface can accumulate charge and form a capacitor.
A capacitor (device) has capacitance (property).

Capacitors store energy in the electric field between its conducting surfaces.

The energy stored in the electric field is:

A capacitor is an open circuit and no current actually passes across the space between the capacitor surfaces.
However when the current is changing there is the appearance of a current through it because charge flows on or off the surfaces.

end of previous semester material

Resistance-Capacitance transients

When a steady potential difference is switched in and out of a circuit with a resistor and a capacitor in series, there is a transient period i.e. a time while the capacitor charges or discharges.

Charging a Capacitor

In the circuit shown below, the switch is initially at position 1 and there is no current in the circuit.

When the switch is moved from 1 to 2 then current will flow to charge the capacitor.

Define the instantaneous current flowing around the circuit as i, and the instantaneous charge that is stored on the capacitor as q. With the switch in position 2, and moving clockwise around the circuit from the battery, the potentials will be:

As the current (i) flows, charge is flowing onto the capacitor, so the charge stored (q) will increase, i.e.

A loop equation for the instantaneous charge stored on the capacitor can now be written:

To solve for the instantaneous charge, the variables are separated.

The charge stored is zero at time zero, and q at time t.

The solution is:  The equation shows that the instantaneous charge stored on the capacitor approaches the final (steady) charge exponentially. This is shown in the graph.

The current flowing around the circuit is given by the following:

The solution is:  The equation shows that the current jumps to the closed circuit value and then decreases exponentially towards the open circuit zero. This is shown in the graph.

The potential difference across the capacitor follows the shape of the charge curve since v(C)=q/C. The potential difference across the resistor follows the shape of the current curve since v(R)=Ri.

Initially all the potential is across the resistor, and finally all the potential is across the capacitor. The two transient potentials add up to the battery potential at all times.

The Capacitative Time Constant

Since the curves are smooth and approach their final values asymptotically some arbitrary measure of the response speed is needed. The product RC is a logical choice because it has the values of both components and the dimension of time.
RC = [V]/[I] x [Q]/[V] = [Q]/[I] = [T]

The capacitative time constant is defined as, τ = RC, seconds so: 

Some representative times:
When t = RC    

When t = 2RC    

When t = 3RC    

As a rule of thumb the transient has ceased by 3 time constants.

Discharging a Capacitor

Initially, with the switch in position 2, the circuit current is zero and the capacitor has charge Q=CV.

When the switch is moved from 2 to 1 then a current will flow to discharge the capacitor.

Moving clockwise around the circuit from the switch, the potentials will be:

Now since charge is flowing off the capacitor, as the current in the circuit increases, the charge stored will decrease, i.e.

The loop equation for the instantaneous charge stored on the capacitor can now be written and solved for q:

The solution is:  The equation shows that the instantaneous charge stored on the capacitor approaches the final (zero) charge exponentially. This is shown in the graph.

The current flowing around the circuit is given by: This is the same shaped graph as the charging current but the current flows in the opposite direction (shown as a negative in the graph).

The total potential is instantly zero and the two transient potentials add up to zero at all times. The capacitor potential follows the charge stored ( v(C)=q/C ) and the resistor potential follows the current ( v(R)=Ri ).

Example

For the circuit shown below, find the charge on the capacitor and the current in the circuit 0.03 s after the switch is closed.

 

                       

Example

In the circuit shown the switch is moved from 1 to 2, and left there until the capacitor is fully charged. Find (a) the energy supplied by the battery during that time, and (b) the energy dissipated by the resistor during that time, and (c) the energy stored in the capacitor during that time.

The energy supplied by the battery:

The energy dissipated by the resistor:

The energy stored in the capacitor:

Notice that the energy supplied by the battery is CV², with half of this dissipated by the resistor, and half of it stored in the capacitor.

From previous semester - Recap Two.

Magnetic Induction

The magnetic induction in a region of space is measured by the force per unit charge divided by the speed of the charge moving at right angles to the field lines ().

The speed has to be included because the measured magnetic field in the region interacts with the induced magnetic field that comes with the moving charge.

Magnetic Flux

This is the magnetic "flow" of induction through the cross-sectional area of a closed conducting loop.

Magnetic flux has the unit Tesla.m²

Faraday’s Law

An emf is generated in a closed conducting loop when the magnetic flux through it changes. The emf is in a direction that would produce a current and its a consequent magnetic flux, to oppose the initial change in flux.

end of previous semester material

Inductors

The simplest inductor is a long insulated conducting wire (ideally with no resistance) wound into a straight coil with N, turns and length l.

A magnetic induction appears inside each turn of the wire when a current flows through the inductor, thus producing an induction through the whole of the interior volume of the coil.

The induction may be taken as constant through most of the length and equal to: 

If the current changes then the induction changes, and from Faraday's law, an opposing emf will be induced.

If the current is increasing then the inductor acts like an emf sending a current backwards, to try and get the current down to what it was.

If the current is decreasing then the inductor acts like an emf sending more current forwards, to try and get the current up to what it was.

Putting the factors together:

The Self Inductance of the coil is defined by: Now: so eliminating the time from each side of the equation, and dividing by current, the unit of inductance will be flux/current or T.m2.A-1. It was given the special name of the Henry

Resistance-Inductance transients

When a steady potential difference is switched in and out of a circuit with a resistor and an inductor in series there is a transient period while the current reaches a steady value.

Increasing the Current in an Inductor

For the circuit shown, the switch is initially at position 1 and there is no current in the circuit.

When the switch is moved from position 1 to 2, since the coil ideally has no resistance, then current will flow till it reaches a steady value of V/R.

Moving clockwise around the circuit from the battery, the potentials will be:

Solving for the circuit current

The solution is: The equation shows that the current through the inductor approaches the final (steady) current exponentially. This is shown in the graph.

The potential difference across the inductor is given by: The equation shows that the potential difference across the inductor jumps to the battery potential and then decreases to zero exponentially. This is shown in the graph.

The potential difference across the resistor will follow the rise of the current since V = RI. The potential difference across the inductor follows how quickly the current changes. This is shown in the graaph.

When the current through an inductor increases from zero: initially the potential is all across the inductor and finally the potential is all across the resistor. The two transient potentials add up to the battery potential at all times.

The Inductive Time Constant

The speed of response is measured with the same concept as the capacitor response. The quotient L/R is a logical choice because it has both component values and the dimension of time.

The inductive time constant is defined as, τ = L/R, seconds so:  ,   and  

Decreasing the Current in an Inductor

Initially, the circuit current is V/R.

When the switch is moved from 2 to 1 the supply potential is cut off and the inductor will produce a forward current to try to make up the decrease.

Moving clockwise around the circuit from the switch, the potentials will be:

Solving for the circuit current:

The solution is: The equation shows that the current will start at the original value and then decrease to zero exponentially. This is shown in the graph.

Using the equation for the current, the potential difference across the inductor is given by: The potential difference switches sign and reduces exponentially to zero as shown in the graph.

The potential difference across the resistor will (as usual) follow the current.

As the total potential is instantly zero, the two transient potentials will add up to zero.

Example

For the circuit shown. Find (a) the time constant τL, and (b) the current flowing at time, τL, and (c) the potential difference across the inductor at time, τL.

Summarising:

Charging a capacitor: and

Discharging a capacitor: and

Capacitative time constant:

Increasing the Current in an Inductor: and

Decreasing the Current in an Inductor: and

Inductive time constant:

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