Inductance and rl circuits

Self-Induction in a single loop

After switch ‘S’ is closed, the current produces a magnetic flux through the area enclosed by the loop. As the current increases toward its equilibrium value, this magnetic flux changes in time and induces an emf in the loop (Lenz’s law).

The battery symbol drawn with dashed lines represents the self induced emf (also called back emf).

Self-Induction in coil

• An induced emf is always proportional to the time rate of change of the current. The emf is proportional to the flux, which is proportional to the B field and the field is proportional to the current I:

• L is a constant of proportionality called the inductance of the coil and it depends mainly on the geometry of the coil. The unit of inductance is the henry (H), equivalent to 1 volt-second per ampere.

• Proof I: Show that the inductance of a coil of N windings depends on the coil’s geometry:

RL circuit with a battery source  (switch is at a)

• An RL circuit contains an inductor and a resistor.

• Assume S₂ is connected to point a.

• When switch S₁ is closed (at time t = 0), the current begins to increase.

• At the same time, a “back” emf is induced in the inductor that opposes the original increasing current.

Proof II(a): RL circuit with a battery source  (switch is at a)

• Applying Kirchhoff’s loop rule to the previous circuit yields:

• Solving this for the current, we find:

RL circuit without a battery  (switch is at b)

• The circuit now contains just the right hand loop;

• The battery has been eliminated.

• The expression for the current becomes:

Magnetic energy stored in a solenoid

Consider the power delivered by the battery

• It means that the power delivered to the coil is:

Alternating Current Circuits

• An AC circuit consists of a combination of circuit elements such as R, C or L (or combinations of them) and a power source.

• The power source provides an alternating voltage, v.

• Notation:

  • Lower case symbols will indicate instantaneous values.

  • Capital letters will indicate fixed values.

• The output of an AC power source is sinusoidal and varies with time according to the following equation:

• Δv is the instantaneous voltage

• ΔV_max is the maximum output voltage of the source -also called the voltage amplitude

• ω is the angular frequency of the AC voltage.

• The voltage is positive during one half of the cycle and negative during the other half.

The angular frequency is:

• ƒ is the frequency of the source (Hz)

• T is the period of the source (s)

• Commercial electric power plants in most of countries use a frequency of 50 Hz, though in US and UK the typical value of frequency is 60 Hz

• This corresponds with an angular frequency of 314 rad/s and 377 rad/s respectively.

Resistor connected to an AC power source

• Δv_R is the instantaneous voltage across the resistor.

• Kirchhoff’s 2^nd law gives:

• Ohm’s Law across R gives:

B oth the current and the voltage reach their maximum values at the same time. They are said to be in phase.

Phasor Diagrams

To simplify the analysis of AC circuits a phasor diagram can be used.

• A phasor is a vector whose length is proportional to the maximum value of the variable it represents.

• The vector rotates counterclockwise at an angular speed equal to the angular frequency ω associated with the variable.

• The projection of the phasor onto the vertical axis represents the instantaneous value of the quantity it represents, (e.g. current i_R or voltage v_R).

Solenoid connected to an AC source

• Kirchhoff’s loop rule can be applied

• 

Note that ≠ 0 at t=0

But V_max / I_max has units of Ohms. Therefore ωL must represent some kind of “resistance” of the solenoid when subjected to AC voltages. Indeed it is called impedance X_L. (Impede = prevent)

Average Power and rms values of i_R and v_R

Consider a resistor R again that is connected to an AC power supply.

• The rate at which electrical energy is dissipated in the resistor R is given by:

• i_R is the instantaneous current.

• The heating effect produced by an AC current with a maximum value of I_max is not the same as that of a DC current of the same value.

• The maximum current occurs for a small amount of time.

• The root mean square value of current i_R is defined as the equivalent direct current that would cause exactly the same average heating effect across the resistor R over a time of one complete period T.

• T he average power P_avrg of an AC generator over one complete cycle dissipated across a resistor R was:

• The mean power dissipated by an inductor L over one period T is zero. Power is absorbed by the solenoid during half the period, and given out in the circuit over the other half. The instantaneous power is:

Similarly, the mean power dissipated by a Capacitor C over one period T is zero.

AC Circuits

• AC power source characteristic:

• Solenoid connected to AC power source:

• Capacitor connected to AC power source:

• AC power source characteristic:

• Solenoid connected to AC power source:

• Capacitor connected to AC power source:

Phasor diagrams

“C.I.V.I.L” Capacitor - I leads V , Inductor – V leads I

C ombining the phasor diagrams

The individual phasor diagrams can be combined.

A single phasor I_max is used to represent the current in each element.

In series, the current is the same in each element

Vector addition is used to combine the voltage phasors.

ΔVL max and ΔVC max are in opposite directions so they subtract.

Their resultant is perpendicular to ΔVRmax

From the vector diagram, ΔV_max can be calculated:

• The total impedance Z of the RLC circuit:

AC power delivered to an RLC series circuit

• We have already proved in L 41 that the average power delivered to an AC circuit is dissipated only on the resistor R as heat.

• The Inductor and Capacitor do not dissipate any power from the source over one period.

• Therefore all the power that is dissipated from the AC source over one period T is :

• But Ohm’s law applies between I_max, V_max and Z so that: I_max = V_max/Z

• The factor cosϕ is called the power factor.

Resonance (Maximum Power Transfer)

• It is clear from (eq. 8) that the average power delivered to an RLC series circuit by the AC source is maximum when:

• Then from (eq.7) we see that X_L = X_C , which is true when ω takes the particular value:

ω₀ - resonant angular frequency of the RLC circuit.

• When the AC source voltage oscillates at ω₀ the impedance Z of the RLC combination reaches a minimum value. Equation 6 gives:

• As a result of Ohm’s law the amplitude of the current I_max reaches its highest value for a given voltage V_max of the AC source.

T ransformers

• An AC transformer consists of two coils of wire wound around a core of iron

• The side connected to the input AC voltage source is called the primary and has N₁ turns

• The other side, called the secondary, is connected to a resistor and has N₂ turns

• The core is used to increase the magnetic flux and to provide a medium for the flux to pass from one coil to the other (but it is not necessary)

• Assume an ideal transformer

  • One in which the energy losses in the windings and the core are zero

• The rate of change of the flux is the same for both coils.

• In the primary coil we have:

• In the secondary coil:

The defining law of transformers becomes:

If N₂ > N₁ – step up transformer, v₂ increases

If N₂ < N₁ – step down transformer, v₂ decreases