Serial LR Circuits (Resistance Coil)

Serial LR Circuits are obtained by serially connecting the resistance and coil to each other.

In our first articleabout inductors , we briefly looked at the time constant of an inductor, indicating that the current flowing from an inductor cannot change instantly, but will increase at a constant rate determined by the self-induced emk in the inductor.

In other words, an inductor in an electrical circuit resists the passing of the current through ( i). Although this is completely true, in the tutorial we hypothesed that it is an ideal inductor without resistance or capacitance associated with coil windings.

However, in real use, coils, solenoids, relays or any bandage component will always have a certain resistance, no matter how small. This is due to the use of actual coil rotations of the wire used to use copper wire, which has a resistance value.

Series LR Circuits consist mainly of an inductor and resistance.

Serial LR Circuits
Serial LR Circuit

The Above Serial LR circuit is connected by a constant voltage source and a switch. Suppose the switch (S) is turned on until t = 0 instantly closes, and then remains permanently turned off by generating a voltage input of type "step response/step response". The current begins to flow from the i circuit, but does not rapidly rise to the maximum Imax value determined by the V/R (Ohm Act) ratio.

This limiting factor is due to the presence of self-induced emf in the inductor as a result of the growth of magnetic influx (Lenz Law). After a while, the voltage source neutralizes the effect of the self-induced emk, the current flow becomes constant, and the induced current and field drops to zero.

We can use Kirchhoff's Voltage Law (KVL) to describe the individual voltage drops that exist around the circuit:

Serial LR Circuits

Voltage drop in resistance is R, = I*R (Ohm Act). You can also easily make your ohm law calculations with the calculator we have created here.

Serial LR Circuits

The voltage drop on the inductor, L is now our familiar expression L(di/dt).

Serial LR Circuits

Next, the final statement for individual voltage drops around the Series LR circuit can be given as follows:

Serial LR Circuits

We can see that the voltage drop in resistance depends on the current, the i, while the voltage drop in the inductor depends on the speed of change of the current, di/dt. When the current is equal to zero, the above expression, which is also a quadratic differential equation at the same time as ( i = 0 ) t = 0, can be rewritten to give the value of any current at any time:

Serial LR Circuits
Current Expression in Serial LR Circuits

Where:
V is in Volts R is in
Ohm
L, Henry t e in seconds
is the base of the Natural
Logarithm = 2.71828

The Time Constant of the Serial LR circuit is given as ( τ ), L/R, where V/R represents the last stable state current value after the five time constant values when it reaches this maximum stable state value in Current 5τ, the coil's inductive acts more like a short circuit, reducing the coil to zero and effectively removing the coil from the circuit.

Therefore, the current flowing from the coil is limited only to the resistant element of the coil windings in the gen. Ohm. It can be presented as a graphical representation of current growth representing the voltage/time characteristics of the circuit.

Serial LR Circuits
Curves for Serial LR Circuit

Since the voltage drop in resistance is equal to VR I*R (Ohm Law), it will have the same exponential growth and shape as the current. However, the voltage drop on the inductor, VL will have a value equal to : And(-Rt/L). Then the voltage on the inductor will have a starting value equal to the source voltage when VL, t = 0 is instantly or when the switch is first turned off, and then it will exponentially reduce to zero, as shown in the curves above.

The time required for the current flowing in the LR series circuit to reach the maximum stable state value is equal to approximately 5 time constants or 5τ. This time constant τ is measured in seconds τ = L/R, where R is the value of resistance in ohm and L is the value of the inductor in Henry. This then forms the basis of an RL charging circuit, 5τ can also be considered as "5*(L/R)" or temporary duration of the circuit.

The transition time of any inductive circuit is determined by the relationship between inductive and resistance. For example, the larger the inducing for a fixed-value resistance, the slower the transition time, and therefore a longer time constant for serial LR circuits. Likewise, the smaller the resistance value for a constant inducing value, the longer the transition time.

However, the resistance value for a fixed-value inducing is increased, shortening the transition time and therefore the time constant of the circuit. This is because as resistance increases, the circuit becomes increasingly resistant, as the value of inductees becomes negligible compared to resistance. If the value of resistance is increased large enough compared to inducing, the transition time will effectively be reduced to almost zero.

Serial LR Circuit Question Example 1

A coil with an induced 40mH and a resistance of 2Ω is connected to form an LR series circuit. If they are connected to the 20V DC feed:

a) What will be the final stable state value of the current:

Serial LR Circuits

b) What will be the time constant of the RL series circuit:

Serial LR Circuits

c) What will be the transition time of the RL series circuit:

Serial LR Circuits

d) What will be the value of the emk induced after 10 ms:

Serial LR Circuits

e) What is the time constant value of the circuit current after the switch is closed?

Serial LR Circuits

The Circuit's Time Constant is calculated as 20ms in τ, question b). Then the current circuit current at this time is calculated as follows:

Serial LR Circuits

You may have noticed that the answer to the question (e), which gives a time constant of 6.32 Amps, is equal to 63.2% of the last stable state current value of 10 Amps that we calculated in question (a). This 63.2% or 0.632 x IMAX also corresponds to the temporary curves shown above.

Power in Serial LR Circuits

Serial LR Circuits

The instantaneous speed of the power distributed by the resistance in the form of heat is given as follows:

Serial LR Circuits

The ratio of energy stored in the inductor in the form of magnetic potential energy is given as follows:

Serial LR Circuits

Then we can find the total power in the RL series circuit by multiplying it by i, and therefore:

Serial LR Circuits