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|16.15||Introduction to RLC Circuits|
In this Physics tutorial, you will learn:
In the previous tutorial, we have discussed about LC circuits. What happens if we add a resistor in this circuit? Recall what we said in the previous tutorial about the resistance of LC circuits and how it affects the energy distribution in the circuit.
Do you think capacitors have resistance? What about inductors? Recall how an inductor behaves after a long time of circuit operation. What can you say about the resistance of inductor at the beginning of process?
Now, let's explain these questions by considering the most inclusive type of circuit - the RLC circuit. For simplicity, here we will consider only the series RLC circuit.
By definition, a RLC circuit is a circuit that contains at least a resistor, a capacitor and an inductor. The simplest RLC circuits are series RLC circuits, like the one shown in the figure below.
As stated in the previous tutorial, when the resistance of circuit is considered, the total energy (electric plus magnetic) in the circuit does not remain constant but it rather decreases with time as some of this energy converts into thermal energy and is therefore dissipated in the form of heat by the resistor. As a result, the energy in the circuit will decrease after each cycle; the oscillations fade and becomes less visible until they disappear completely.
Due to this decrease in the electromagnetic energy of the system, all related quantities such as current, charge and potential difference will decrease too, as there is a continuous decrease in the amplitude of oscillations. We say the oscillations are damped, exactly as occurs in a real block-and-spring oscillation system, in which the amplitude of oscillations decreases after each cycle because of friction.
In tutorial 10.2 "Energy in Simple Harmonic Oscillations" we have provided a general overview on what damped oscillations are. Let's recall a passage from the core paragraph of that tutorial:
"If we make a system oscillate, the amplitude of oscillations would decrease until it becomes zero if energy is not provided continuously to the system. In this case, the oscillations fade away with time. Such oscillations are known as damped oscillations the amplitude of which decreases with time. Look at the graph below. ,/i>
The "envelope" that surrounds the graph of damped oscillations represents a decreasing exponential function of the form x(t) = A0 ∙ e-γ ∙ t where A0 is the initial amplitude and γ is a constant. Despite the amplitude decreases with time, this is still a simple harmonic motion because the other quantities such as period and frequency remain unchanged. This kind of SHM has the equation
On the other hand, in sustainable SHM, the amplitude of oscillations does not change with time. The envelope shows a horizontal function of the type x(t) = A0."
In the next paragraph, we will see that a RLC circuit without a sustainable energy supply from outside, represents a system of damped oscillations because the energy of system decreases with time. This occurs because part of the total energy of the system converts into thermal energy and is therefore dissipated in the form of heat from the resistor to the environment. This brings a continuous decrease in the total energy of the system.
In the previous tutorial, we have explained that the total electromagnetic energy in a LC circuit is
When a resistor is added in this circuit, the total electromagnetic energy will decrease at a rate of
because some of the energy in the system turns into thermal energy of resistor and is dissipated in the environment in the form of heat energy. The negative sign in the equation means that the total energy of the system decreases.
Differentiating the above equation with time, we obtain
Since i = dQ/dt, and di/dt = d2 Q/dt2 we obtain
Simplifying both sides of the last equation by dQ/dt, we obtain
This is the differential equation for the damped oscillations in a RLC circuit.
The charge decay in such a circuit is calculated through an expression, which is a combination of exponential and sinusoidal equation, as occurs in all types of damped oscillations. Thus, the charge left in a RLC circuit after a given time t of operation is found by:
is the angular frequency of damped oscillations and
is the angular frequency of undamped oscillations.
In addition, you can see that the amplitude also contains an exponential decaying term e-R ∙ t/2L. This means the amplitude of every successive oscillation is smaller than the previous one as the power of Euler's Number e here is negative.
The potential difference between the plates of a 5μF capacitor connected in an alternating 50Hz RLC circuit shown in the figure is 20V and the resistance of resistor in the circuit has a value of 0.25Ω.
Take the initial phase as zero.
From the above results, we draw the following conclusions:
For small values of resistance, ω' ≈ ω, so we can neglect the (R/2L)2 factor and focus only on the value of ω.
We can use a similar approach when calculating the energy decay in a RLC circuit operating without an external energy supply. For this, we observe what happens to the electric energy in the capacitor. Since
we can write this expression as a function of time, i.e.
This means the energy of the electric field in a RLC circuit oscillates in a cos2 fashion while the amplitude decreases exponentially with time.
A 50Hz RLC circuit contains a 10Ω resistor, a 0.4H inductor and a 2nF capacitor connected in series. The capacitor initially stores a charge of 5μC. Calculate:
Take the initial phase equal to zero.
R = 10 Ω
L = 0.4 H
C = 2nF = 2 × 10-9 F
Q0 = 5μC = 5 × 10-6 C
f = 50 Hz
a) We initial = ?
b) W(2) = ?
As stated in the previous paragraph, a RLC circuit needs an external sustainable source of emf to make it operate for a long time at steady values. This is made possible by connecting the circuit to an AC power supply, which on the other hand is supplied by an AC generator. An AC generator consists on a current-carrying loop placed inside an external magnetic field, as discussed in the tutorial 16.3 "Magnetic Force on a Current-Carrying Wire. Ampere's Force." Such a system produces forced oscillations in the circuit, which makes it operate for a long time with constant periodic values.
A simplified version of an AC generator is shown in the figure below. A conducting loop is placed inside an external magnetic field. The slip rings are used to create the contact with multiple coils. Each ring is connected to one end of the loop and to the rest of the circuit through metal brushes. As a result, the rings slip against the metal brush and are free to rotate. This brings an induced current I in the circuit.
As the conducting loop of the generator is forced to rotate in the external magnetic field B, an emf is induced in the loop. The equation of this induced emf is
where εmax is the amplitude of the induced emf in the generator (usually the amplitude is the initial induced emf), while ωd is called the "driving angular frequency" because it drives an induced current in the circuit, the equation of which is given by
where imax is the amplitude of the driven current in the circuit. The induced current may not be in phase with the induced emf, so the inclusion in the formula of the initial phase φ is necessary.
In addition, we may express the driven angular frequency as
where fd is the driven frequency of the induced current.
Thus, in circuits with no or very small resistance, the induced current (and emf) oscillate at angular frequency
which is known as the "natural angular frequency". Such oscillations are known as free oscillations.
On the other hand, when a resistor is present is the circuit, the oscillations are known as damped (when no external source is present) or forced (when an external source is needed to keep the values of the induced current and emf constant). The following rule is true for the forced oscillations:
"The induced current and emf in a circuit always occur according the angular frequency of the forced oscillations, regardless the value of the natural angular frequency."
To make the understanding of a RLC circuit more digestible, we will discuss separately three simple circuits, each of them containing only an external emf and one of the three circuit components: resistor, capacitor or inductor, which produce a load in the circuit. Let's start with the circuit that produces the resistive load.
Let's consider a circuit containing only an alternating emf source and a resistor as shown in the figure.
From the Law of Energy Conservation, we have
where ΔVR is the potential difference across the resistor. We can write this equation as
When neglecting the resistance of conducting wire, we obtain
Moreover, we have for the current in the circuit:
Giving that in this type of circuit the current is in phase with the potential difference, we have φ = 0. The graph below shows one cycle of induced current and potential difference in an AC circuit containing a resistive load:
To make the graph plotting easier, we use arrows similar to vectors even though neither current nor voltage are vectors. These diagrams are known as phasor diagrams. The angle formed by the arrows and the horizontal (time) axis gives the ωt term. A phasor diagram pertaining the above graph is shown below:
The voltage of an AC generator is given by the equation ΔV(t) = 90 sin ωt and the frequency of generator is 60 Hz. Calculate:
Now, let's consider a circuit supplied by an AC source, which contains only a capacitor C as shown in the figure.
Using a similar approach as we did when dealing with the resistive load, we obtain for the potential difference at any instant across the capacitor:
From the definition of capacitance, we have for the charge stored in the capacitor at any instant t:
and for the current at any instant t:
is called capacitive reactance of capacitor. It has the unit of resistance (Ohm).
From experiments, it results that current leads by one quarter of a cycle the voltage in such a circuit. If we replace the cos ωd ∙ t term with a phase-shifted sine expression of + π/2 rad, we obtain
Hence, we obtain for the current in the circuit:
In addition, we have for the maximum potential difference in the circuit
Since there is a shift in phase by one quarter of a period (π/2 = 2π/4 = T/4), the graphs of potential difference and current versus time for one complete cycle are:
The corresponding phasor diagram for this circuit is
The current is π/2 (a quarter of a cycle) in advantage to potential difference. Therefore, we say: "the current leads the voltage by π/2".
Remark! The capacitive reactance behaves as an AC resistance. As the frequency of current approaches zero, the capacitive reactance raises to infinity and as a result, the circuit behaves as a DC circuit. However, the current flow in this way (in one direction only) is prevented from the high resistance between the plates of capacitor (at the gap between the plates), which does not allow the current to flow between the plates and to close therefore the cycle.
A circuit containing a 60 Hz AC power source that oscillates according the equation ΔVC(t) = 50 ∙ sin ωd ∙ t and a 20 μF capacitor as shown in the figure.
The reasoning is the same even when we have a circuit in which there is only an AC source and an inductor as shown in the figure.
The potential difference across the inductor is
From Faraday's Law, we have
Thus, combining the above equations, we obtain
The current flowing at any instant in the circuit is obtained through integration techniques. Thus,
is called inductive reactance and is measured in Ohms, similarly to capacitive reactance discussed in the previous paragraph.
Using the trigonometric identity
we obtain for the current flowing in a circuit containing an inductive load:
From this equation, we can see that for a purely inductive load, the current is delayed (is out of phase) by π/2 (a quarter of a cycle) to the potential difference. Therefore, we obtain the graph below:
Again, we use the phasor concept to simplify the understanding of the above graph. The phasor diagram that corresponds the above graph is shown below:
A 0.2 H inductor is connected to an AC source of voltage ΔVC(t) = 40 ∙ sin(100 π ∙ t).
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