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|16.16||The Series RLC Circuit|
In this Physics tutorial, you will learn:
In the previous tutorial, we discussed about some general features of RLC circuits such as the induced current and voltage or energy in such circuits. In addition, we gave the definition of inductive and capacitive reactance, which were analogue to the resistance provided by resistors.
In this tutorial, we will extend the study of RLC circuits in the interaction between components, especially in regard to the total resistance of the circuit. In addition, we will provide the equations of current amplitude and other quantities not discussed so far.
As explained in the previous tutorial, a series RLC circuit contains all three elements - a resistor, an inductor and a capacitor - connected in series in the same conducting wire. This means the electromotive force produced by the source produces three separate voltages across each component, as shown in the figure.
We denote the maximum voltages across each component as ΔVR(max), ΔVL(max) and ΔVC(max) respectively.
In the previous tutorial, we have also explained that the current is in phase with voltage only at the resistor; the current in the inductor leads the voltage by one quarter of a cycle (π/2) while in the capacitor, current is behind the voltage by a quarter of a cycle (π/2). Since there is the same current flowing throughout the circuit (i.e. the current is in phase in all components), the maximum voltage produced by the source will not be the arithmetic sum of the three maximum individual voltages, i.e.
This is because voltages are not in phase with each other. However, when considering the instantaneous voltages ΔV(t) for each component, we have
The last equation is always true and you can consider the loop rule to convince yourself about this.
As for the current flowing in the circuit at any instant, we can write
In this tutorial we will explain how to find the current amplitude imax and the phase constant φ, and investigate how they depend on the driving angular frequency ωd. For this, we will use the phasor diagrams, which - as explained in the previous tutorial - represent a simplified version of current and voltage representation, especially in the actual context, where these values change continuously with time.
First, let's express the current in a series RLC circuit through a phasor diagram like the one shown below.
We can draw the three phasors of voltage for the above position of the current phasor. Thus, since current and voltage across the resistor are in phase, the phasor arrow of the resistive voltage will be collinear with that of current.
On the other hand, the current in the capacitor leads the voltage by π/2 (a quarter of a cycle, or rotation). Therefore, the capacitive voltage phasor is displaced by π/2 radians anticlockwise to the current phasor because capacitive voltage is quarter a cycle behind the current.
Finally, since the current is behind by π/2 to the voltage (it lags voltage by quarter of a cycle), the inductive voltage phasor is displaced by π/2 clockwise to the current phasor.
The following figure shows all four phasors discussed above.
The projections of each voltage phasor in the vertical axis give the instantaneous values of the corresponding voltages. They are not shown in the diagram to avoid making it too much crowded.
Since ΔVC(max) and ΔVL(max) have opposite directions, it is better to subtract them, just as we do when subtracting two vectors. Then, we find the resultant of (ΔVL(max) - ΔVC(max)) and ΔVR(max), which represents the net maximum voltage Vnet(max) by applying the rules of vectors addition.
Giving that at any instant the phasors obey the rule
we obtain for the amplitudes of the above quantities (when phasors are taken as vectors):
Let's use the rules of vector addition to find the net maximum voltage in terms of the other three voltages. Using the notation (ΔVL(max) - ΔVC(max)) instead of their separate notation, we obtain for the net voltage:
The last equation is obtained by applying the Pythagorean Theorem. Using the Ohm's Law for each component, we obtain
where XL and XC are the inductive and capacitive reactances in the circuit respectively.
Rearranging the last equation for the maximum current, we obtain
The expression √R2 + (XL-Xc )2 is known as impedance Z of the RLC circuit for the given driving angular frequency ωd. It represents the total opposition a RLC circuit presents to current flow. The unit of impedance is Ohm, Ω. Hence, we have
Thus, we can write
If we substitute the reactances XL and XC with their corresponding expressions found in the previous tutorial, we obtain
The voltage in the series RLC circuit shown in the figure oscillates according the expression ε(t) = 150 sin (120π ∙ t).
The values of resistance, inductance and capacitance of the corresponding circuit elements are 20Ω, 50mH and 0.4mF respectively. Calculate:
We have explained earlier that the only voltage in phase with current is the resistive voltage. This means for resistive voltage the phase constant is zero (φ = 0). As for the other two voltages, the phase constant is -π/2 for ΔVC and + π/2 for ΔVL.
However, we are more interested about the phase constant of the total voltage, than for individual voltages. In this regard, the general phase constant φ will be the angle formed by the resistive voltage and the total voltage phasors, as shown in the figure.
Applying the trigonometry rules, we have:
This formula obtained above is very important, as we don't have to know the amplitudes of current and potential difference in a RLC circuit to calculate the initial phase. It is enough knowing the values of resistance and the two reactances for this.
The following cases are present when considering the last equation of phase constant φ in a series RLC circuit:
As special cases, in purely inductive circuits (XL ≠ 0 and XC = R = 0), we have the maximum value possible of phase constant (φ = π/2); in purely capacitive circuits (XC ≠ 0 and XL = R = 0), the phase constant is minimum (φ = π/2); while in purely resistive circuits (R ≠ 0 and XL = XC = 0) the phase constant is zero because φ = 0 and therefore, tan φ = 0.
The equation of voltage in a series RLC circuit is
and the values of resistance, inductance and capacitance of the corresponding circuit elements (resistor, inductor and capacitor) are 10Ω, 40mH and 80μF respectively.
First, let's write some useful clues. Thus, from the equation of voltage, we see that
εmax = 40 V
ωd = 2π ∙ f = 100π rad/s = > f = 50 Hz
R = 10 Ω
L = 40 mH = 4 × 10-2 H
C = 80 μF = 8 × 10-5 F
As stated earlier, resonance is a state of a RLC circuit in which the current and net emf are in phase. This occurs when XL = XC. The graph of voltage and current versus time in a situation involving resonance for a complete cycle is shown below.
The amplitudes may be different from those shown in the above graph; they depend on the corresponding values. This means the current graph may be above the voltage graph or they may have the same height, no matter which graph is higher. The graph above is only for illustration purpose.
We will show below the corresponding phasor diagram as well.
In absence of resonance, the graphs of current and voltage do not have their peak at the same time because either current is ahead of voltage (when the phase constant is negative, i.e. when the circuit is more capacitive than inductive), or it is behind the voltage (when the phase constant is positive, i.e. when the circuit is more inductive than capacitive). In such cases, the voltage and current phasors are not collinear, as discussed earlier.
Giving that during resonance XL = XC, we have after substitutions:
This is the known equation obtained earlier for natural angular frequency of free oscillations. In this case, the oscillations are forced (driven) by the power source. This makes possible the generation of long-lasting (sustainable) oscillations, for which we can apply the approach used in free oscillations, as they do not fade with time.
As for the frequency of RLC circuit oscillations, we have (giving that ωd = 2π ∙ f):
From all discussed so far about the current and voltage in an alternating circuit, we can point out two important features:
If we are asked to find other related quantities in an AC circuit such as power or energy, we cannot use the maximum values of current and voltage as this gives a considerable error; all values would be higher than actually they are. We cannot consider other known methods to find the average values such as the arithmetic mean, or approximations of the sinusoids to obtain a series or rectangles for example. Therefore, the only available method remains the root mean square method, similar to that discussed in the kinetic theory of gases, in which we calculated the rms speed of an ideal gas molecule.
we obtain for the rms current
Since from trigonometry it is known that
we can write
We have used this value earlier in some exercises but now you know what does it mean.
Likewise, for the average voltage, we have
A series RLC circuit is shown in the figure.
εmax = 250 V
R = 5 Ω
L = 5 mH = 5 × 10-3 H
C = 4 μF = 4 × 10-6 F
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