Physics Lesson 16.16.2 - The Current Amplitude

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Welcome to our Physics lesson on The Current Amplitude, this is the second lesson of our suite of physics lessons covering the topic of The Series RLC Circuit, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

The Current Amplitude

First, let's express the current in a series RLC circuit through a phasor diagram like the one shown below.

Physics Tutorials: This image provides visual information for the physics tutorial The Series RLC Circuit

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

ε_source = ∆V_R + ∆V_L + ∆V_c

we obtain for the amplitudes of the above quantities (when phasors are taken as vectors):

ε_source(max) = ∆V_net(max) = ∆V_R(max) + ∆V_L(max) + ∆V_C(max)

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:

ε_source(max)² = ∆V_R(max)² + (ΔV_L(max) - ΔV_C(max) )²

The last equation is obtained by applying the Pythagorean Theorem. Using the Ohm's Law for each component, we obtain

ε_source(max)² = (i_max ∙ R)² + (i_max ∙ X_L - i_max ∙ X_c)²

where X_L and X_C are the inductive and capacitive reactances in the circuit respectively.

Rearranging the last equation for the maximum current, we obtain

i_max = ε_source(max)/√R² + (X_L-X_c )²

The expression √R² + (X_L-X_c )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

Z = √R² + (X_L-X_c )²

Thus, we can write

i_max = ε_source(max)/Z

If we substitute the reactances X_L and X_C with their corresponding expressions found in the previous tutorial, we obtain

i_max =ε_source(max)/√R² + (ω_d ∙ L-1/ω_d ∙ C)²

Example 1

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:

The frequency in the circuit
The impedance in the circuit
The maximum current flowing in the circuit

Solution 1

Since the equation of voltage in a RLC circuit has the form
ε(t) = ε_max ∙ sin⁡(ω_d ∙ t)
It is clear that ε_max = 150V and ωd = 120π rad/s. Also, we have
ω_d = 2 ∙ π ∙ f = 120π
2f = 120
f = 60Hz
From the equation of impedance Z, we have (giving that L = 50mH = 0.05 H and C = 0.4mF = 0.0004 F):
Z = √R² + (X_L-X_c )²
= √R² + (ω_d ∙ L-1/ω_d ∙ C)²
= √20² + (120 ∙ 0.05 - 1/120 ∙ 0.0004)²
= √20² + (6 - 20.8)²
= √20² + (-14.8)²
= √400 + 219
= √619
= 24.9Ω
The maximum current flowing through the circuit is given by:
i_max = ε_max/Z
= 150 V/24.9Ω
= 6.02A

You have reached the end of Physics lesson 16.16.2 The Current Amplitude. There are 5 lessons in this physics tutorial covering The Series RLC Circuit, you can access all the lessons from this tutorial below.

More The Series RLC Circuit Lessons and Learning Resources

Magnetism Learning Material
Tutorial ID	Physics Tutorial Title	Revision Notes	Revision Questions
16.16	The Series RLC Circuit
Lesson ID	Physics Lesson Title	Lesson	Video Lesson
16.16.1	Recap on the Series RLC Circuit
16.16.2	The Current Amplitude
16.16.3	The Phase Constant
16.16.4	Resonance in a Series RLC Circuit
16.16.5	Effective Values of Alternating Current and Voltage

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