The Series RLC Circuit
Pre-lab Questions
1. At resonance, which is bigger – the inductive or capacitive reactance?
2. Sketch the shape of the graph you expect for VR/VR, max vs frequency.
3. What is the phase angle between the current and voltage for a circuit that is purely
resistive? Purely capacitive? Purely inductive?
4. Why can’t you simply sum up the measured voltages to get the total voltage?
5. What are some uses for inductors in the real world?
AC Circuits
Commercial power plants supply time-varying voltage and current (AC). The relationship
between voltage and time can be written as:
(1)
𝑣(𝑡) = 𝑉 sin 𝜔𝑡
where V is the maximum voltage, t is time, and  is the angular frequency, which is
related to the frequency of the source by 𝜔 = 2𝜋𝑓.
Before looking at the RLC circuit, however, we will first look at three simpler circuits
containing each of the elements individually and an alternating voltage source. Please
answer the included questions directly on your Series RLC Group Worksheet…
Intro Part 1:
Resistive Loads
The simple AC circuit shown here consists of an AC generator
and a resistor. If the emf of the generator is given by equation (1),
then the voltage drop across the resistor is:𝑣𝑅 (𝑡) = 𝑉𝑅 sin 𝜔𝑡
where VR is the maximum voltage across the resistor. Using
Ohm’s Law, an expression for the current can be written:
𝑖𝑅 (𝑡) =
𝑣𝑅 (𝑡)
𝑅
=
𝑉𝑅
𝑅
sin 𝜔𝑡 = 𝐼𝑅 sin 𝜔𝑡
R
(2)
where IR is the maximum current through the circuit. Note that in the expressions for
𝑣𝑅 (𝑡)and 𝑖𝑅 (𝑡), the arguments of the sine functions are the same; the maximum voltage
and the maximum current occur at the same instant in a resistor.

In the circuit above, suppose the resistance is changed and R is increased to R’.
Describe how this will affect the peak current. (record your answer on the
worksheet)

Suppose instead the frequency is increased. Describe how this would affect the
peak current and the phase difference between current and voltage. (again, record
your answer on the worksheet)
Intro Part 2:
Alternating Current in a Capacitor
The circuit shown consists of a generator and a single capacitor.
If the emf of the generator is given by equation (1), then the voltage
drop across the capacitor is:
C
𝑣𝐶 (𝑡) = 𝑉𝐶 sin 𝜔𝑡
where VC is the max voltage across the capacitor. Recall that the charge stored in a
capacitor is defined to be Q=CV. For a capacitive circuit, the amount of charge stored in
the capacitor at time t is therefore:
𝑞𝐶 (𝑡) = 𝐶𝑣𝐶 (𝑡) = 𝐶𝑉𝐶 sin 𝜔𝑡
and since current is the time derivative of charge, the equation for current can be
written as:
𝑖𝐶 (𝑡) =
𝜕𝑞𝐶 (𝑡)
𝜕𝑡
= 𝜔𝐶𝑉𝐶 cos 𝜔𝑡 = 𝐼𝐶 cos 𝜔𝑡
(3)
𝑉
where 𝐼𝐶 = 𝜔𝐶𝑉𝐶 = 𝜒𝐶
𝐶
The term 𝜒𝐶 = 1⁄𝜔𝐶
is called the capacitive reactance.
By rearranging equation (3), you see C is the ratio of peak voltage to peak current,
which is similar to Ohm's law, R = V/I. It is equivalent to a resistance for a capacitor and
has units of ohms. It is very important to note that this reactance is frequency
dependent and, unlike resistance, decreases as  increases.
How is the current related to the voltage across the capacitor? From the trigonometric
property relating sines and cosines, you can rewrite the current in the capacitor as:
𝑖𝐶 (𝑡) = 𝐼𝐶 sin(𝜔𝑡 + 𝜋⁄2)
(4)
The current is /2 radians (or 90) out of phase with the voltage. It reaches its maximum
value before the voltage, by one-quarter cycle.

We’ve already investigated RC circuits in class, but not with alternating voltages.
Let’s start with just a capacitive circuit. Construct a v vs t and i vs t graph for a
simple C circuit paying special attention to any phase differences between the
voltage and current.

Suppose that the capacitance is increased, still with no resistance. How does this
affect the peak current (peak current always means maximum current at a given
frequency, not the resonance current) and the phase difference between the
current and voltage? What if there is resistance?

What if the capacitance is decreased? What happens to the phase difference and
the peak current?

Now suppose the frequency is increased, with the capacitance unchanged.
Describe how this would affect the peak current and phase difference between the
current and voltage.
Intro Part 3:
Alternating Current in an Inductor
L
The circuit shown consists of a generator and an
inductor. If the emf of the generator is given by equation (1),
then the voltage drop across the inductor is:
𝑣𝐿 (𝑡) = 𝑉𝐿 sin 𝜔𝑡
where VL is the maximum voltage drop across the inductor. From the definition of
inductance we can also write:
𝑣𝐿 (𝑡) = 𝐿
𝜕𝑖𝐿 (𝑡)
𝜕𝑡
This isn’t the most helpful form – a quick rearranging of algebra yields
𝜕𝑖𝐿 (𝑡)
𝜕𝑡
=
𝑣𝐿 (𝑡)
𝐿
=
𝑉𝐿
𝐿
sin 𝜔𝑡
By integrating, we can obtain an equation for the current in an inductor:
𝑉
𝑖𝐿 (𝑡) = − 𝜔𝐿𝐿 cos 𝜔𝑡 = −𝐼𝐿 cos 𝜔𝑡
𝑉
(5)
𝑉
where 𝐼𝐿 = 𝜔𝐿𝐿 = 𝜒𝐿
𝐿
The term χL = ωL is called the inductive reactance. It is the ratio of peak voltage to
peak current in an inductor and has units of ohms. It is the equivalent of resistance for an
inductor and (like the capacitive reactance and resistance, has unites of ohms). Unlike
resistance, χL increases as  increases.
By again using a trigonometric relationship, the expression for current can be written:
𝑖𝐿 (𝑡) = 𝐼𝐿 sin(𝜔𝑡 − 𝜋⁄2)
(6)
Again, the current is /2 out of phase with the voltage, but in this case current reaches its
maximum value after the voltage by one-quarter cycle.
Reactance and Impedance
The term impedance, given the symbol Z, is a general term that includes both
resistance and reactance of a set of circuit elements. You should see now that it is
incorrect to talk about the total resistance of a circuit that contains different circuit
elements, because the term resistance implies no frequency dependence. Most real
circuits are best described by the total impedance, which can include resistances and
reactances.

For a purely inductive circuit (only inductors, no resistors or capacitors), what
does the v(t) and iL(t) vs t graph look like? Sketch them below.

Suppose the inductance increases, still with no resistance. How does this affect
the peak current and the phase difference between the current and voltage

As the frequency is increased in the circuit, what happens? How does this affect
the phase difference and peak current?
The Series RLC Circuit
L
C
Now consider a circuit with all three circuit elements
connected to a sinusoidal voltage source described by equation
(1). The total voltage drop across the combination equals the
voltage supplied by the source:
𝑣𝑡𝑜𝑡𝑎𝑙 (𝑡) = 𝑣𝑅 (𝑡) + 𝑣𝐶 (𝑡) + 𝑣𝐿 (𝑡)
R
The resulting alternating current is given without proof as:
𝑖(𝑡) = 𝐼 sin(𝜔𝑡 − 𝜙)
(7)
where I is the maximum current, ω is the frequency of the voltage source and ϕ is the
phase angle, given by:
𝜙 = tan−1 [
(𝜒𝐿 − 𝜒𝐶 )⁄
𝑅]
(8)
The phase angle changes with frequency , since the reactances change with
frequency . It may be positive or negative depending on the relative sizes of the
reactances, so that the total current through the RLC circuit may lead or follow the
applied voltage.
Since the circuit elements are in series, the current through each element at
any time t is the same: but the voltages are not. For a given current, VR is in
phase with the current, the current leads VC by 90° and follows VL by 90°. All of this
information can be summarized on a phasor diagram.
I
V
R
I
V
R
V
R
V
T
o
ta
l

V
C
V
L

t

t

V
C
(V
L-V
C)
V
L
(a)
(b)
Diagram (a) shows both the phases of the maximum voltages with respect to the
current and the magnitudes of the instantaneous voltages, v(t), across each circuit element
at some arbitrary time t (a “snapshot”). As time elapses, imagine the current and
maximum voltages (bold arrows) spinning about the origin, and the instantaneous
voltages, which are shown as the y component of the maximum voltages, changing
periodically.
This is complicated to visualize with all the phasors changing
simultaneously. Later in the activity you will have the opportunity to watch a simulation
display the changing phasors with time.
Diagram (b) shows that the maximum voltages VR and (VL - VC) add as vectors to
give Vtotal for some arbitrary positive phase angle . From the diagram and simple
trigonometry, we can write:
2
𝑣𝑡𝑜𝑡𝑎𝑙
= 𝑣𝑅2 + (𝑣𝐿 − 𝑣𝐶 )2
(9)
From Ohm's Law and equations (3) and (5), this expression can be written as:
2
𝑣𝑡𝑜𝑡𝑎𝑙
= (𝐼𝑅)2 + (𝐼𝜒𝐿 − 𝐼𝜒𝐶 )2
Since the maximum current I is a common factor:
𝐼=
𝑉𝑡𝑜𝑡𝑎𝑙
√𝑅 2 + (𝜒𝐿 − 𝜒𝐶 )2
The denominator of this expression is the total impedance of the circuit, Z. It is
dependent on frequency and we can rewrite this equation to include the frequency: (10)
𝐼=
𝑉𝑡𝑜𝑡𝑎𝑙
2
√𝑅 2 + (𝜔𝐿 − 1 )
𝜔𝐶
Resonance
According to equation (10), the current through our series RLC circuit can change
merely by changing the frequency, f, of the source. The current will be at a maximum
value when the denominator is a minimum. This will occur when the second term in the
denominator is zero, or when the inductive reactance equals the capacitive reactance.
If 𝜔𝐿 = 1⁄𝜔𝐶 then, 𝜔0 = 2𝜋𝑓0 = 1⁄
(11)
√𝐿𝐶
This is called the resonance frequency of the circuit. Also note that when the circuit is
at resonance, the phase angle ϕ is zero (see equation 8) and the current is in phase with
the applied voltage Vtotal.
Procedure
Typically an oscilloscope is used to measure the voltages and probe an AC circuit
because of the fast frequencies involved. But an oscilloscope itself is a powerful (and
complicated) lab tool that that can take some experience to become familiar with. For the
sake of not having to learn the details of the oscilloscope at the same time as the details of
the RLC circuit, we will be using some online simulations for the following circuits.
Please download the free simulation from wolfram.com (following the online
instructions) and run the simulation located at:
http://demonstrations.wolfram.com/SeriesRLCCircuits/
The graph displayed on this is the voltage from the AC generator and to the current
through the system. Please note that the y-axis automatically rescales as necessary and
therefore you should watch the values on the y-axis carefully. In controlling the variables
for the simulation, notice the small plus signs at the end of the sliders next to each term
on the left. If you click on that small plus, you will have more detailed control over the
values.
1. Set the components in the simulation to R = 5Ω, C = 250µF, and L = 50mH.
Predict what the resonant frequency of the circuit will be by using equation (11).
Find the resonant frequency by changing the top slider for the AC generator and
watching for the maximum current. Does the simulation match your prediction?
2.
If the inductance is increased a small amount from the starting values in question
1, with the resistance and capacitance unchanged, how does this affect the peak
current and phase angle between the current and the voltage? How does this
affect the resonant frequency and why? Test your predictions with the simulation.
3. If the capacitance is increased a small amount from the starting values in question
1, with the resistance and inductance unchanged, how does this affect the peak
current and phase angle between the current and the voltage? How does this
affect the resonant frequency and why? Test your predictions with the simulation
4. Take Data! For varying ω , record IR values (which allow you to calculate VR).
(please note the V displayed by the simulation is the voltage from the generator).
5. Plot
VR
VR Max
vs. frequency. What does the shape of this graph indicate? How is
that information useful? What factors would change the overall shape of this
curve.
6. At some ω from your data in question 4 below ωo where I is appreciably smaller
than at resonance, calculate VR using I=VR/R. Calculate the reactance of both the
inductor and the capacitor using the calculated current and equations (3) and (5).
Calculate the phase angle from equation (8). Now open the supporting
simulation at
http://demonstrations.wolfram.com/PhasorDiagramForSeriesRLCCircuits/
Recreate your parameters for the circuit off resonance with this simulation. How
do your predictions compare to the simulated values?
7. At some ω from your data in question 4 above ωo where I is appreciably smaller
than at resonance, calculate VR using I=VR/R. Calculate the reactance of both the
inductor and the capacitor using the calculated current and equations (3) and (5).
Calculate the phase angle from equation (8) and compare to the phasor diagram in
the simulation.
8. OPEN EXPLORATION: Set up some other values for the components in the
circuit and see how these values affect both the resonance frequencies, the phase
shift and the phasor diagrams. What situations would you design a circuit with a
large capacitor? A small capacitor? A large inductor? A small inductor?