High and Low Pass Filters and Resonance in an AC circuit
In this lab an oscilloscope will be used to study the frequency dependent nature of
sinusoidal (AC) driven series RC and series RLC circuits. In the RC circuit, high pass
filter behavior is exhibited when taking voltage out at the R, low pass taking voltage out
at the C. In the RLC circuit, resonance is exhibited when taking voltage out at the R.
Theory
In an RLC series circuit with applied AC voltage ε = ε max cos ωt , the current I is
expected to be
=
I I max cos(ωt − φ )
where
I max =
Z=
ε max
Z
R 2 + ( χ L − χC )
2
Z is impedance, and χ L = ω L and χ C = 1/(ωC ) are the inductive and capacitive reactance,
respectively. Impedance has the units of Ω, and thereby becomes the frequency
dependent resistance of an AC-series circuit. The quantity
χ − χC
φ = tan −1 L
R
is the phase angle between ε and I ; that is, the current which is in phase with the voltage
out of the resistor is out of phase with ε by ϕ . This has implications concerning the
power that is deliverable to the resistor which is the consumer of power (average power is
reduced from that where by cosϕ ). When χ L > χ C the circuit is more inductive than
capacitive, making the phase angle is positive so that the current lags the applied voltage;
on the other hand when χ C > χ L the circuit is more capacitive than inductive, making the
phase angle is negative so that the current leads the applied voltage.
Note that when χ L = χ C , ϕ = 0. This occurs at the resonant frequency ω = 1/ LC , i.e.,
at the un-damped frequency of a pure LC circuit. This makes Z = Z (ω) a minimum so
current becomes a maximum; that is, the maximum use (or power) is delivered to the
circuit by making the voltage oscillate at the natural frequency of the circuit.
As one might expect from the respective single element AC circuits, the voltages across
R, L, and C are
∆vR =
∆VR cos ωt
∆vL =
∆VL cos(ωt − π / 2)
∆vC =
∆VC cos(ωt + π / 2)
where the amplitudes are
1
I max R
∆VR =
I max χ L
∆VL =
I max χ C
∆VC =
The maximum voltage ε max = I max Z is related to the respective voltages by phasors
(complex vectors rotating at angular frequency ω about a complex origin) so that the
amplitudes do not just add arithmetically, but add as complex vectors, so
ε max =
∆VR2 + ( ∆VL − ∆VC )
2
which is the origin of the impedance Z.
One can make a variety of circuits with this general approach. For example, if we take
out the resistor we have an LC-circuit; if we take out the capacitor we have an RL-circuit;
if we take out the inductor we have an RC-circuit.
The RC-circuit is of particular interest because it is easy to make a high-pass and lowpass filter simply by taking the voltage out from the resistor and capacitor, respectively.
Consider that in an RC-circuit there is no inductor so
=
Z
R 2 + χC 2
Taking the Δvout at the resistor
1 , ω → ∞
∆Vout ∆VR I max R R
=
=
=
→
∆Vin ε max I max Z Z
ωτ RC , ω → 0
where τRC = RC is the RC-time constant. This allows high frequencies to pass, and
damps out low frequencies. Now, Taking the Δvout at the capacitor
1/ωτ RC , ω → ∞
∆Vout ∆VR I max χ C
1
= =
=
→
∆Vin ε max
, ω →0
I max Z
Z ωC
1
This allows low frequencies to pass, and damps out high frequencies.
Part I – High Pass - Low Pass in an RC circuit.
A. Procedure:
• Set R = 500 Ω and C = 1μF and probe the R (remember to place R next to the
function generator in the circuit). Remember to add in the resistance rGenerator = 50
Ω of the function generator for the total R.
• Find the amplitude of the voltage as the frequency f ( = ω/2π) is varied from 50 to
1000 or so in increments of 50 Hz. Record the values in a table.
• Plot the theoretical curve where the high frequency limit voltage is set equal to that
observed.
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B. Procedure:
• Use the settings as in “A”, but now probe the C (remember to place C next to the
function generator in the circuit).
• Find the amplitude of the voltage as the frequency f is varies from 50 to 1000 or so in
increments of 50 Hz. Record the values in a table.
• Plot the theoretical curve where the low frequency limit voltage is set equal to that
observed.
Part II – Resonance in an RLC circuit.
A. Procedure:
• Make a series RLC circuit taking voltage out at R.
• Select the same R and C as above, and add a 100mH inductor. Don’t forget to
include the rInductor in the total R of the circuit.
• Calculate the resonant frequency. Find the amplitude of the voltage at R as the
frequency f is varied from in increments of from 50Hz to about 500Hz beyond the
resonant frequency or so in increments of 50 Hz. Record the values in a table.
• Plot the expected power ∆VR2 / R by making the maximum value of the expected and
that observed identical.
• Estimate the width at half-maximum of the resonance and record it. Compare it to the
expected width R/L
• Now change the resistance to a small value (set the resistor box to “10Ω” say) and
repeat. This time, the resonance is much narrower, so take smaller increments in Hz
near the resonance to get a good resonance curve. Plot the theoretical power curve,
again making the maximum value the same as observed.
• Estimate the width at half-maximum of the new resonance and record it. Compare it
to the expected width R/L
B. Procedure:
•
Repeat this for a new set of L or C (to get a new resonant frequency).
Note: Plot A and B each on separate graphs, where both A and B have two graphs, one
for each total resistance used.
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