High-pass and low-pass filters
TEP
Related Topics
Circuit, resistance, capacitance, inductance, capacitor, coil, phase displacement, filter, Kirchhoff’s law,
Bode diagram
Principle
A sinusoidal voltage is applied to circuits with coil, capacitor and ohmic resistance. The oscillating circuits, i.e. their filter characteristics are studied as functions of frequency.
Material
1
1
2
1
1
1
2
1
Coil, 300 turns
Resistor (case1), 1 W, 50 Ω
Carbon resistor, 1 W, 1 kΩ
Capacitor (case 2), 1 µF
Capacitor (case 1), 22 µF
Capacitor (case1), 47 µF
Connection plug
Difference amplifier
06513-01
06056-50
39104-19
39113-01
39105-44
39105-45
39170-00
11444-93
1
1
1
4
1
1
1
1
25 MHz Digital storage oscilloscope
with colour display,
Digital function generator
Connecting cord, blue, 𝑙 = 100 mm
Connecting cord, red, 𝑙 = 250 mm
Connecting cord, blue, 𝑙 = 250 mm
Connecting cord, red, 𝑙 = 500 mm
Connecting cord, blue, 𝑙 = 500 mm
Connection box
11456-99
13654-99
07359-04
07360-01
07360-04
07361-01
07361-04
06030-23
Fig. 1: Experimental set-up
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TEP
High-pass and low-pass filters
Tasks
Study the following circuits, determining the ratios of output voltage to
input voltage:
1. 𝑅𝐶/𝐶𝑅 network,
2. 𝑅𝐿/𝐿𝑅 network,
3. 𝐶𝐿/𝐿𝐶 network,
4. 2 𝐶𝑅 networks in series.
Fig. 2: schematic circuit for task 1
For the 𝑅𝐶/𝐶𝑅 network and the double 𝐶𝑅 network also determine the phase displacement. For the
𝐶𝐿/𝐿𝐶 network try to estimate the natural frequency and confirm the measurement with the theoretical
value.
Set-up
Perform the experimental set-up according to Fig. 1. For the measurements connect the input voltage to
channel A of the difference amplifier and the output voltage to channel B. Connect the ground of the difference amplifier with the ground of the digital function generator. The output voltage is always measured across the considered circuit component. For task 4, an example of the circuit is shown in Fig. 4.
Procedure
Use a sinusoidal voltage signal as input with an amplitude of
approximately 5 V. For all measurements use a frequency
range of 10 Hz to 10 kHz with suitable steps.
Task 1: The schematic circuit is shown in Fig. 2 where 𝑅
and 𝐶 denote the resistor and the capacitance respectively.
Use 𝑅 = 1 𝑘𝛺 and 𝐶 = 1 µF. 𝑈𝑅 is the voltage across the
resistor, 𝑈𝐶 across the capacitance and 𝑈0 is the input volt- Fig. 3: schematic circuit for task 4
age.
Task 2: Replace in Fig. 2 the capacitance with the inductance where instead of 𝑈𝐶 one now
measures 𝑈𝐿 . Use the resistor with resistance = 50 Ω .
Task 3: Replace in Fig. 2 the resistance with the inductance where instead of 𝑈𝑅 one now measures
𝑈𝐿 . Use the capacitor with capacitance = 47 µF .
Task 4: The schematic circuit is shown in Fig. 4. Use 𝑅1 = 50 Ω, 𝑅2 = 1 kΩ and 𝐶1 = 22 µF, 𝐶2 =
47 µF. 𝑈out denotes the voltage that is to be measured and compared to the input voltage 𝑈0 .
For this experiment the use of an oscilloscope is preferable to voltmeters
as the latter do not take phase relationships into account and only measure
rms-values. The peak-to-peak values of input and output voltage can be
read off the oscilloscope screen directly. For better accuracy always
choose the highest gain settings that are still suitable. In order to determine
the phase displacement, set the oscilloscope to show both channels and
choose channel one as trigger source. The phase displacement can be determined either by reading the time shift between the maxima of the different signals or by directly determining the phase shift. For the latter, choose
the time setting in such manner that exactly one half-wave of each signal is
shown. You may use the variable sweep rate to achieve better accuracy.
Then the phase displacement can be read off directly in cm where 1 cm
corresponds to a shift of 18 °.
Fig. 4: Example circuit for task 4
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High-pass and low-pass filters
TEP
Theory
Task 1:
̂ cos 𝜔𝑡 which consists of a capacitor of capacitance 𝐶 and a
For a circuit with the supply voltage 𝑈0 = 𝑈
resistor of resistance 𝑅, the network rule (see Fig. 3) reads:
𝑄
𝑈0 = 𝐼𝑅 + 𝐶
(1)
where 𝐼 is the current and 𝑄 is the charge on the capacitor. If, taking account of the fact that 𝐼 =
eq. (1) is differentiated and solved for 𝐼, one obtains
𝐼 = 𝐼̂ cos(𝜔𝑡 − 𝜑)
̂𝑅 −1 (1 +
𝐼̂ = 𝑈
𝑑𝑄
,
𝑑𝑡
(2)
−1⁄2
1
)
𝜔²𝑅²𝐶²
.
The phase displacement 𝜑 is given by
tan 𝜑 = (𝜔𝑅𝐶)−1 .
(3)
From eq. (2), the voltage which can be tapped off across 𝑅 (high-pass) is
−1⁄2
̂ (1 + 1 )
𝑈𝑅 = 𝑅𝐼 = 𝑈
𝜔²𝑅²𝐶²
cos(𝜔𝑡 − 𝜑)
(4)
and the voltage across 𝐶 (low-pass) is
̂(1 + 𝜔²𝑅²𝐶²)−1⁄2 cos(𝜔𝑡 − 𝜑) .
𝑈C = (𝜔𝐶)−1 = 𝑈
(5)
Task 2:
If a coil of inductance 𝐿 and an ohmic resistance 𝑅 are in the circuit, the network rule reads
𝑑𝐼
𝑈0 = 𝐼𝑅 + 𝐿 𝑑𝑡 .
Solving for 𝐼 gives eq. (2) where now
−1⁄2
̂ (𝑅² + 1 )
𝐼̂ = 𝑈
𝜔²𝐿²
.
The phase displacement 𝜑 is given by
tan 𝜑 =
𝜔𝐿
𝑅
.
(6)
From eq. (4), the voltage across 𝑅 (low-pass) is
−1⁄2
̂ (1 + 𝜔²𝐿²)
𝑈𝑅 = 𝑅𝐼 = 𝑈
𝑅²
cos(𝜔𝑡 − 𝜑) ,
(7)
and across the coil (high-pass):
−1⁄2
̂ (1 + 𝑅² )
𝑈𝑅 = 𝜔𝐿𝐼 = 𝑈
𝜔²𝐿²
cos(𝜔𝑡 − 𝜑) .
(8)
Task 3:
If there are a coil of inductance L and a capacitor of capacitance C in a circuit, the network rule reads:
𝑑𝐼
𝑄
𝑈0 = 𝐿 𝑑𝑡 + 𝐶 .
Differentiation and solving for 𝐼 one obtains again eq. (2) with
̂ (𝜔𝐿 −
𝐼̂ = 𝑈
1 −1
)
𝜔𝐶
.
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High-pass and low-pass filters
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The phase displacement 𝜑 is given by
1
tan 𝜑 = 𝜔𝐿 − 𝜔𝐶 .
(9)
From eq. (2), the voltage which can be tapped off across the coil (high-pass) is:
1
̂ (1 −
𝑈L = 𝜔𝐿𝐼 = 𝑈
)
𝜔²𝐿𝐶
−1
cos(𝜔𝑡 − 𝜑)
(10)
and across the capacitor (low-pass):
𝐼
̂ (1 − 𝜔²𝐿𝐶)−1 cos(𝜔𝑡 − 𝜑)
𝑈C = 𝜔𝐶 = −𝑈
(11)
In eqs. (10) and (11) a singularity occurs for
𝜔 = 1⁄√𝐿𝐶 (Thomson equation)
(12)
with 𝜔 = 2𝜋 ∙ 𝜈 which is the natural frequency of the circuit.
Task 4:
If two 𝐶𝑅 networks are connected in series (see Fig. 4), the current source with the first 𝐶𝑅 network can
be regarded as a new voltage source with voltage 𝑈𝑅1 as per eq. (4). The voltage which can be tapped
off across 𝑅2 is thus
̂(1 + (𝜔2 𝐶12 𝑅12 )−1 )−1⁄2 ∙ (1 + (𝜔2 𝐶22 𝑅22 )−1 )−1⁄2 cos(𝜔𝑡 − 𝜑1 − 𝜑2 )
𝑈=𝑈
(12)
where 𝜑1 and 𝜑2 are the phase displacements of the individual networks in accordance with eq. (3).
Evaluation and results
In the following the evaluation of the obtained values is
described with the help of example values. Your results
may vary from those presented here.
To obtain the rms-values of the measured voltages, the
peak-to-peak values read off the oscilloscope have to be
divided by 2√2. The resulting values should be normalized with the input voltage for better comparison.
Fig. 5: Voltage ratios of the 𝑹𝑪/𝑪𝑹 network as a function of frequency with 𝑹 = 𝟏 𝐤𝛀 and = 𝟏 µ𝐅 .
Task 1:
Fig. 5 and 6 show the measured values (cross and circle)
and theoretical values (red lines) for comparison, calculated from eqs. (4) and (5). For very low voltage ratios the
measured values tend to deviate significantly from the
expected values. The reason for this behavior lies in the
equally low absolute voltage and an corresponding worse
signal-to-noise ratio which makes accurate measurements especially of phase shifts very hard.
Fig. 6: Phase displacement of the 𝑹𝑪/𝑪𝑹 network as a
function of frequency.
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High-pass and low-pass filters
TEP
Task 2:
Fig. 7 shows the voltage ratios tapped off the resistor
(cross) and the coil (circle) respectively. Especially for
small values of the coil’s voltage ratio the values strongly
deviate from the calculated values (red line). The reason
lies in the very low signal-to-noise ratio especially for the
voltage across the coil at low frequencies. The theoretically expected values were calculated from equations (7)
and (8).
Task 3:
Fig. 8 shows the voltage ratios tapped off the capacitance
(cross) and the inductance (circle) respectively. As can be
clearly seen, the natural frequency of this oscillating circuit lies somewhere near 2.5 kHz. From eq. (12) one obtains the natural frequency at approximately 2.4 kHz. The
theoretical values (red line) are calculated only at the
measured frequencies to avoid the singularity at this frequency.
Fig. 7: Voltage ratios of the 𝑳𝑹/𝑹𝑳 network as a function of frequency with 𝑹 = 𝟓𝟎 𝛀.
Task 4:
Fig. 9 shows the measured voltage ratio (cross) and
phase shift (circle) as well as the expected values (red
line). The theoretical values were calculated from eqs.
(12) and (3) respectively. The measured values correspond well to the expected values.
Fig. 8: Voltage ratios of the 𝑪𝑳/𝑳𝑪 network as a function of frequency with 𝑪 = 𝟐. 𝟐 µ𝐅.
Fig. 9: Voltage ratio (left axis) and phase displacement
(right axis) of the double CR network as a function of
frequency with 𝑹𝟏 = 𝟓𝟎 𝛀, 𝑹𝟐 = 𝟏 𝐤𝛀 and 𝑪𝟏 = 𝟐𝟓 µ𝐅,
𝑪𝟐 = 𝟓𝟎 µ𝐅.
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