EE 110L
Experiment #6: Step Response and Advanced Filter Topics
Name: Kenny Lu
Student ID: 903801866
Lab Partner: Michael Chow
Prof: Hassan Babaie
TA: Neha Sinha
Objectives
The goal of this lab was to investigate the step response of capacitive and inductive circuits.
Additionally, to design and measure advanced RC filters.
Theory
Step Response
If a step input is applied to a resistive divider circuit, the output voltage will also be a step.
However, if the circuit contains a reactive element (e.g. a capacitor or an inductor), the result is a nonstep output. The voltage across an element at time t is given by:
𝑡
𝑣(𝑡) = 𝑣(0) + [𝑣(∞)– 𝑣(0)] (1 – 𝑒 − 𝜏)
where v(0), v(∞) are the initial and final voltage, respectively. And τ is the time constant.
The time constant τ is defined as the time required for the response of a reactive element to fall
to 1/e (37 %) of the original output voltage, or to rise to 63 % of its final value, in response to a step
input. For a capacitive circuit, τ = RTh*C. For an inductive circuit, τ = L/RTh. Rth is the Thevenin
resistance
Twin-T Notch Filter
The twin-t notch filter is a filter used to reject signals at a particular frequency, called the null
frequency
R2
R1
O
VO
C1
C2
Vi
R3
C3
Figure 7: Twin-T Notch Filter
The general equation for the gain of a Twin-T Notch Filter can be expressed as
𝜔 𝜔
𝑏
[( 𝑤𝑜 – 𝜔 ) + 𝑗 (𝑏 + 𝑘 − 1)]
𝑉𝑜
𝑜
=
𝑉𝑖 [(𝜔𝑜 – 𝜔 ) + 𝑗 (𝑏 + 𝑏 + 1 + 1)]
𝜔 𝜔𝑜
𝑘 𝑘 𝑏
Where 𝜔𝑜 =
1
𝑅𝐶
At 𝜔𝑜 = 𝜔, the gain of the filter reduces to
𝑉𝑜
𝑏 2 (𝑘 + 1)– 𝑏𝑘
=
𝑉𝑖𝑛 𝑏 2 (𝑘 + 1) + 𝑏 + 𝑘
𝑘
The filter can have a gain of zero at its notch frequency whenever 𝑏 = 𝑘+1
Procedure
1. The first step was to measure the internal resistance of the function generator as it produces square
waves. We did this by first setting the function generator to 1 KHz and the amplitude to
approximately 2 volts peak to peak. We put a resistor with resistance RL parallel with the function
generator so that it acts as a voltage divider. We measured the output voltage and then used the
following formula to calculate the internal resistance. Then we change to voltage peak to peak to 20
volts and repeat.
𝑅𝐿
𝑉𝑜 = 𝑉𝑖 ∗
𝑅𝑆 + 𝑅𝐿
𝑉𝑖
𝑅𝑠 =
∗ 𝑅𝐿 − 𝑅𝐿
𝑉𝑜
RS
O
Vi
RL
VO
O
Figure 1: Internal Resistance
2. The next step was to measure the step response of various RC and RL circuits. We used a capacitor
value of 10 nF and a resistor of around 1KΩ. We calculated the theoretical time constant τ by either
τ = RC or τ = L/R, depending on the circuit. We set the function generator to a square wave, with the
frequency of 10% of 1/τ. This was to make sure that the function had enough time to decay when
the next cycle came. At t = 5τ, the decay is almost 100% of the initial value at t = 0. We measure the
time it takes the circuit to reach 100% decay and record it as 5τ for each of the 4 circuits.
C
R
O
O
C
VI
VO
R
VI
VO
O
O
Figure 3: Circuit B
Figure 2: Circuit A
L
R
O
O
VI
L
VO
R
VI
VO
O
O
Figure 5: Circuit D
Figure 4: Circuit C
3. In this step, we created a twin-T notch filter. We first measured the capacitor and resistor values,
and then constructed the circuit. We used capacitors in parallel to create 2*C. We then measured
the null frequency of the circuit by using sine waves and comparing Vin and Vout. The equation to
measure the null frequency is given by:
1
2𝜋𝑅𝐶
R=R average; C=C average
𝑓𝑜 =
We then measured Vo/Vi at theoretical null frequency, and then we measured Vo/Vi at our
measured null frequency. The final step was to measure the rejection ratio, by dividing the gain at
null frequency by the gain at high frequency.
R2
R1
O
VO
C1
C2
Vi
R3
C3
Figure 6: Twin-T Notch Filter
Data and Data Analysis
1. Internal Resistance
20V pk-pk
2V pk-pk
101.8
RL (Ω)
101.8
20.6
Vi (V)
2
13.2
Vo (V)
0.512
1
f (kHz)
1
57.07
RS (Ω)
295.86
To find the sourcer resistance RS, we plug in the measured values into the equation in the theory
section:
𝑉𝑖
𝑅𝑠 =
∗ 𝑅𝐿 − 𝑅𝐿
𝑉𝑜
As you can see that we obtained a value for the internal resistance exactly as expected.
2. Step Response of RC and RL circuits
Circuit A
Values
R(Ω)
993
C(nF)
10.03
Vi pk-pk (v)
22.4
Vo pk-pk (v)
20.6
5τtheoretical (s)
5.24x10-5
5τmeasured (s)
4.56x10-5
5τtheoretical is five times the result of the theoretical time constant calculated using the formula:
τtheoretical = RTC
The total resistance RT is sum of the internal source and series resistor resistance.
Circuit B
R(Ω)
C(nF)
Vi pk-pk (v)
Vo pk-pk (v)
5τtheoretical (s)
5τmeasured (s)
Values
993
10.03
21.2
42
5.24x10-5
4.68 x10-5
5τtheoretical is the same as circuit a) because the same resistor and capacitors are used.
Circuit C
R(Ω)
L(mH)
Vi pk-pk (v)
Vo pk-pk (v)
5τtheoretical (s)
5τmeasured (s)
Values
993
3.38
21
42
1.6x10-5
1.7x10-5
5τtheoretical is five times the result of the theoretical time constant calculated using the formula:
τtheoretical = L/RT
The total resistance RT is sum of the internal source and series resistor resistance.
Circuit D
R(Ω)
L(mH)
Vi pk-pk (v)
Vo pk-pk (v)
5τtheoretical (s)
5τmeasured (s)
Values
993
3.38
21
42
1.6x10-5
1.7x10-5
5τtheoretical is the same as circuit c) because the same resistor and inductor are used.
Twin-T Notch Filter
R1 (KΩ)
R2 (KΩ)
R3 (KΩ)
R4 (KΩ)
Measured
281.2
277.5
278.2
276.4
C1 (nF)
C2 (nF)
C3 (nF)
C4 (nF)
fo(Hz)
32
34
36
41
45
50
56
59
65
70
76
80
85
90
96
1000
fo (Hz)
Measured
10.03
10.22
10.12
10.36
𝑓𝑜 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 =
1
= 57.25𝐻𝑧
2𝜋𝑅𝐶
|v1| (volts)
21
21
21
21
21
21
21
21
21.2
21
21
21
21
21
21
21
|v2| (volts)
6
5.2
4.8
3.6
2.8
1.6
0.296
1
2
2.8
3.6
4
4.6
5.4
6
21.02
theoretical measured
57.25
56
|v2/v1|
0.29
0.24
0.23
0.17
0.13
0.076
0.014
0.048
0.095
0.13
0.17
0.19
0.21
0.257
0.288
0.96
Error (theoretical vs. measured)
2.18%
From fo, we can find the following:
𝑄𝑙𝑜𝑤 =
𝑓𝑜
= 1.166
2(𝑓𝑜 − 𝑓1)
𝑄ℎ𝑖𝑔ℎ =
𝑓𝑜
= 1.166
2(𝑓2 − 𝑓𝑜)
Discussion and Error Analysis
1. Internal Resistance
Our device had a high resistance at 2 volts peak to peak, because it was once sent it for repair.
While the resistance was 52.44 ohms at 20 volts peak to peak, it was 294.66 ohms at 2 volts
peak to peak.
2. Step Response
In the second part of the experiment, we created two capacitive and two inductive
circuits and measured the step response for each circuit. We calculated the theoretical time
constant and verified the constant by measuring where the output voltage reached almost 100%
of the amplitude. The time it took the signal to reach this mark was 5 times of the experimental
time constant, 5τ.
For circuit A, had an error of 13% when comparing the theoretical time constant and
measured time constant. This might be because we were not sure when to exactly to place the
time cursor. As the time approaches 5τ, the output voltage almost approaches a horizontal line
and it is hard to tell where exactly it reaches the maximum value. This is an error due to
eyeballing measurements.
For circuit B, we had an error of 10.7%, the time constants should be the same because
the same circuit elements are used. And indeed, the two measurements are very close. For a), it
was 45.6μs and b) was 46.8μs.
In circuits C and D, the theoretical 5τ = 16μs. And for both circuits, we measured the
result to be 17μs. This is an error of 5.9%
3. Twin-T notch filter
When designing the Twin-T Notch Filter, the calculated null frequency was 57.25 Hz.
Through measurement, the null frequency was found to be approximately 56Hz, where it
showed the smallest gain. The error for the measured value was 2.2% which is not too far off
from what it should be and so it is an acceptable value.
Conclusion
The objectives set for this experiment were accomplished and this lab was successful because
we were able to perform the experiment with small errors between what we calculated (theoretical)
and what was measured. In addition, we are now familiar with the step response of first-order filters
and Twin-T Notch Filters.