RC Circuit Laboratory Experiment Low Pass Filter and Elmore Time Delay
Mochamad Prananda Department of Electrical Engineering University of Washington Seattle, WA
[email protected]
Abstract This laboratory (lab) report emphasizes on — This analyzing the characteristics of RC circuits, such as RC time constant, rise time, and fall time. Because of its characteristics, especially RC time constant, RC circuit can be modeled as a passive low pass filter that is important in electronics, such as radios that can select some audio frequencies over others. To determine the roles of its characteristics, students can design an RC circuit that cascaded with another RC circuit. After students designed the cascaded RC circuit with the same capacitor and resistor values, the total RC time constant of this cascaded RC circuit resulted to be 3RC. This time constant is very different from a single cascaded RC w hose time constant is RC. From other research studies, this total RC time constant of a cascaded RC circuit can be determined using the Elmore Time Delay formula, a technique that makes RC time constant calculations calculations easier. After looking how RC time constant changes, students can also determine the rise time and fall time that are directly proportional proportional to RC time constant. These characteristics determine determine how fast the RC circuit responses to an input voltage source, which will be used in more advanced electrical engineering classes. After knowing all the important characteristics characteristics of RC circuit, a simple and effective low pass filter circuit can be designed. Using at least 16 points, students can plot the theoretical and experimental low pass filter gains and frequencies to see how a low pass filter works.
circuits easily, many electrical engineering institutions decided to use RC circuit in laboratory experiments. As an example, this laboratory report represents how students can experiment with RC circuits. In this laboratory report, students will report the analyses of RC circuit’s important characteristics, such as RC time constant rise time, and fall time to understand how RC circuit is important in electronics. A. The Purpose of the RC Circuit Experiment The purpose [2] of this laboratory is to challenge students so that: Students can measure the output waveform of simple RC circuits excited by step functions.
Keywords — RC RC circuits, time constant, Elmore time delay, rise time, fall time, cut-off cut-off frequency, low pass filter
I. I NTRODUCTION RC circuits are very popular amongst electrical engineers because of RC circuits’ circuits’ broad application. Many electrical engineers use RC circuits as a base to build electrical devices that include windshield wipers, strobe lights, flashbulbs, and radio, for example [1]. Inside these devices, electrical engineers heavily used RC circuits for digital logic gates and fil ter designs [1], two elements are very important in designing elegant and powerful circuits in both digital and analog circuit. Out of the applications of RC circuit that includes logic gates and filter designs, electrical engineers use RC circuit most commonly as a passive low pass filter. Alongside its wide use in many electronic devices, RC circuit as a passive low pass filter is also very easy for undergraduate electrical engineering students to design and use. Because of this reason that many students can design RC
Students can calculate and measure various timing parameters of switching waveforms (RC time constant, rise time, and fall time) in computer systems. Students can compare theoretical and experimental results, and explain the differences between the two results. Students can design a passive low pass filter and utilize it to see how a low pass filter selects input frequencies. Students can utilize the Elmore Delay Time Method to analyze cascaded RC circuits. II. BACKGROUND
To analyze RC circuits, students need to know circuit theories, such as voltage division method, the basics of differential equations (including Laplace transform), the definition of transfer function, definition of rise time and fall time, the definition of low pass filter, and Elmore Time Delay method (including linear approximation technique) [2]. A. Voltage Division [3] In circuit theory, the voltage division method determines the voltage across an impedance if there are two impedances in series. For example, if there are two impedances Z 1 and Z 2 in series with a voltage source V in in, students can get the voltage across Z across Z 1 from this equation:
V R = 1
Z 1 Z 1 + Z 2
(1)
V in
The equation above can be used to further design a simple RC circuit. B. Basics of Differential Equations [3] Another method that students must know is the method to solve differential equations, such as the Laplace transform method. Students use differential equations extensively in circuit theory [4], including the RC circuits in this lab. In this lab, students will apply what they learned in class to laboratory together with the voltage division method that they learned in introductory electrical engineering class. To get a sense on how differential equations and voltage division play a role in solving RC circuits, students should consider the first picture of the following circuit diagrams:
voltage input are represented in the s domain, students can utilize the voltage division to look for the value of voltage V out out . Applying the voltage division through the capacitor and the resistor, students should get the equation for V out as: out as: Where (4) is the simplified equation for (3) so that the equations are in the standard form of Laplace transform. From (4) that is already in standard form, students can easily transform (4) back to the time domain using the Laplace transform table [3]. After transforming (4) back to the time domain, the relationship between R between R1 , C 1 , V in is: in, and V out out is:
V out (t) (t )= L -1 {V in (s)} = V in (1 e
(2)
sC 1
Alongside the transformation of the capacitor, students should also transform the input voltage V in /s. The reason in to V in in /s. of this transformation is that V in in is a step function where V in in steps from 0 V to a value that is bigger than 0 V at time equals to 0 second. After knowing what capacitors, resistors, and the
1 / (sC 1 )
V in (s)
R1 + 1 / (sC 1 )
s
1 / (R1C 1 )
V in (s)
1 / (R1C 1 ) + s
s
(3)
(4)
V out (s)=
-1 {V (s)} = V e -t/(R1C 1 ) in in
L
(5)
C. Definition of Rise Time and Fall Time [3]
In Fig 1a, the input voltage V in in can relate to the voltage output, V out out , across C 1 by using a relationship between a resistor and a capacitor. From circuit theory class, students can derive this relationship by using the Laplace transform method. In Laplace transform, impedances, currents, and voltages can be written in term of s, s, the term that makes the equation to be in s-domain instead of time domain [4]. In s In s term term of the Laplace transform method, the resistor R1 does not transform to another term. However, the capacitor C 1 transforms to:
V out (s)=
=
(5) and (6) are very useful for finding the characteristics of RC circuit theoretically. Amongst all the characteristics of RC circuit, RC time constant for Fig. 1a can be found from the equation above as R as R1C 1. After determining the RC time constant of an RC circuit theoretically, students can then conduct experiments to find the actual RC time constant by doing this lab.
Fig. 1a. A simple RC circuit with a resistor and a capacitor (left side) Fig. 1b. Two cascaded RC ci rcuits (right side)
1
(5)
On the other hand, when V in in already stagnates to a value greater than zero, V in in does not change in form in Laplace transform. In this case, the inverse Laplace transform becomes:
V out (t)
{C 1 } = C 1 (s)= L {C
-t/(R1C 1 ) )
To conduct the experiments, it is important for students to understand what rise time and fall time are in an RC circuit. Rise time time τr is is the time interval between the 10 %-point and the 90 %-point of the waveform when the signal makes the transition from low voltage (L) to high voltage (H). In this case, rise time makes finding points easier to finally find RC time constant in RC circuits. On the other hand, fall time time τf is the time interval between the 90 %-point and the 10 %-point of the waveform when the signal makes the transition from high voltage (H) to low voltage (L). From here, students can calculate theoretically what the estimated rise time and fall times are from (5) and (6). As for purpose, rise time and fall time will be important for more advanced electrical engineering classes when it comes to dealing with high speed signal processing. D. Definition of Transfer Function [2] Another theory that students must know is the definition of transfer function [3]. Transfer function is the most important tool for filter design, especially a simple RC circuit. The transfer function is defined as the gain of the circuit H(s). H(s). More specifically, the gain is output voltage over input voltage. However, this gain is not defined in the time domain anymore, it is in t he Laplace s Laplace s domain. The purpose of the gain to be in the s domain is because transfer functions can be transformed conveniently to Fourier Transform [3] that students will learn in higher level electrical engineering classes. Because of Fourier Transform, s Transform, s equals to j to jω ω in transfer function where j where j denotes
the imaginary number operator and ω is the frequency of the AC input of a circuit. After understanding what s is defined in the transfer function, students can see easily that the gain of a circuit is related to the frequency because the gain of where the gain can be written in terms of frequency. Because of this relationship, the gain of the circuit and the frequency can control a filter function. Students should be able to define what the transfer function of the RC network in Fig. 1. Using the definition of transfer function, the transfer function of the RC network in Fig. 1 can be written as:
H(s) =
1/(R1C 1 ) 1/(R1C 1 ) + s
=
1/(R1C 1 )
(6)
F.
1/(R1C 1 ) + j
This equation can be used to find the magnitudes of of H(s) H(s).. Then, the magnitudes can be graphed in respect to frequency instead of time. Since now the transfer function is in the frequency domain, frequency can get very large. Therefore, students can use the logarithmic scale of frequency to see the result. In addition to the logarithmic scale, students must also know how to take an absolute value of an imaginary number. The absolute value of an imaginary number is defined as:
| a + bj |= a 2 + b 2
Alongside the cut-off frequency, students should also know that there are two types of low-pass filter: passive and active. A passive low pass pass filter is a filter circuit that uses uses components components that are passive, such as resistors, capacitors, and inductors. Most of the time, in passive low pass filter, the maximum gain (before cut-off frequency) of a circuit will always be 1. On the other hand, an active low pass filter is a filter circuit that uses the passive elements, elements, usually resistors resistors and capacitors, capacitors, in addition to operational amplifiers, an active element. In active low pass filter, the maximum gain of a filter varies, a characteristic that makes many electrical engineering wants. However, in this lab, students should only care about passive low pass filters because students will experiment RC circuits and see how RC circuit can be a low pass filter. Elmore Time Delay and Least-Square Technique
Since solving the time domain equation for a cascaded RC circuit, such as Fig. 1b, is time consuming, students should know to do this laboratory experiment is the Elmore Time Delay technique [1,2]. This method only applies to welldamped circuits that usually only involve resistors and capacitors like RC circuits. In RC circuit analysis, students should use this method to approximate the time constant of a cascaded RC circuit. To easily find the RC time constant, the Elmore Time Delay is defined as:
(7)
This relationship can be used to find the magnitudes of H(s) where s where s is now j now jω ω. After knowing the techniques on how to find a magnitude and frequency of a transfer function, students should be able to design a passive low pass filter. E. Definition of Low Pass Filter[3] To design a passive low pass filter, it is necessary for students to understand the definition of a low pass filter. In circuit theory, low pass filter is a filter design that passes low-frequency signals and attenuates signals with frequencies higher than the cut-off frequency. In this lab, the frequencies relate strongly to AC voltage signals in the form of Acos( of Acos( ωt +θ +θ ) Volts ) Volts (V) where A where A is the amplitude, ω is the frequency, and θ is the phase angle of a voltage signal. These characteristics of an AC voltage signal let students differentiate between one voltage signal to the others. To vary these voltage sources, students should set the cut-off frequency of the low pass filter. In low pass filter, the cut off frequency (in Hertz, Hz) is the frequency when the circuit gain H(s)| |H(s)| equals to the maximum circuit gain with half power, |H max is 1.. On the other hand, the cutmax|/ 2 . In our case, |H max max| is 1 off frequency in low pass filter as Fig. 1a is denoted by:
=
N
N
n 1
m n
∑ Rn ∑C m =
(9)
=
Where N is the number of cascaded RC circuit. In our case, there are only two cascaded RC circuits as shown in Fig. 1b. There is also an approximation technique to find the RC constant from a graph: linear least-square-fit technique. This technique is useful for finding the approximation of an RC circuit after students conduct the experiments. The idea behind this technique is to derive the equation with amplitude A A and the slope slope of the the followin following g equation equation in terms terms of .
1
V out A
= e -t/
(10)
Now, students can write the equation e quation for ln{1 — V Vout out (t)/A} as (t)/A} as a function of t . This equation only works for V out out on (5) and the rising output for RC cascaded circuits. III.M III. MATERIALS AND METHODS A. Electrical Components and Other Materials
f c =
1 2
=
1
(8)
2 π RC
Students can notice that the RC time constant term from the simple RC circuit appears in the equation, the reason why students should analyze the RC time constant to design a low pass filter.
For this lab, students will need three types of resistors and one type of capacitor. For resistors, resisto rs, students will need a 50 Ω resistor, two 10 KΩ KΩ resistors, resistors, and a 27 KΩ resistor. For the capacitor, student will need two 0.01 µF capacitors. Students will also need two types of cables for this lab: the cable that connect to the function generator and t wo cables that connect to the oscilloscope. For the function generator, there
are two branches with a tail that connects to the input slot of the function generator. One branch only has a black hook, and another branch has a red hook. The branch with a red hook connects to the end of the input resistor in Fig. 1, and the branch with a black hook connects to the ground connections of the circuits. Once students connect the branches in the ri ght places, the circuit should receive an input voltage signal. The other cable is the one that connects the oscilloscope to the circuits. This cable enables the oscilloscope to read voltage values or signals in various parts of the circuit. However, the parts that students should care are the input and the output voltages. To connect to the parts of the cir cuit, the cable has two branches with a tail that connects to output channels. The branches are are the one with a long handle and the one with a black hook, and the branch with only a black hook. The one with the long handle connects to the parts of the circuit whose values students want to know. On the other hand, the other branch, the one with only a black hook, connects to the ground the ground connections connections of the circuits. After the cable with a black hook connects to ground connections and the cable with a long handle connects to the output of the circuit, students should see the output waveform in the oscilloscope. Another thing that students will need is a breadboard. Students should make sure that the breadboard is big enough to put both a single RC circuit and two cascaded RC circuits. B. Laboratory Instruments There are three two types of laboratory instruments that students will need in this lab: the function generator and the oscilloscope. The function generator (Tektronix AFG 3021) provides the input voltage of the circuit that students students will make. In the function generator, students will only need an input voltage as a sine function and a square function. From the function generator, students can change the amplitude, frequency, and the phase angle of a sine function, and change the amplitude and the period of a square function. These functions will be the input for the circuits. For RC circuits in this lab, the input voltage will be across a resistor. Besides controlling the input voltage and the circuits, students must keep the internal impedance of the function generator as “High“High Z” to get the actual input voltage. This impedance imp edance determines whether the oscilloscope will read the input voltage correctly. Besides the function generator, another important laboratory instrument is the oscilloscope (Tektronix TDS 2004). This instrument is very helpful for reading the output values of the circuits. Unlike a multimeter, the oscilloscope can only read voltage values (sometimes current values). However, the oscilloscope can read different voltage values or functions, ranging from sine function to sawtooth function. From the two functions that students should consider: the square function a nd the sine functions, students can easily get the important characteristics (amplitude, frequency, period, rise time, and fall time) of the output waveforms resulted from the input waveform and the RC circuit. The oscilloscope automatically measures the amplitude, frequency, and the period of t he output waveform. However, to look for fall time and rise time, one way
to measure the two characteristics of the function is to use the cursors in the oscilloscope to locate two points in the waveforms. Alongside the tools to measure output waveforms, students can also check the input voltage waveforms in the oscilloscope to make sure that students set the input voltages correctly from the function generator. One way to make sure sure that the input voltage is correct is by comparing the amplitude, and the frequency for sine waveforms, and period and amplitude for square waveforms in the oscilloscope. C. Procedures: Simple RC circuit This part of the lab is where students analyze the characteristics (rise time and fall time) of a simple RC circuit to find the RC time constant of the circuit. The reason why students should analyze the RC time constant is to compare theoretical and experimental RC time constant. Furthermore, students will use this analyzed RC time constant to design a low pass filter. In this part of the lab, students should theoretically calculate the RC time constant τ1 with R with R 1 = 10 k Ω and C 1 = 0.01 µF as shown in Fig. 1a. From these values, students should get that the RC time constant students should be should be τ1 = 100 µs. This time constant can then be compared to the experimental time constant. To find the experimental time constant, students should get some data points from the output waveform (such as the points to find rise time and fall time) by using the cursor tools in the oscilloscope. After students attain the data points, students can then use the square-linear-fit technique to find time constant τ1ex. To analyze the data points for finding the time constant, students need to set up the RC circuit. Students should design the RC circuit in a breadboard using R using R1 = 10 k Ω and C 1 = 0.01 µF as shown in Fig. 1a. From here, students should apply the input voltage across R1 as a square function with a 5 V amplitude and a period T ≥ 3 milliseconds (ms). (ms). After students set the input signal, students should make sure that the input signal waveform appears on the oscilloscope’s screen. Also, students should should see the output signals from the oscilloscope’s screen. The output signals should look like an exponential increase, and then exponential decay after a half of the input period T if if students set the cables and the circuits correctly. To get the information for finding RC time constant, it is enough to make the oscilloscope screen to show only about one period of the output waveforms. From here, students should find the four points (10 % and 90 % L to H, 90 % and 100% H to L) to find rise time and fall time. Using the points to find rise time, students should be able to find the RC time constant using the least-square-technique. Students will get two time constants from (10). The best way is to average all the time constant results, and compare the averaged result to the expected time constant. D. Cascaded RC Circuit In this part of the lab, students will follow the exact same procedure as the simple RC circuit. However, the only difference is that students will have two resistors R resistors R2 and R and R3 with
the same value (10 KΩ), and two resistors C 2 and C 3 with the same value (0.01 µF). After we know all the resistor and capacitor values, the RC time constant for this circuit is expected to be 300 µs. After knowing the estimated time constant, students will only compare the time constant values, not the rise time and fall time values due to level of complexity. However, students will find 10% and 90% L to H values to find the time constant by using (10). The purpose of finding the exact time constant for the cascaded circuit is to show that the Elmor e Time Delay method works for finding cascaded circuits. This method is useful for finding the cut-off frequency of higher order filters that students will learn later. E. Low Pass Filter Design
R ISE ISE TIME, FALL TIME AND TIME CONSTANT VALUES (CASCADED RC CIRCUITS) RC Simple Circuit Time Constant (Using leastsquare-fit) Ti me M easur easur ed (µs) (µs) Ti me Calcu lat ed (µs) (µs)
10% (L to H)
35 (at ~500 mV)
31.6 (at 500 mV)
90% (L to H)
710 (at ~4.5V)
757.71 (at 4.5 V)
332.2
300
308.3
300
10% (L to H) RC time constant 90% (L to H) RC time constant Average RC Time Constant
320.1 µs (6.7 % error)
Time Constant Values
400
This part of the lab is where RC time constant plays a very important role: cut-off frequency approximation. To find the expected cut-off frequency, students can use (8). The values of the resistor and capacitor to find the expected cut-off frequency would be the same as the simple RC circuit since students will use the simple RC circuit as a low pass filter in this lab. On the other hand, to test how the magnitude of the circuit gain differs due to the functionality of the low pass filter, students can differ the frequency when testing the frequencies from 10 Hz to 1 MHz with the following manner: 10 Hz, 20 Hz, 50 Hz, 100 Hz, 200 Hz, …, …, 1 MHz (a total of 16 points). After getting the magnitude of the circuit gains fro m these different frequencies, students can plot the magnitude values and co mpare them to the actual magnitude values calculated from (6) and (7) using computer software sets, such as Matlab or Mircosoft Excel. To see a clearer graph, students should use the log 10 10 scale for frequency and dB scale for the magnitude values that is defined as 20log 10 10(|H(s)|) dB. IV.RESULTS IV. RESULTS TABLE I.
TABLE II.
R ISE ISE TIME, FALL TIME AND TIME CONSTANT VALUES (SIMPLE RC CIRCUIT)
300 200 100 0 S im imple RC
C as ascaded RC
Fig. 2. Error bars and the average for time constant values of simple RC and cascaded RC circuits (Above)
Fig. 3a. Oscilloscope graph depicting a simple RC circuit and its rise time (upleft side) Fig. 3b. Oscilloscope graph depicting two cascaded RC circuits and their total fall time (up-right side)
RC Simple Circuit Fall Time, Rise Time, and Time Constant (Using least-square-fit) Ti me M easur easur ed (µs) (µs) Ti me Calcu lated (µs)
10% (L to H)
10 (at ~500 mV)
10.5 (at 500 mV)
90% (L to H)
250 (at ~4.6 V)
230.3 (at 4.5 V)
90 % (H to L)
11.2 (at ~4.6 mV)
10.5 (at 4.5 V)
10% (H to L)
245 (at ~500 mV)
230.3 (at 500 mV)
Rise Time
240.1 (9% error)
219.8
Fall Time
235.7 (7% error)
219.8
94.9
100
98.9
100
10 % (L to H) RC time constant 90 % (L to H) RC time constant Average RC Time Constant
96.9 µs (3% error)
Fig. 4. Time domain osci lloscope graph of when f = 10 Hz (Yellow), and when f = 1 MHz (Blue) for Low Pass Filter circuit (above)
constant. The cut-off frequency can be seen visually in Fig. 5 that it is the point where the gain is about -3 dB or when the power is one half of the power of the maximum gain. This power is the reason why the cut-off frequency is called the 3 dB frequency. Also, students can see that the cut-off frequency is in the middle of 10 2 Hz and 10 4 Hz that indicates about 1000 Hz and close to 1591.51 Hz. From this frequency, students can see from Fig. 5 that the gain decreases to 0 after the cut-off frequency, which can be noted as bigger negative numbers in dB.
Fig. 5. Frequency domain graph of circuit gain values (both experimental and theoretical) with % error bars for Low Pass Filter circuit (above)
V. DISCUSSIONS From the simple RC circuit, the time constant least-squarefit approximation works well in this lab. The time constant average from the 10% and 90% rising values is 3% in error comparing to the calculated value. This result is also still reasonable because the error is still under 10%. In this lab in finding the time constant, students should notice that students can even use one point to find the RC time constant via the least-square-fit technique. However, more data will make the approximation better because individual error will have less effect on the overall error. In this lab, students would only need two points for simplification. The rise time and fall time for the simple RC circuit is also close to correct when compared to the calculated rise time and fall time. The rise time and fall time have about 9% and 7% error comparing to the calculated values. These results are good because the errors are still under 10%. Students can also see that the average time constant for the cascaded RC circuits is about 6.7% error that is under 10%, a reasonable error. After reviewing the errors of time constant, students should notice that the Elmore Time Delay technique works well for approximating the time constant because the experimental and the calculated Elmore time constant agree with an error of less than 10%. From the Elmore Time Delay technique, students can also differentiate visually from Fig. 3a and Fig. 3b that the simple RC time constant rises and decays much faster than the cascaded RC time constant. This difference is due to the bigger time constant (3 R1C 1 ) ) of the cascaded RC circuit [1]. For the last part of the lab, the 16 points received from the lab data approximated the theoretical low pass filter graph in Fig. 5 quite well. The only aspect to notice is that the errors get bigger after the cut-off frequency, which is a phenomenon to consider. However, the errors of each data are still about 5% to 7%, which are still very good. Student should notice the cut-off frequency that can be calculated as 1591.54 Hz from (8) is related to the time
Students should also see the definition of attenuation in Fig. 4 in time domain where the gain of 10 Hz attenuates from 1 to 0 in 1 MHz. And, this phenomenon describes how low pass filter works (the bigger the frequency from the cut-off frequency, the smaller the gain). The data that students attained have errors that are small enough, about less than 10%, to consider that the st udents have done the lab successfully when compared to the theoretical values. Some errors to consider might be the inaccuracy of putting cursors in the function generator, bad connections between cables, and the quality of the elements. For justification, the students who conducted these experiments noticed some corrosion mark on the cables that might contribute to the error, thus making the cables have bad connection due to bad quality. Furthermore, students also noticed that the values on the oscilloscope screen did not stay still when students tried to keep the cursors stay. VI.C VI. CONCLUSION The data attained from this lab showed that the theoretical data and the experimental data matched very well when students do the lab correctly, such as the cascaded RC’s time constant has three times the amount of simple RC’s time constant which is an important discovery in this lab. After knowing the data matched, students can use the data, such as the 16 points data for low pass filter, to verify that the techniques to find RC time constant, rise time, fall time, and cut-off frequency work well in real life application. These parameters of an RC low pass filter are very important in real life. For example, the RC time constant can determine the cutoff frequency of a low pass filter. On the other hand, rise time and fall time can be applicable high speed signal processing [1]. Finally, this lab prepares students to be able to use their skills about low pass filter in more advanced electrical engineering topics, such as digital circuits and signal processing. EFERENCES R EFERENCES [1]
Gupta, R., Tutuianu, B., Pileggi, L. (1997). The Elmore Delay as a Bound for RC Trees with Generalized Input Signals . IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 16, No. 1, 95-104. [2] Nilsson, J. W., W., & Riedel, Riedel, S. A. A. (2011). Electric (2011). Electric Circuits (9th ed.). ed.). Boston: Prentice Hall. [3] Step Response of RC Circuit (2013). EE (2013). EE 233: Circuit Theory Labortatory Labortatory #1. #1. University of Washington Department of Electrical Engineering.
