In this lab the the first two of the following four fundamental first order RC and RL circuits are considered.
The differential equation that describes circuit #1 is:
with initial condition vO(0) = V0.
The response of circuit #1 to a step input vS(t) = VAu(t) is
For a more detailed derivation, see the Review of 1'st Order Circuit Analysis notes.
Using phasor analysis, the sinusoidal steady-state response of circuit #1 is
Define the transfer function H of a circuit with phasor input VS and phasor output VO as
The quantity H is complex valued in general and can be written in magnitude/phase form as
The magnitude of H is called the magnitude of the frequency response of a circuit and the phase of H is called the phase of the frequency response of a circuit.
For circuit #1 H is computed as
See again Review of 1'st Order Circuit Analysis for a more detailed derivation.
The step response of circuit #2 for an input vS(t) = VAu(t) is
In this case the initial condition is V0 = vC(0) = vS(0) - vO(0).
Using phasor analysis, the transfer function H (and thus the frequency response) of circuit #2 is found to be
For a more detailed derivation see Review of 1'st Order Circuit Analysis.
E1. Step Response of RC LPF. Build RC circuit #1 on your breadboard using R = 3.3 kOhm and C = 100 nF. A quick reminder: A 3.3 kOhm resistor has color code orange-orange-red (= 33×102 Ohm = 3.3 kOhm) and a 100 nF capacitor is usually labeled as 104 (= 10×104 pF = 100 nF). You can also use two 47 nF (usually labeled as 473) capacitors in parallel.
Measure the step response using a 50% duty cycle rectangular pulse with frequency about 100 Hz and amplitude about 500 mVpp as input signal vS(t). Display the input and the output signals on the oscilloscope. The following figure shows approximately what this should look like. A quick reminder: Use the oscilloscope probe(s) in you kit to connect the oscilloscope to your circuit.
Sketch and label the output signal and determine the RC time constant from the graph. Determine also the rise time which is defined as the amount of time it takes for the output to go from 10% to 90% of the step size. If you change C from C = 100 nF to C = 200 nF, how does the rise time change?
E2. Frequency Response of RC LPF. The magnitude and the phase of the frequency response of a circuit were defined earlier in terms of the transfer function H = VO / VS where VS and VO are the input and the output of the circuit in phasor form.
Measure the magnitude and the phase (in degrees) of the frequency response of RC circuit #1 (with R = 3.3 kOhm and C = 100 nF as in E1) at frequencies f = 50, 70, 100, 200, 300, 500, 700, 1000, 2000, 3000, 5000 Hz as follows. Use a sinusoidal signal at each of the specified frequencies and an amplitude of about 500 mVpp as the input signal vS(t). Display both vS(t) and vO(t) on the screen of the oscilloscope. This should look similar to the following figure.
The magnitude of H is the ratio VO / VS of the amplitudes of vO(t) and vS(t). Referring to the figure above, VO = vb - va and VS = vd - vc. The phase of H is the phase difference between the output signal vO(t) and the input signal vS(t). In terms of the above figure, this phase difference is computed as -360*(t2 - t1) / (t3 - t1) in degrees, or as -2*pi*(t2 - t1) / (t3 - t1) in radians. Thus, if the output signal lags behind the input signal the phase difference is negative, and it is positive otherwise. Note that the times t1 and t3 are obtained from the (positive going) zero crossings of vS(t), whereas t2 corresponds to a (positive going) zero crossing of vO(t). For human interpretation phase in degrees is usually more intuitive, but for mathematical computations phase in radians is generally preferred.
Of special importance is the so called half-power frequency or -3dB frequency where the amplitude of the output is equal to 1 / sqrt(2) = 0.707 times the value of the maximum output amplitude at any frequency. Determine the approximate value of the half-power frequency of RC circuit #1.
E3. PSpice Simulation of RC/RL LPF. Simulate the step response and the frequency response of circuit #1 (with R = 3.3 kOhm and C = 100 nF as in E1) in PSpice.
Note: While using PSpice turn off Realtime Virus Scanning (right-click on EKG-like icon in lower right corner tray in Windows). When you are done with PSpice, turn Virus Scanning back on again.
For the step response, use the circuit shown below. Step by step instructions for the simulation of the step response are given here.
Use the cursor in the graph of the simulation to determine the time constant of the circuit and to determine the rise time (from 10% to 90% of the step size of the response).
For the frequency response, use the circuit shown below. Step by step instructions for the simulation of the frequency response are given here.
Use the cursor in the graph of the simulation to determine the half-power (or -3dB) frequency fc. What are the phase values (in degrees) at 0.1 fc, fc, 10 fc? Make a logarithmic plot (in dB) of the magnitude of the frequency response. In this plot the magnitude of the frequency response should decay following approximately a straight line above fc. What is the slope of this line in dB per decade of frequency?
E4. Step and Frequency Response of RC HPF.
Repeat E1, E2 and E3 for circuit #2.
Use again R = 3.3 kOhm and C = 100 nF. This circuit acts as a highpass filter (HPF) or differentiator, so you will have to adapt the wording of some of the questions in E1, E2, and E3 accordingly.
Lab worksheet in PDF format: lws01.pdf
Note: Each student needs to turn in a lab worksheet. The raw measured data for each student in a lab team will be the same, but the conclusions drawn from it are individual to each partner in the team.
©2003-2007, P. Mathys. Last revised: 1-23-07, PM.
