- To study the step response of first order circuits.
- To understand the concept of the time constant.
- Function generator
- Digital multimeter (DMM)
First-order transient circuits are described by a first order differential equation. First-order circuits contain a resistor and only one type of storage element, either an inductor or a capacitor, i.e. RL or RC circuits.
For a step voltage/current source input, the output can be expressed as
Where, is the circuit response at , and is the response at . The parameter is called time constant of the circuit and gives the time required for the response (i) to rise from zero to 63% (or ) of its final steady value as shown in Figure 4 – 1 (a), or (ii) to fall to 37% (or ) of its initial value as shown in Figure 4 – 1 (b). Therefore, the smaller the value of , the faster the circuit response is.
For a RC circuit
For a RL circuit
tau = L/R
Applying the equations above, the voltage responses across the capacitor and the resistor in Figure 4-1 can be written as:
Figure 4 – 1 A first order circuit and its responses. (a) voltage over the capacitor; (b) voltage over the resistor.
Figure 4 – 2 and Figure 4 – 3 show an RC and an RL circuit.
For all circuits, R = 1 kΩ, C = 0.1 uF, L = 100 mH.
A. Step voltage input
- For the circuits in Figure 4 – 2 using step voltage sources, derive the analytical expression for , when .
- Sketch or plot for each circuit.
B. Square wave input
- For the circuits in Figure 4 – 3 use a square wave input. Assume that is a symmetric square wave with amplitude and period .
- Sketch or plot for each circuit using superposition. (Show at least five cycles.)
Hint: the square wave can be broken up into a series of step functions with displacement of and alternate polarities. Each of these step function inputs generates an output. Thus the total output response is the summation of all the individual output.
Figure 4 – 2 – Circuits with step voltage sources
Figure 4 – 3 – Circuits with square wave input
Build and simulate the circuits in Figure 4 – 3 using any simulation software such as MultiSim or Multisim Live. Set the input voltage to with a frequency of 1 kHz. Display on the oscilloscope. Compare this result with the plot from PREPARATION step B.
Use the same component values as in the PREPARATION and the same input settings used in the SIMULATION, and build the circuits shown in Figure 4 – 3 (a) – (d). Complete the measurements described below. Refer to Experiment #2 for how to use function generator and oscilloscope.
A. Square wave output
On the oscilloscope, connect Ch1 to the input and Ch2 to the output so that both the input and the output are displayed on the screen. Save the screen image for both the input and the output, preferably to a USB drive. Use the ‘Menu’ button on the ‘Save/Recall’ section on the bottom of the oscilloscope screen, then use ‘File Utilities’ to select or create a folder to save the image. Press the ‘Save Screen Image’ button and use the associated buttons next to the screen to select the format or edit file name etc. Compare these waveforms with the results from PREPARATION and SIMULATION.
B. Time constant measurement
- Turn off the input (Channel 1) by pressing the channel number button. Now only the output is displayed on the screen.
- Zoom in on the output curve on the oscilloscope such that a large portion (at least half a cycle) of the rise/drop of a cycle is displayed on the screen. Consider the rise and drop over one half cycle only for each circuit. Use cursors to determine the maximum voltage difference E for the output. The time it takes for the output to rise from the minimum value to 63% of E or to drop from the maximum to 37% of E is the time constant of the circuit. Record this time constant for each circuit.
Prepare your report as per the guidelines given in APPENDIX III.