RC Flters TEP Related Topcs Hgh-pass, low-pass, Wen-Robnson brdge, parallel-t flters, dfferentatng network, ntegratng network, step response, square wave, transfer functon. Prncple Resstor-Capactor (RC) crcuts serve as flters for frequences. The frequency response of the most commonly used RC flters s recorded by pont-by-pont measurements as well as the frequency sweep method, and dsplayed on the osclloscope. The results are plotted and verfed usng the measure analyss software. Equpment Plug- n board, 4 mm plugs 633- Resstor n plug- n box, 5 Ohms 657-5 4 Capactor/case / nf 395-4 5 Carbon resstor W, kohm 394-9 5 Connect. plug whte 9 mm ptch 397- Dfference amplfer 444-93 Dgtal functon generator 3654-99 5 MHz Dgtal storage osclloscope 456-99 Alternatve for 5 MHz Dgtal storage osclloscope: Osclloscope 3 MHz, channels 459-95 Adapter; BNC- plug/socket 4 mm 754-6 Adapter, BNC T-plug/socket 4 mm 754- Connectng cord, l mm, yellow 7359- Connectng cord, l 5 mm, red 736-3 Connectng cord, l 5 mm, blue 736-4 Measurement software Measure 45- Screened cable, BNC, l 3 mm 754- Screened cable, BNC, l 5 mm 754- Fg. : Expermental setup: the Wen-Robnson brdge. P448 PHYWE Systeme GmbH & Co. KG All rghts reserved www.phywe.com
TEP RC Flters Tasks - Record the frequency response of the output voltage of. a hgh-pass flter. a low-pass flter 3. a band-pass flter 4. a Wen-Robnson brdge 5. a parallel-t flter, pont by pont and to dsplay the sweep on the osclloscope. - Investgate the step response of 6. a dfferentatng network 7. an ntegratng network - Analyze and verfy the measurements usng the measure analyss software. Set-up and Procedure - Set up the experment n accordance wth Fg. or the crcut dagrams n Fgs. to 6. To measure the frequency response pont by pont, the nput and output voltages and are measured wth the osclloscope. Splt the generator sgnal wth the BNC T-plug, as seen n Fg.. Connect one output drectly to one of the osclloscope channels, and the other through the analyzed crcut to the other osclloscope channel. Adjust the osclloscope channel settngs to properly see both sgnals on the dsplay. O Fg. : Hgh-pass flter. Fg. 3: Low-pass flter. P448 PHYWE Systeme GmbH & Co. KG All rghts reserved
RC Flters TEP Fg. 4: Band-pass flter. Fg. 5: Wen-Robnson brdge. Fg. 6: Parallel T-flter. P448 PHYWE Systeme GmbH & Co. KG All rghts reserved 3 www.phywe.com
TEP RC Flters - The frequency response can also be dsplayed on the osclloscope by sweepng the generator frequency. In ths case the ~f-output of the generator s connected to the X-nput of the osclloscope n X-Y operaton (channel ), whle the sgnal passng through the crcut setup s connected to the Y-nput of the osclloscope (channel ). Set the generator functon to the f f mode and a logarthmc range wth adequate sweep factor (f n f n *sweep factor). Enter the desred values for the frequency range (f and f ) and choose the f f mode from the ~f menu. The menu contans some preset frequency ntervals whch you can use to quckly change the frequency range you want to use. Set the dsplay mode of the osclloscope to persst-on and adjust the volts/dv for each channel to obtan proper resoluton of the sgnal on the screen. You wll see that as the sgnal travels along the x-axs (hence frequency s ncreasng), the ampltude changes accordng to the flter characterstcs. Example of nput values for the hgh- and low-pass flters are: f khz, f 5 khz, sweep.3, osclloscope channel (~f sgnal) Volts/dv Volt, osclloscope channel (flter sgnal) Volts/dv Volt. - In the case of the Wen-Robnson brdge, the output voltage s not connected to earth although the generator and osclloscope are and have the same earth potental, and we have to use the dfferental amplfer (Note: there are several ways to set up the Wen-Robnson brdge wth the provded el ements). It should be noted that the upper cutoff lmt of the dfferental amplfer s khz when the nput voltage s V pp. - The nf capactor requred n the parallel-t flter s made up from two nf capactors connected n parallel. - To nvestgate the square-wave behavor of the dfferentatng network (hgh-pass) and the ntegratng network (low-pass), square-wave frequences of., and tmes the cut-off frequency f c are fed to the nput. Set the sgnal n the functon generator to square-wave. Set the frequency to low (hgh) for the low-pass (hgh-pass) flter. Increase (decrease for hgh-pass flter) the sgnal frequency and observe the gradual change of the flter sgnal. The low-pass flter ntegrates hgh nput frequences (Fg. ), whle the hgh-pass flter dfferentates low nput frequences (Fg. ). Theory and Evaluaton. The hgh-pass flter (Fg. ) s a frequency-dependent voltage dvder, whose behavor s descrbed n complex terms by A( w) o R () R + + wc wrc where s the magnary unt, ω s the frequency, and are the output and nput voltage, respectvely, R s the resstance and C s the capactance. From eq. () t follows that the ampltudefrequency dependence s gven by A () + w R C 4 P448 PHYWE Systeme GmbH & Co. KG All rghts reserved
RC Flters TEP The 3 db cut-off frequency f c s the frequency at whch A (3) By substtutng (3) nto () we obtan f c wc π πrc (4). The low-pass flter (Fg. 3) s characterzed by the followng behavor A( w) wc R + wc + wrc (5) Hence the dependence of the rato of the sgnal ampltudes on frequency s gven by eq. (6) A + w R C and the 3 db cut-off frequency fc s f c wc π πrc (7) Fg. 7: Example of a measured frequency response of the hgh- and low-pass flters. P448 PHYWE Systeme GmbH & Co. KG All rghts reserved 5 www.phywe.com
TEP RC Flters 3. If we connect a hgh-pass and a low-pass flters together, we obtan a band-pass flter (Fg. 4). The followng equaton, wth the abbrevaton Ω wrc, apples to the unloaded crcut: A( w) + wc R + R + wc + wc R Ω + 3Ω Ω (8) Hence we obtan for the ampltude A (9) 9 + Ω Ω The output voltage has ts maxmum value for Ω, hence the md-band frequency s f m wm () π πrc Fg. 8: Example of a measured frequency response of the band-pass flter. 6 P448 PHYWE Systeme GmbH & Co. KG All rghts reserved
RC Flters TEP 4. If we add a parallel-connected voltage dvder consstng of ohmc resstors n the rato : to the band-pass flter, we obtan a Wen-Robnson brdge (Fg. 5): whch gves A( w) Ω 3 + 3Ω Ω () Ω A () 3 9Ω + ( Ω ) The output voltage has ts mnmum value at f m wm (3) π πrc Fg. 9: Example of a measured frequency response of the Wen-Robnson brdge. 5. nlke the Wen-Robnson brdge, the parallel-t flter (Fg. 6) has the advantage that the output voltage s connected to earth. sng the nodal equaton we obtan A( w) Ω + 4Ω Ω (4) Hence A Ω (5) 6Ω + ( Ω ) P448 PHYWE Systeme GmbH & Co. KG All rghts reserved 7 www.phywe.com
TEP RC Flters The output voltage has ts mnmum value at f m wm (6) π πrc Fg. : Example of a measured frequency response of the parallel T-flter. 6. Investgaton of the step response of a dfferentatng network. The nodal equaton d C ( ) (7) dt R s appled to the unloaded output to calculate the step response of the dfferentatng network (hghpass). If nput voltages are used wth frequences f << f c then << and t follows from eq. (7) that d RC (8) dt Low-frequency nput voltages are therefore dffe-rentated (Fg. ). 8 P448 PHYWE Systeme GmbH & Co. KG All rghts reserved
RC Flters TEP Fg. : Square-wave behavor of the hgh-pass flter at varous frequences (low nput voltages are dfferentated). 7. Investgaton of the step response of an nte-gratng network. Applyng the nodal equaton to the unloaded output of the low-pass flter gves R d C dt If nput voltages are used wth frequences f >> fc then << and from eq. (9) we obtan therefore d dt RC () dt () RC P448 PHYWE Systeme GmbH & Co. KG All rghts reserved 9 www.phywe.com
TEP RC Flters Hence, hgh nput voltages are ntegrated (Fg. ). Fg. : Square-wave behavor of the low-pass flter at varous frequences (hgh nput voltages are ntegrated). Data analyss The measurement results can be plotted and analyzed wth the measure software. To enter the data manually, choose Enter data manually from the Measurement menu (Fg. 3). Fg. 3: Enter data manually opton n the Measure software. P448 PHYWE Systeme GmbH & Co. KG All rghts reserved
RC Flters TEP Choose the adequate number of channels (for example, f you measured ampltude vs. frequency, choose channels, f you want to plot the frequency response for two flters on the same plot, lke Fg. 7, choose 3 channels, etc.). The x-data feld creates nteger numbers of the specfed length (number of values) startng wth. Enter the number of values that corresponds to the number of measured data ponts, and enter the Ttle, Symbol, nt and Dgts for each channel. Note: you can adjust these values later, as well as specfy whch channel s plotted on the x and y axes. When done, clck Contnue. A table opens, where you can nput the measurement values for each channel (Fg. 4). You can add and remove measurement ponts usng the symbols at the bottom of the table. Clck OK when done. A plot of the data appears. Fg. 4: Enterng the data n the Measure software. Go to Channel manager n Measurement menu to specfy the data channels to be plotted as the x and y axes. In the left column you can see the data channels you entered. The rght column wll be the new arrangement of the x and y axes. Notce the feld hghlghted n blue n the rght column (default s the Destnaton x-axs feld, but you can change that by double-clckng on a requred feld). Mark a data channel n the left column, and then clck on the transfer arrow between columns. The channel now appears n the hghlghted feld n the rght column. Hence, to create plots of frequency vs. ampltude rato, choose frequency as the new x-axs, and ampltude as the y-axs. Clck OK when done. You can easly access the data table and dsplay optons by clckng on the plot wth the rght mouse button. For example, to obtan a plot lke Fg. 7 from the frequency and ampltude data, go to dsplay optons, choose the X-data tab, and change scalng to logarthmc. You can adjust other dsplay optons, lke the x and y ranges and labels etc. It s recommended you save the measurement after you enter the data, you can do that at any tme from the Fle menu. You can load saved data wth the Open experment command n the Fle menu. Note: many commands are also accessble drectly through correspondng cons located n the taskbar. To analyze the plotted data, scroll down the Analyss menu and choose the requred opton, for example Show extrema. A wndow appears, showng the extrema of the plot (Fg. 5). Mark the Vsualze results opton to dsplay the values on the plot. You can export a plot to a fle usng the Export data opton from the Measurement menu. Mark Save to fle and Export as btmap to save the plot as a btmap mage. P448 PHYWE Systeme GmbH & Co. KG All rghts reserved www.phywe.com
TEP RC Flters Fg. 5: Analyzng the data wth the Measure software. You can also use the measure software to generate and modfy theoretcal data, for example, to check what the dfferental and ntegral of a square-wave functon look lke. Go to the Measurement menu and choose Functon generator (Fg. 6). Choose the square-wave functon and set the correspondng parameters (you can use those seen n Fg. 6). Clck Calculate. A plot of the square-wave functon appears. Note: you can generate any functon by clckng the Equaton tab and enterng a formula. Now choose Channel modfcaton from the Analyss menu. Mark dfferentate (ntegrate) and clck Calculate. The dfferental (ntegral) of the sgnal s added onto the plot. Fg. 6: Generatng a theoretcal dataset n the Measure software.. P448 PHYWE Systeme GmbH & Co. KG All rghts reserved