LTI system - RC low-pass filter

 


Peak-to-average power ratio (PAPR) of OFDM systems

LTI system example: RC low-pass filter

Linear time-invariant (LTI) systems can be represented by the transfer function. It determines the output signal of an LTI system for a given input signal in the frequency domain. An RC low-pass filter serves as example to examine amplitude and phase of this complex valued frequency response.

Transfer function
Y(f)=H(f)X(f) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaacI cacaWGMbGaaiykaiabg2da9iaadIeacaGGOaGaamOzaiaacMcacqGH flY1caWGybGaaiikaiaadAgacaGGPaaaaa@429B@
LTI system's output is related to the input by the transfer function H(f) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiaacI cacaWGMbGaaiykaaaa@3908@ .

In this briefing and the subsequent experiment an RC low-pass filter serves as example for an LTI system. The transfer function, amplitude response and phase response are derived. An RC low-pass filter is a potential divider circuit containing a resistor and a capacitor. It implements a first order low-pass.

RC lowpass   Transfer function of RC lowpass

Recalling the capacitor impedance the transfer function results to

H(f)= 1 j2πC R+ 1 j2πC = 1 1+j2πRC MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiaacI cacaWGMbGaaiykaiabg2da9maalaaabaWaaSGaaeaacaaIXaaabaGa amOAaiaaikdacqaHapaCcaWGdbaaaaqaaiaadkfacqGHRaWkdaWcca qaaiaaigdaaeaacaWGQbGaaGOmaiabec8aWjaadoeaaaaaaiabg2da 9maalaaabaGaaGymaaqaaiaaigdacqGHRaWkcaWGQbGaaGOmaiabec 8aWjaadkfacaWGdbaaaaaa@4E46@
or
H(f)= 1 1+j(f/ f C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiaacI cacaWGMbGaaiykaiabg2da9maalaaabaGaaGymaaqaaiaaigdacqGH RaWkcaWGQbGaaiikamaalyaabaGaamOzaaqaaiaadAgadaWgaaWcba Gaam4qaaqabaGccaGGPaaaaaaaaaa@41A8@ where f C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGdbaabeaaaaa@37D5@ is the cutoff frequency f C = 1 2πRC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGdbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikda cqaHapaCcaWGsbGaam4qaaaaaaa@3DC8@ .

The rearrangement in polar coordinates leads to:

H(f)= 1 1+(f/ f C ) 2 e jarctan(f/ f C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiaacI cacaWGMbGaaiykaiabg2da9maalaaabaGaaGymaaqaamaakaaabaGa aGymaiabgUcaRiaacIcadaWcgaqaaiaadAgaaeaacaWGMbWaaSbaaS qaaiaadoeaaeqaaOGaaiykamaaCaaaleqabaGaaGOmaaaaaaaabeaa aaGccqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeyyXIC TaciyyaiaackhacaGGJbGaaiiDaiaacggacaGGUbGaaiikamaalyaa baGaamOzaaqaaiaadAgadaWgaaadbaGaam4qaaqabaWccaGGPaaaaa aaaaa@531B@

Thus the absolute value | H(f) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca WGibGaaiikaiaadAgacaGGPaaacaGLhWUaayjcSdaaaa@3C29@ yields the amplitude response, and the argument arg(H(f)) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaack hacaGGNbGaaiikaiaadIeacaGGOaGaamOzaiaacMcacaGGPaaaaa@3D28@ yields the phase response.

| H(f) |= 1 1+(f/ f C ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca WGibGaaiikaiaadAgacaGGPaaacaGLhWUaayjcSdGaeyypa0ZaaSaa aeaacaaIXaaabaWaaOaaaeaacaaIXaGaey4kaSIaaiikamaalyaaba GaamOzaaqaaiaadAgadaWgaaWcbaGaam4qaaqabaGccaGGPaWaaWba aSqabeaacaaIYaaaaaaaaeqaaaaaaaa@44D4@

φ(f)=arctan(f/ f C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaai ikaiaadAgacaGGPaGaeyypa0JaeyOeI0IaciyyaiaackhacaGGJbGa aiiDaiaacggacaGGUbGaaiikamaalyaabaGaamOzaaqaaiaadAgada WgaaWcbaGaam4qaaqabaGccaGGPaaaaaaa@45C1@

Amplitude response |H(f)| of RC low-pass with fc = 10MHz
Amplitude response |H(f)| of RC low-pass with fc = 10MHz

Calculate the missing values in the table!

f (frequency) [MHz] 5 10 15 20 40
y ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaja aaaa@3705@ (output amplitude) [V]             0.707             0.447            
φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOgaaa@37B4@ (phase shift) [ ° MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiSaalaaa@37E3@ ] 45.00 63.44
Example output amplitude and phase shift for fc =10MHz and x ^ =1V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaja Gaeyypa0JaaGymaiaadAfaaaa@39A0@
LTI system example: low-pass filter - labAlive experiment

In this laboratory experiment you will analyze the amplitude and phase response of a RC low-pass filter. In a sine-wave analysis the amplitude and phase response is measured.

LTI system example: RC low-pass filter
LTI system example: RC low-pass filter

Start

The simulation starts with a sine input signal at amplitude 1V and frequency of 10MHz. The oscilloscope shows both the input and output signal.

LTI system sine input and output phase shift oscilloscope
Sine input signal (blue) at 1V amplitude and output signal (black). The amplitude, delay and thus the phase shift of the output signal can be measured. The input frequency of 10MHz causes an output amplitude of 0.707 V, hence marking the cutoff frequency.

Experiment

Adjust the input frequency and measure the respective amplitudes and phase delay times of the output signal. Compare the results with the values calculated in the tutorial.

Signal Generator
f (frequency) [MHz] 5 10 15 20 40
y ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaja aaaa@3705@ (output amplitude) [V]             0.707                                    
t (phase delay time) [ns] 12.5
φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOgaaa@37B4@ (phase shift) [ ° MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiSaalaaa@37E3@ ] 45

Note that, with the input signal starting at zero, the phase delay time can be metered at the zero crossing of the output signal (black). The phase shift is the phase delay time divided by the cycle duration.

φ= t360° T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaey ypa0ZaaSaaaeaacaWG0bGaeyyXICTaaG4maiaaiAdacaaIWaGaeyiS aalabaGaamivaaaaaaa@4109@

Note

Stop the simulation after a complete oscilloscope sweep by pressing pause on your keypad and click on the point to measure. Ctrl + click yields the local maximum or zero crossing of the measured curve.

Next steps