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SIGNAL CONDITIONING: FILTERING
Frequency Response The frequency response of a circuit is its steady-state response to a sinusoidal input as the frequency of the sinusoidal input varies.
V i = Vi∠θi and V o = Vo∠θo
Recall in Laplace analysis, the transfer function is obtained by taking the ratio between the output and input. The system's frequency response can be found by substituting jω for s.
TF (s = jω) = TF∠θTF
The relationship between input and output is
Vo∠θo = TF∠θTF Vi∠θi
A circuit's frequency response is a phasor relationship involving magnitude and phase..
- The magnitude response is the ratio of output and input magnitudes. TF is a function of ω.
- The phase response is the phase shift between input and output. That is θTF = θo - θi. θTF is a function of ω.
OUR INTEREST IN FILTERING IS USUALLY CONFINED TO THE MAGNITUDE RESPONSE.
Find the frequency response of this system.
The nodal equations read:
Use Maple to find Vo(s) = TF(s) Vi(s)
restart; eqns:={v1=vi, (v2-v1)s/8+v2/(2s)-2*(v1 -v2)+v2/4=0}: soln:=solve(eqns,{v1,v2}): assign(soln); vo:=v2: TF:=vo/vi;
s (s + 16) TF := ------------- s^2 + 4 + 18 s
This is the system transfer function. To obtain the system frequency response, substitute jω for s.
x 1 2
1 i 2 1 2 2 1 2
Define control variable: V = V - V nodal equations voltage source equation: V = V
KCL at node 2: V^ - V^ + V^ - 2 (V - V ) + V = 0 (^8) 2s 4 s
Active Low Pass This lowpass filter has a break frequency (in rad/s) of 1/RfC and a dc gain of Rf/Rin.
There is also inversion since this is the inverting configuration.
Passive High Pass High pass filters pass high frequencies from input to output and attenuate low frequencies.
The capacitance impedance increases at low frequencies. By voltage division, Vo/Vin will decrease at low frequencies.
Active Band Pass Find the transfer function of this circuit.
