lecture 14 outline 14-1 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. Vi = Vi∠θi and Vo = 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.. 1. The magnitude response is the ratio of output and input magnitudes. TF is a function of ω. 2. 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.lecture 14 outline 14-2 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/(2*s)-2*(v1-v2)+v2/4=0}: > soln:=solve(eqns,{v1,v2}): > assign(soln); > vo:=v2: > TF:=vo/vi; s (s + 16) TF := ------------- s2 + 4 + 18 s This is the system transfer function. To obtain the system frequency response, substitute jω for s. x121i212212Define control variable: V = V - Vnodal equationsvoltage source equation: V = VV - VVVKCL at node 2: + - 2 (V - V) + = 082s4slecture 14 outline 14-3 Filtering Filters are frequency selective systems. In filtering, the magnitude response is often of most interest. Looking at the filter as an input/output relation, filters are classified by how the input and output magnitudes are related at different frequencies. We’ll use four types of filters: lowpass, highpass, bandpass, and band-reject. Another way to categorize filters is to classify them passive or active. Passive filters consist solely of passive components (R’s, C’s, L’s, transformers, etc.) Passive filters receive energy only from the input. Active filters also use active components such as transistors or op-amps. Active filters can provide a gain >1. They receive energy from the input and from external power supplies. Passive Low Pass Low pass filters pass low frequencies from input to output and attenuate high frequencies. The capacitance impedance decreases at high frequencies. By voltage division, Vo/Vin will go down as well.lecture 14 outline 14-4 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.lecture 14 outline 14-5lecture 14 outline 14-6 Active High Pass This highpass filter has a break frequency (in rad/s) of 1/RinC and a high frequency gain of Rf/Rin. There is also inversion since this is the inverting configuration. Passive Band Pass Band pass filters only pass a band of frequencies from input to output. The capacitance's impedance grows large at low frequency, and the inductance's impedance grows large at high frequency. By voltage division Vo/Vin will grow small at high or low frequencies The peak amplitude of the transfer function is 1 at = 1LCω .lecture 14 outline 14-7lecture 14 outline 14-8 Active Band Pass Find the transfer function of this