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## 9. Frequency Response

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**9. Frequency Response**CIRCUITS by Ulaby & Maharbiz**Transfer Function**Transfer function of a circuit or system describes the output response to an input excitation as a function of the angular frequency ω. Other Transfer Functions Voltage Gain Magnitude Phase**RC Low Pass**To determine corner frequency:**Filter Terminology**Zin1= R + jωL. Im [Zin1] = 0 when ω = 0 Im [Zin2] = 0 requires that ZL = −ZC or, equivalently, ω2 = 1/LC**Scaling**Scaling is used to configure a prototype version of the intended practical scaled circuit such that in the prototype circuit, element values are on the order of ohms, henrys and farads.**RL Filter --Magnitude**Log scale for ω and dB scale for M**RL Filter--Phase**Log scale for ω and linear scale for φ(ω)**Bode Plots: Straight line approximations**Bode Magnitude Slope= 20N dB per decade Bode Phase Slope= 45N degrees per decade 1 decade 1 decade**Bode Plots**Bode Magnitude Slope= 40dB per decade Bode Phase Slope= 90 degrees per decade**Example 9-4: Bode Plots**Standard form Numerator: simple zero of second order with corner frequency 5 rad/s Denominator: pole @ origin, and simple pole with corner frequency 50 rad/s**Bandpass RLC Filter (cont.)**Quality Factor Q: characterizes degree of selectivity of a circuit where Wstoris the maximum energy that can be stored in the circuit at resonance (ω = ω0), and Wdissis the energy dissipated by the circuit during a single period T.**Bandpass RLC Filter (cont.)**• Derivation of Q Resonant frequency Bandwidth**Highpass Filter**Lowpass Filter**Superheterodyne receiver**Frequency of received signal is “down-converted” to a lower intermediate frequency, while retaining the modulation ( which contains the message information) intact**Tech Brief 17: Bandwidth and Data Rate**Signal-to-noise ratio**Tech Brief: Bandwidth and Data Rate**Shannon-Hartley Theorem Channel capacity (data rate) in bits/s Bandwidth in Hz Note: A high data rate can be achieved even if the signal power is smaller than the noise, so long as sufficient bandwidth is available.