1EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu1Announcements Midterm #1 Tuesday July 12, 11:30 am – 1pm in 145 Dwinelle Non-programmable calculators allowed 1 double-sided cheat sheet allowed. Must be hand made Material up to and including lecture 7 Midterm Review Session Monday July 11, 5 – 8pm in 277 Cory Attend only your second lab slot next weekEE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu2Review Phasors Source vs. Impedence representation First Order Circuits Initial and Final conditions Second Order Circuits Solution2EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu3Lecture #8OUTLINE Decibels Transfer function First-order lowpass filter Cascade connection and Logarithmic frequency scales Bode PlotsReading Chap 6-6.5EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu4Bel and Decibel (dB) A bel (symbol B) is a unit of measure of ratios of powerlevels, i.e. relative power levels. The name was coined in the early 20th century in honor of Alexander Graham Bell, a telecommunications pioneer. The bel is a logarithmic measure. The number of bels for a given ratio of power levels is calculated by taking the logarithm, to the base 10, of the ratio. one bel corresponds to a ratio of 10:1. B = log10(P1/P2) where P1 and P2 are power levels. The bel is too large for everyday use, so the decibel (dB), equal to 0.1B, is more commonly used. 1dB = 10 log10(P1/P2) dB are used to measure Electric power, Gain or loss of amplifiers, Insertion loss of filters.3EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu5Logarithmic Measure for Power To express a power in terms of decibels, one starts by choosing a reference power, Preference, and writingPower P in decibels = 10 log10(P/Preference) Exercise: Express a power of 50 mW in decibels relative to 1 watt. P (dB) =10 log10(50 x 10-3) = - 13 dB Exercise: Express a power of 50 mW in decibels relative to 1 mW. P (dB) =10 log10(50) = 17 dB. dBm to express absolute values of power relative to a milliwatt. dBm = 10 log10(power in milliwatts / 1 milliwatt) 100 mW = 20 dBm 10 mW = 10 dBmEE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu6Aside About Resonant Circuits When dealing with resonant circuits it is convenient to refer to the frequency difference between points at which the power from the circuit is half that at the peak of resonance. Such frequencies are known as “half-power frequencies”, and the power output there referred to the peak power (at the resonant frequency) is 10log10(Phalf-power/Presonance) = 10log10(1/2) = -3 dB.4EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu7From the expression for power ratios in decibels, we can readily derive the corresponding expressions for voltage or current ratios.Suppose that the voltage V (or current I) appears across (or flows in) a resistor whose resistance is R. The corresponding power dissipated, P, is V2/R (or I2R). We can similarly relate the reference voltage or current to the reference power, asPreference= (Vreference)2/R or Preference= (Ireference)2R.Hence,Voltage, V in decibels = 20log10(V/Vreference)Current, I, in decibels = 20log10(I/Ireference)Logarithmic Measures for Voltage or CurrentEE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu8Note that the voltage and current expressions are just like the power expression except that they have 20 as the multiplier instead of 10 because power is proportional to the square of the voltage or current.Exercise: How many decibels larger is the voltage of a 9-volt transistor battery than that of a 1.5-volt AA battery? Let Vreference= 1.5. The ratio in decibels is20 log10(9/1.5) = 20 log10(6) = 16 dB.5EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu9Transfer Function Transfer function is a function of frequency Complex quantity Both magnitude and phase are function of frequencyTwo Port filter networkVinVout()()()outout ininVfVHfθθθ==∠−=∠outinVHVH(f)EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu10Filters Circuit designed to retain a certain frequency range and discard othersLow-pass: pass low frequencies and reject high frequenciesHigh-pass: pass high frequencies and reject low frequenciesBand-pass: pass some particular range of frequencies, reject other frequencies outside that bandNotch: reject a range of frequencies and pass all other frequencies6EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu11Common Filter Transfer Function vs. Freq()HfFrequencyHigh Pass()HfFrequencyLow Pass()HfFrequencyBand PassFrequencyBand Reject()HfEE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu12First-Order Lowpass Filter()()12121( ) 1 1tan1( ) 11112()1() , tan1BBBBjCRCjC R jRCRCLet and fRC RCHffHffffωωωωωωπθθ−−=== ∠−+++===∠⎛⎞==−⎜⎟⎝⎠⎛⎞+⎜⎟⎝⎠CVH(f) =VH(f)R+-CVVC+-1/210 101() 22()120log 20( )log 2 3(0) 2BBHfHfdBH−===− =−7EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu13First-Order Highpass Filter()()()1212tan1( ) 1 21() , tan21RBBBRCRjRCRCjC R jRCRCfffHffffωωπωωωωπθ−−⎡⎤=== ∠−⎢⎥++⎣⎦+⎛⎞⎜⎟⎛⎞⎝⎠==−⎜⎟⎝⎠⎛⎞+⎜⎟⎝⎠VH(f)=VR+-CVVC+-1/210 101() 22()120log 20( )log 2 3(0) 2BBHfHfdBH−===− =−VREE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu14First-Order Lowpass Filter121211tan112()1() , tan1RBBBBLjLRLRRRRLet and fLLHffHffffωωωωπθθ−−⎛⎞== ∠−⎜⎟⎝⎠⎛⎞++⎜⎟⎝⎠===∠⎛⎞==−⎜⎟⎝⎠⎛⎞+⎜⎟⎝⎠VH(f) =VH(f)R+-LVVL+-VR8EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu15First-Order Highpass Filter1212tan2112()() , tan21LBBBBBjL LLRRjLRLRRRRLet and fLLHffffHffffωωπωωωωπθπθ−−⎡⎤⎛⎞== ∠−⎜⎟⎢⎥⎝⎠⎣⎦⎛⎞++⎜⎟⎝⎠===∠⎛⎞⎜⎟⎛⎞⎝⎠==−⎜⎟⎝⎠⎛⎞+⎜⎟⎝⎠VH(f) =VH(f)R+-LVVL+-VREE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu16First-Order Filter CircuitsL+–VSCRLow PassHigh PassHR= R / (R + jωL)HL= jωL/ (R + jωL)+–VSRHigh PassLow PassHR= R / (R + 1/jωC)HC= (1/jωC) / (R + 1/jωC)9EE40 Summer 2005: Lecture 8 Instructor: Octavian Florescu17Gain or Loss Expressed in
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