Announcements 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 week EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 1 Review Phasors Source vs Impedence representation First Order Circuits Initial and Final conditions Second Order Circuits Solution EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 2 1 Lecture 8 OUTLINE Decibels Transfer function First order lowpass filter Cascade connection and Logarithmic frequency scales Bode Plots Reading Chap 6 6 5 EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 3 Bel and Decibel dB A bel symbol B is a unit of measure of ratios of power levels 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 EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 4 2 Logarithmic Measure for Power To express a power in terms of decibels one starts by choosing a reference power Preference and writing Power 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 dBm EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 5 Aside 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 EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 6 3 Logarithmic Measures for Voltage or Current From 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 as Preference Vreference 2 R or Preference Ireference 2R Hence Voltage V in decibels 20log10 V Vreference Current I in decibels 20log10 I Ireference EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 7 Note 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 is 20 log10 9 1 5 20 log10 6 16 dB EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 8 4 Transfer Function Transfer function is a function of frequency Complex quantity Both magnitude and phase are function of frequency Vin H f Two Port filter network Vout Vout Vout out in Vin Vin H f H f EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 9 Filters Circuit designed to retain a certain frequency range and discard others Low pass pass low frequencies and reject high frequencies High pass pass high frequencies and reject low frequencies Band pass pass some particular range of frequencies reject other frequencies outside that band Notch reject a range of frequencies and pass all other frequencies EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 10 5 Common Filter Transfer Function vs Freq H f H f Low Pass High Pass Frequency Frequency H f H f Band Reject Band Pass Frequency EE40 Summer 2005 Lecture 8 Frequency Instructor Octavian Florescu 11 First Order Lowpass Filter H f VC 1 j C 1 V 1 j C R 1 j RC 1 1 RC 2 tan 1 RC 1 1 and f B 2 RC RC H f H f Let B H f 1 f 1 fB 2 f tan 1 fB 1 H fB 2 1 2 2 H fB 1 20 log10 2 3 dB 20 log10 H 0 2 EE40 Summer 2005 Lecture 8 V Instructor Octavian Florescu R VC C 12 6 First Order Highpass Filter H f VR R j RC V 1 j C R 1 j RC H f f fB f 1 fB 2 2 RC 2 1 RC tan 1 RC 2 f tan 1 fB VR 1 2 1 2 2 H fB 1 20 log10 2 3 dB 20 log10 H 0 2 H fB EE40 Summer 2005 Lecture 8 V R VC C Instructor Octavian Florescu 13 First Order Lowpass Filter L tan 1 R L 1 R R R Let B and f B 2 L L H f H f H f 1 VR j L V 1 R H f 1 f 1 fB 2 EE40 Summer 2005 Lecture 8 1 2 f tan 1 fB VR V Instructor Octavian Florescu R VL L 14 7 First Order Highpass Filter j L VL R H f j L V 1 R L L tan 1 2 R L 1 R R R Let B and f B L 2 L H f H f H f R 2 VR f fB f 1 fB 2 2 f tan 1 fB EE40 Summer 2005 Lecture 8 R V VL L Instructor Octavian Florescu 15 First Order Filter Circuits High Pass VS R C Low Pass Low Pass VS R L High Pass HR R R 1 j C HR R R j L HC 1 j C R 1 j C HL j L R j L EE40 Summer 2005 Lecture 8 Instructor Octavian Florescu 16 8 Gain or Loss Expressed in Decibels The gain produced by an amplifier or the loss of a filter is often specified in decibels The input voltage current or power is taken as the reference value of voltage current or power in the decibel defining expression Voltage gain in dB 20 log10 Voutput Vinput Current gain in dB 20 log10 Ioutput Iinput Power gain in dB 10 log10 Poutput Pinput Example The voltage gain of an amplifier …
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