EECS 105 Fall 2003 Review Prof J S Smith EECS 105 Fall 2003 Review Sinusoidal stimulus z z Sin in Sin at every node We are going to analyze circuits for a single sinusoid at a time which we are going to write It is especially interesting because any voltage or current in our circuit if this is the only input must also be sinusoidal with the same frequency and so can also be written in this form vin t Vi sin t But we are going to use exponential notation 1 vany t Vany e j e j t C C 2 1 iany t I any e j e j t C C 2 vin t Vi sin t Vi e j t e j t 2 vin t Vi e j e j t Vi e j e j t 2 1 vin t Vi e j e j t C C 2 Complex conjugate Because our equations will be linear the same things will happen to the complex conjugate terms as happen to the first terms so they will just tag along same as first term but with j j wherever it occurs Department of EECS University of California Berkeley EECS 105 Fall 2003 Review Prof J S Smith Prof J S Smith Department of EECS EECS 105 Fall 2003 Review University of California Berkeley Prof J S Smith Phasors z Each of the voltages between nodes and each of the currents can then be represented by a single complex number remember this is for a single frequency input of a particular phase and amplitude 1 vany t Vany e j e j t C C V any 2 V any z z You can not use phasor notation without added precautions if you need to multiply voltages and currents such as in a power calculation because that is not linear However you must not take the real part or add the complex conjugate before you put back in the time dependence e j t 1 iany t I any e j e j t C C I any 2 I any Department of EECS University of California Berkeley Department of EECS University of California Berkeley 1 EECS 105 Fall 2003 Review Prof J S Smith EECS 105 Fall 2003 Review Prof J S Smith Complex Transfer Function Impedances of resistors capacitors inductors z z Z r r Z c z 1 C j Excite a system with an input voltage vin Define the output voltage vany to be any node voltage branch current For a complex exponential input the transfer function from input to output or any voltage or current can then be written Z L L j H To find the equivalent impedance for a network Use series or parallel connections Thevenin equivalents Or as a last resort Kerchoff s laws and algebra Department of EECS just multiply top and bottom by ej t sufficient times University of California Berkeley EECS 105 Fall 2003 Review Prof J S Smith Department of EECS University of California Berkeley EECS 105 Fall 2003 Review Prof J S Smith Impedance z Admittance Suppose that the input is defined as the voltage of a terminal pair port and the output is defined as the current into the port v t Arbitrary LTI Circuit i t z Suppose that the input is defined as the current of a terminal pair port and the output is defined as the voltage into the port v t Ve j t V e j t v v t i t Ie j t I e j t i z Department of EECS Arbitrary LTI Circuit i t v t Ve j t V e j t v i t Ie j t I e j t i The impedance Z is defined as the ratio of the phasor voltage to phasor current self transfer function Z H n1 n2 j n3 j 2 L d1 d 2 j d 3 j 2 L z The admittance Z is defined as the ratio of the phasor current to phasor voltage self transfer function Y H V V j v i e I I University of California Berkeley Department of EECS I I e j i v V V University of California Berkeley 2 EECS 105 Fall 2003 Review Prof J S Smith EECS 105 Fall 2003 Review Voltage and Current Gain z Arbitrary LTI Circuit i1 t z z Transimpedance admittance The voltage current gain is just the voltage current transfer function from one port to another port v1 t i2 t z z v2 t V2 V2 j 2 1 e V1 V1 Gi I 2 I 2 j 2 1 e I1 I1 EECS 105 Fall 2003 Review z University of California Berkeley Prof J S Smith To directly calculate the transfer function impedance transimpedance etc we can generalize the circuit analysis concept from the real domain to the phasor domain With the concept of impedance admittance we can now directly analyze a circuit without explicitly writing down any differential equations Use KVL KCL mesh analysis loop analysis or node analysis where inductors and capacitors are treated as complex resistors University of California Berkeley or v2 t J V2 V2 j 2 1 e I1 I1 K I 2 I 2 j 2 1 e V1 V1 S Department of EECS University of California Berkeley EECS 105 Fall 2003 Review Prof J S Smith Bigger Example no problem z Consider a more complicated example Z eff H Vo ZC 2 Vs Z eff Z C 2 H Department of EECS i2 t Direct Calculation of H no DEs z Arbitrary linear Circuit v1 t or i1 t If G 1 the circuit has voltage current gain If G 1 the circuit has loss or attenuation Department of EECS z Current voltage gain are unitless quantities Sometimes we are interested in the transfer of voltage to current or vice versa Gv Prof J S Smith Department of EECS Z eff R2 R1 Z C1 ZC 2 R2 R1 Z C1 Z C 2 University of California Berkeley 3 EECS 105 Fall 2003 Review Prof J S Smith EECS 105 Fall 2003 Review Finding the Magnitude quickly z The magnitude of the response can be calculated quickly by using the property of the mag operator H G0 j K G0 K z Finding the Phase z 1 j z1 1 j z 2 L 1 j p 2 1 j p 2 L the phase can be computed quickly with the following formula 1 j z1 1 j z 2 L p H p G0 j K 1 j p 2 1 j p 2 L p G0 p j K p 1 j z1 p 1 j z 2 L p 1 j p1 p 1 j p 2 L 1 j z1 1 j z 2 L 1 j p 2 1 j p 2 L z The magnitude at DC depends on G0 and the number of poles zeros at DC If K 0 gain is zero If K 0 DC gain is infinite Otherwise if K 0 then gain is simply …
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