HW7_solutions[1]

# HW7_solutions[1] - EECS 314 Winter 2007 HW 07 Overview ©...

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Unformatted text preview: EECS 314 Winter 2007 HW 07 Overview © 2007 Alexander Ganago Overview In this HW 07, we keep studying steady-state responses of circuits that include capacitors and inductors to sinusoidal input signals, both at varied frequencies and at fixed frequencies. The main analytic tool for solving these problems is again the use of impedances for the inductors and capacitors Z L = j " # " L Z C = 1 j " # " C Algebraically, we use impedances in the same way we use resistances: each impedance has the units of ohms, both KCL and KVL apply, and we can use all methods of circuit analysis, such as voltage division, current division, node voltages, mesh currents, etc. The main distinctions between impedance and resistance include: 1) Impedance depends on the frequency, while resistance does not. Note that the frequency ω in the equations above is measured in radians/sec, not in Hz. 2) Impedance carries the information about the ratio of the magnitude of voltage to the magnitude of current and about the phase shift between voltage and current; thus we use complex numbers for impedances, while resistance is a real number. Note that all measurable parameters (voltages, currents, phase shifts) are always real. At the fixed frequency ω , each impedance is a complex number, purely imaginary and positive for an inductor, such as +j5 Ω , and purely imaginary and negative for a capacitor, such as – j15 Ω . Note the units of measure: use the frequency ω in rad/sec, inductance in henrys, and capacitance in farads, then the impedance will be in ohms. In problems of this type, we have to go from the time domain to the frequency domain (technically, calculate the impedance for each capacitor and each inductor at the given frequency), solve the circuit equations in the frequency domain (obtain the voltages and currents in the phasor form such as I 1 = I 1, max ∠ϕ 1 and V 2 = V 2,max ∠ ϕ 2 ), and then go back to the time domain in order to express the voltages and currents in the form i 1 = i 1, max ⋅ cos( ω t + ϕ 1 ), and v 2 = v 2,max ⋅ cos( ω t + ϕ 2 ) If we expect the frequency ω to vary, as in the case of filters that work with signals of many frequencies, we keep expressions for Z L and Z C as stated above and consider equations that should be solved for the frequency (for example, in search of the resonance) or limits such as ω → and ω → ∞ . EECS 314 Winter 2007 HW 07 Problem 1 Student's name ___________________________ Discussion section # __________ (Last name, first name, IN INK) © 2006 Alexander Ganago Consider the circuit shown on the diagram: Depending on which connectors (A – F) are used in the particular experiment, this circuit can act as a variety of filters....
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HW7_solutions[1] - EECS 314 Winter 2007 HW 07 Overview ©...

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