54 Ch 12 Phasors - Electrical engineers kept looking for...

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Unformatted text preview: Electrical engineers kept looking for ways to keep using Ohm’s law in circuits with inductors and capacitors (avoiding the pain of solving differential equations) – and found the conditions, under which a generalized Ohm’s law in the form V=I ⋅ Z applies. The conditions are: 1. All sources in the circuit should be sinusoidal, at the same frequency ω 2. All circuit elements should be linear 3. The circuit should be in a sinusoidal steady state (all transients already vanished). In general, the impedance Z is a complex number, which means that voltage V and current I may have different phase angles. • For resistors, Z R = R (the impedance is the same as resistance) • For inductors, Z L = j ⋅ω⋅ L : the impedance is the product of j= √ (-1) , the frequency ω expressed in radians per second, and the inductance L in henry. With these units, the impedance is in ohms • For capacitors Z C = 1/(j ⋅ω⋅ C) : the impedance is reciprocal of the product of j = √ (-1) , the frequency ω in radians per second, and the capacitance in farads. With these units, the impedance is in ohms. Making sense of EE // 2nd ed Phasors © 2008 A. Ganago Page 1 of 18 In circuit equations with impedances, the voltage V and current I also become complex. Of course, all voltages and currents in the lab are real, but math with complex numbers allows us to keep track of both the magnitude and the phase of each voltage and current in the given circuit at the given (fixed) frequency. The complex number that expresses both the magnitude and the phase of a voltage or a current in the circuit at the given frequency is called a phasor (note that the handy weapon used by Star Trek heroes is called a phas e r with an e ). With phasors, we can use all convenient methods of circuit analysis such as voltage and current division, node voltage and mesh current equations (which can be solved with matrix methods), etc....
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54 Ch 12 Phasors - Electrical engineers kept looking for...

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