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Single-Node-Pair Circuits (2.4); Sinusoids (7.1);

Single-Node-Pair Circuits (2.4); Sinusoids (7.1);. Dr. S. M. Goodnick September 5, 2002. Example: 3 Light Bulbs in Parallel. How do we find I 1 , I 2 , and I 3 ?. +. I 1. I 2. I 3. I. R 1. R 2. R 3. V. –. Apply KCL at the Top Node. I= I 1 + I 2 + I 3. Solve for V. R eq.

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Single-Node-Pair Circuits (2.4); Sinusoids (7.1);

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  1. Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002 ECE201 Lect-5

  2. Example: 3 Light Bulbs in Parallel How do we find I1, I2, and I3? + I1 I2 I3 I R1 R2 R3 V – ECE201 Lect-5

  3. Apply KCL at the Top Node I= I1 + I2 + I3 ECE201 Lect-5

  4. Solve for V ECE201 Lect-5

  5. Req Which is the familiar equation for parallel resistors: ECE201 Lect-5

  6. Current Divider • This leads to a current divider equation for three or more parallel resistors. • For 2 parallel resistors, it reduces to a simple form. • Note this equation’s similarity to the voltage divider equation. ECE201 Lect-5

  7. Example: More Than One Source How do we find I1 or I2? + I1 I2 Is1 Is2 R1 R2 V – ECE201 Lect-5

  8. Apply KCL at the Top Node I1 + I2 = Is1 - Is2 ECE201 Lect-5

  9. Multiple Current Sources • We find an equivalent current source by algebraically summing current sources. • As before, we find an equivalent resistance. • We find V as equivalent I times equivalent R. • We then find any necessary currents using Ohm’s law. ECE201 Lect-5

  10. In General: Current Division Consider N resistors in parallel: Special Case (2 resistors in parallel) ECE201 Lect-5

  11. Class Examples • Learning Extension E2.11 ECE201 Lect-5

  12. Sinusoids: Introduction • Any steady-state voltage or current in a linear circuit with a sinusoidal source is a sinusoid. • This is a consequence of the nature of particular solutions for sinusoidal forcing functions. • All steady-state voltages and currents have the same frequency as the source. ECE201 Lect-5

  13. Introduction (cont.) • In order to find a steady-state voltage or current, all we need to know is its magnitude and its phase relative to the source (we already know its frequency). • Usually, an AC steady-state voltage or current is given by the particular solution to a differential equation. ECE201 Lect-5

  14. The Good News! • We do not have to find this differential equation from the circuit, nor do we have to solve it. • Instead, we use the concepts of phasorsand complex impedances. • Phasors and complex impedances convert problems involving differential equations into simple circuit analysis problems. ECE201 Lect-5

  15. Phasors • A phasor is a complex number that represents the magnitude and phase of a sinusoidal voltage or current. • Remember, for AC steady-state analysis, this is all we need---we already know the frequency of any voltage or current. ECE201 Lect-5

  16. Complex Impedance • Complex impedance describes the relationship between the voltage across an element (expressed as a phasor) and the current through the element (expressed as a phasor). • Impedance is a complex number. • Impedance depends on frequency. ECE201 Lect-5

  17. Complex Impedance (cont.) • Phasors and complex impedance allow us to use Ohm’s law with complex numbers to compute current from voltage, and voltage from current. ECE201 Lect-5

  18. Sinusoids • Period: T • Time necessary to go through one cycle • Frequency: f = 1/T • Cycles per second (Hz) • Angular frequency (rads/sec): w = 2pf • Amplitude: VM ECE201 Lect-5

  19. Example What is the amplitude, period, frequency, and angular (radian) frequency of this sinusoid? ECE201 Lect-5

  20. Phase ECE201 Lect-5

  21. Leading and Lagging Phase x1(t) leads x2(t) by q- x2(t) lags x1(t) by q- On the preceding plot, which signals lead and which signals lag? ECE201 Lect-5

  22. Class Examples • Learning Extension E7.1 • Learning Extension E7.2 ECE201 Lect-5

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