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Phasors Complex numbers LCR Oscillator circuit: An example Transient and Steady States

Lecture 18. System Response II. Phasors Complex numbers LCR Oscillator circuit: An example Transient and Steady States. Introduction. Any steady-state voltage or current in a linear circuit with a sinusoidal source is also a sinusoid

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Phasors Complex numbers LCR Oscillator circuit: An example Transient and Steady States

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  1. Lecture 18. System Response II • Phasors • Complex numbers • LCR Oscillator circuit: An example • Transient and Steady States

  2. Introduction • Any steady-state voltage or current in a linear circuit with a sinusoidal source is also 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. • 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

  3. 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

  4. Phasors • Recall that a phasor is a complex number that represents the magnitude and phase of a sinusoidal voltage or current x(t) = XM cos(ωt+θ) ↔ X = XMθ Time domain Frequency Domain • For AC steady-state analysis, this is all we need---we already know the frequency of any voltage or current

  5. Phasors & Complex Numbers • Phasor (frequency domain) is a complex number: • X = zq = x + jy • Sinusoid is a time function: • x(t) = z cos(wt + q) • x is the real part • y is the imaginary part • z is the magnitude • q is the phase imaginary axis y z q real axis x

  6. More Complex Numbers • Polar Coordinates: A = z q • Rectangular Coordinates: A = x + jy imaginary axis y z q real axis x

  7. Examples Find the time domain representations of X = -1 + j2 V = 104V - j60V A = -1mA - j3mA

  8. Impedance • AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks like Ohm’s law: V = IZ • Z is called impedance(units of ohms, W) • Impedance is (often) a complex number, but is not technically a phasor • Impedance depends on frequency, ω

  9. Phasor Relationships for Circuit Elements • Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor • A complex exponential is the mathematical tool needed to obtain this relationship

  10. I-V Relationship for a Resistor + i(t) v(t) R –

  11. I-V Relationship for a Capacitor + i(t) v(t) C –

  12. I-V Relationship for an Inductor + i(t) v(t) L – EEE 202

  13. Impedance Summary

  14. Class Examples • P8-1, 8-4, P8-7, P8-5 • Remember: sin(ωt) = cos(ωt–90°)

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