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

2. 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. 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 phasors and complex impedances Phasors and complex impedances convert problems involving differential equations into simple circuit analysis problems

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

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

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

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

8. 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 = I Z Z is called impedance (units of ohms,  ) Impedance is (often) a complex number, but is not technically a phasor Impedance depends on frequency, ω

9. 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. 10 I-V Relationship for a Resistor R v(t)v(t) + – i(t)i(t)

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

12. F06-Lect17EEE 20212 I-V Relationship for an Inductor L v(t)v(t) + – i(t)i(t)

13. 13 Impedance Summary

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