Phasor Diagrams and Impedance

Lecture 21

Set Phasors on Stun1.Sinusoids-amplitude, frequency and phase (Section 8.1)2.Phasors-amplitude and phase (Section 8.3 [sort of])2a. Complex numbers (Appendix B).

Lecture 21

Set Phasors on Kill3.Complex exponentials-amplitude and phase 4.Relationship between phasors, complex exponentials, and sinusoids5.Phasor relationships for circuit elements (Section 8.4)5a. Arithmetic with complex numbers (Appendix B).

Lecture 21

Set Phasors on Vaporize6.Fundamentals of impedance and admittance (some of Section 8.5)7.Phasor diagrams (some of Section 8.6)

Lecture 21

Phasor DiagramsA phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes).A phasor diagram helps to visualize the relationships between currents and voltages.

Lecture 21

An Example-1mFVC+-2mA 401kWVR++-V

Lecture 21

An Example (cont.)I = 2mA 40VR = 2V 40 VC = 5.31V -50V = 5.67V -29.37

Lecture 21

Phasor DiagramReal AxisImaginary AxisVRVCV

Lecture 21

ImpedanceAC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohms law:V = I ZZ is called impedance.

Lecture 21

ImpedanceResistor:

The impedance is R

Inductor:

The impedance is jwL

Lecture 21

ImpedanceCapacitor:

The impedance is 1/jwL

Lecture 21

Some Thoughts on ImpedanceImpedance depends on the frequency w.Impedance is (often) a complex number.Impedance is not a phasor (why?).Impedance allows us to use the same solution techniques for AC steady state as we use for DC steady state.

Lecture 21

Impedance Example:Single Loop Circuitw = 377Find VC20kW+-1mF10V 0VC+-

Lecture 21

Impedance ExampleHow do we find VC?First compute impedances for resistor and capacitor:ZR = 20kW= 20kW 0 ZC = 1/j (377 1mF) = 2.65kW -90

Lecture 21

Impedance Example20kW 0+-2.65kW -9010V 0VC+-

Lecture 21

Impedance ExampleNow use the voltage divider to find VC:

Lecture 21

What happens when w changes?w = 10Find VC20kW+-1mF10V 0VC+-

Lecture 21

Low Pass Filter:A Single Node-pair CircuitFind v(t) for w=2p 3000

Lecture 21

Find Impedances1kW-j530kW5mA 0+-V

Lecture 21

Find the Equivalent Impedance

Lecture 21

Parallel Impedances

Lecture 21

Computing V

Lecture 21

Change the FrequencyFind v(t) for w=2p 455000

Lecture 21

Find Impedances

Lecture 21

Find an Equivalent Impedance

Lecture 21

Parallel Impedances

Lecture 21

Computing V

Lecture 21