1 ECE 3336 Introduction to Circuits & Electronics Note Set #8 Phasors Spring 2013 TUE&TH 5:30-7:00 pm Dr. Wanda Wosik

2/22 AC Signals (continuous in time) Voltages and currents v(t) and i(t) are functions of time now. We will focus on periodic functions f(t) t Periodic cos(x) Even functions y(x)=x 2 Periodic sin(x) Odd functions y(x)=x 3 -x

3 AC Circuit Analysis (Phasors) AC signals in circuits are very important both for circuit analysis and for design of circuits It can be very complicated to analyze circuits since we will have differential equations (derivatives & integrals from v-i dependences) Techniques that we will use will rely on complex numbers to solve these equations, and on Fourier’s Theorem to represent the signals as sums of sinusoids. Periodic signal waveforms

4 ml Sine waves Amplitude change Frequency change Phase shift +Amplitude and DC shift

STEPPED FREQUENCIES C-major SCALE: successive sinusoids –Frequency is constant for each note IDEAL 5© 2003, JH McClellan & RW Schafer

SPECTROGRAM EXAMPLE Two Constant Frequencies: Beats 6© 2003, JH McClellan & RW Schafer Modulating frequency Frequencies Fo±Fm:660Hz±12Hz

Periodic Signals EXAMPLES: 7 Fourier series is used to represent periodic functions as sums of cosine waves. Fundamental frequency in Fourier series corresponds to signal frequency and added harmonics give the final shape of the signal.

8 AC Circuit Analysis What are Phasors? A phasor is a transformation of a sinusoidal voltage or current. Using phasors and their analysis makes circuit solving much easier. It allows for Ohm’s Law to be used for inductors and capacitors. While they seem difficult at first our goal is to show that phasors make analysis so much easier.

Transformation – Complex Numbers 9 ω 0 means rotation frequency of the rotating phasor Solving circuits: Results: 4 Notice the phase shift  Static part } Drawing by Dr. Shattuck Continuous time dependent periodic signals represented by complex numbers  phasors

t=0 Corresponds to the time dependent voltage changes Graphical Correlation Between CT Signals and Their Phasors At t=0  Rotation of the phasor (voltage vector) V with the angular frequency  In general, the vector’s length is r (amplitude) so V=a+jb in the rectangular form: in the polar form: 10 orappletindex.htm

Current lagging voltage by 90° Current leading voltage by 90° Inductance Capacitance For resistance R both vectors V R (j  t) and I R (j  t) are the same and there is no phase shift! 11 Phasors

Impedances Represented by Complex Numbers Current leading voltage by 90° Current lagging voltage by 90° 12

Transformation of Signals from the Time Domain to Frequency Domain 13 Euler identity

14 Complex Numbers - Reminder Equivalent representations Rectangular Polar

15 Complex Numbers – Reminder Example: Use complex conjugate  and multiply 

16 The Limitations The phasor transform analysis combined with the implications of Fourier’s Theorem is significant. Limitations. The number of sinusoidal components, or sinusoids, that one needs to add together to get a voltage or current waveform, is generally infinite. The phasor analysis technique only gives us part of the solution. It gives us the part of the solution that holds after a long time, also called the steady-state solution.

17 Phasors Used to Represent Circuits Steady state value of a solution the one that remains unchanged after a long time is obtained with the phasor transform technique. Sinusoidal source v s. What is the current that results for t > 0? Kirchhoff’s Voltage Law in the loop: This is a first order differential equation with constant coefficients and a sinusoidal forcing function. The current at t = 0 is zero. The solution of i(t), for t > 0, can be shown to be Will disappear=transient Steady State – use only that

18 More on Transient and Steady State The solution of i(t), for t > 0 is Decaying exponential with Time constant  = L/R. It will die away and become relatively small after a few . This part of the solution is the transient response. This part of the solution varies with time as a sinusoid. It is also a sinusoid with the same frequency as the source, but with different amplitude and phase. This part of the solution is the steady- state response.

19 “Steady State solution” for Phasors Frequency of i ss is the same as the source’s Both the Amplitude and Phase depend on: , L and R Finding the phasor means to determine the Amplitude and Phase Frequency dependence is very important in ac circuits. Euler identity Phasors It was input voltage Calculated current