ECE 3336 Introduction to Circuits & Electronics

1. ECE 3336 Introduction to Circuits & Electronics Note Set #9 Phasors, cont. Spring 2013, TUE&TH 5:30 – 7:00 pm Dr. Wanda Wosik

2. Phasor Analysis A phasor is a transformation of a sinusoidal voltage or current. Using phasors, and the techniques of phasor analysis, solving circuits with sinusoidal sources gets much easier. We have already defined phasors. We will show how to obtain the solution with phasors. This will be in the frequency domain. To return to time domain i.e. physical operation of circuits, we will need to transform phasor results to time domain results.

3. Phasor - Match with a Sinusoid Time domain (t) A phasor is a complex numberrepresenting a cos signal (amplitude and phase) at given frequency – here static. Notice various notations! Phasors () P{x(t)} indicates transformation The time domain function is not equal to the phasor.

4. Phasors – Things to Remember • A phasor is a complex number • whose magnitude is the magnitude of a corresponding sinusoid, • and whose phase is the phase of that corresponding sinusoid. • A phasor is complex - does not exist. • Voltages and currents are real - do exist. • A voltage v(t)≠V(jw) its phasor. • A current i(t)≠V(jw) its phasor. • A phasor is a function of frequency, w. • A sinusoidal voltage or current is a function of time, t. Different domains Different domains

5. Circuit Elements in the Phasor Domain To solve circuits all elements have to go through transformation Independent Sources The phasor transform of an independent voltage source is an independent voltage source, with a value equal to the phasor of that voltage. The phasor transform of an independent current source is an independent current source, with a value equal to the phasor of that current. is(t)=Imcos(t+) Im()=Imej

6. Phasor Transforms of Dependent Sources

7. Phasor Transforms of Resistors The phasor transform of a resistor is just a resistor. The ratio of phasor voltage to phasor current is called impedance, with units of [Ohms], or [W], and using a symbol Z. The ratio of phasor current to phasor voltage is called admittance, with units of [Siemens], or [S], and using a symbol Y. For a resistor, the impedance and admittance are real. Ir()/Vr()=YR=G Vr()/Ir()=ZR=R

8. Phasor Transforms of Inductors The phasor transform of an inductor is an inductor with an impedance of jwL. This impedance (in the phasor domain) increases with frequency For an inductor, the impedance and admittance are purely imaginary. The impedance is positive, and the admittance is negative.

9. Phasor Transforms of Capacitors Time domain Phasor domain Impedance decreases With frequency 

10. Table of Phasor Transforms The phasor transforms can be summarized below. Voltages transform to phasors, currents to phasors, and passive elements to their impedances.

11. “Steady State solution” for Phasors • Frequency of iss 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 Example: Circuit response (previous) Phasor – use them to solve circuits Euler identity Frequency dependence is very important in ac circuits. 

12. Imagine the circuit here has a sinusoidal source. What is the steady state value for the current i(t)? Example Solution the Hard Way – time domain Let’s solve this circuit in time domain. We will only do this once, to show that we will never wantto do it again. If the source is sinusoidal, it must have the form, Applying Kirchhoff’s Voltage Law around the loops we get the differential equation, This is a differential equation, first order, with constant coefficients, and a sinusoidal forcing function. The solution will be a sinusoid with the same frequency as the forcing function. go to #18 to phasors solution From D. Shattuck

13. Example Solution the Hard Way Use the Known Solution Our circuit is here again. Still looking for the steady state value for the current i(t)? We can substitute the forcing function into the KVL equation, and get, From D. Shattuck

14. Example Solution the Hard Way Use Euler Identity Next, we take advantage of Euler’s relation, which is Still looking for the steady state value for the current i(t)? This allows us to express our cosine functions as the real part of a complex exponential, So our equation will include common term From D. Shattuck

15. So, now we have, Example Solution the Hard Way More Transformations (math) So, now we can take the derivative and put it inside the Re statement. We can do the same thing with the constant coefficients. This gives us Next, we note that if the real parts of a general expression are equal, the quantities themselves must be equal. So, we can write that We can perform the derivative, and get From D. Shattuck

16. So, now we have, Example Solution the Hard Way ejt will go away So, now we recognize that and divide by it on both sides of the equation to get Next, we pull out the common terms on the left hand side of the equation, Finally, we divide both sides by the expression in parentheses, which again cannot be zero, and we get From D. Shattuck

17. So, now we have, Example Solution the Hard Way Now we will Have Phasors Phasors Our original problem will be solvedsoon in time domain.  This is the solution. We assumed that we know R and L as well as the vS(t) source  so we know Vm, w and . So the only quantities that are unknown are Im and q. Is this sufficient? Do we have everything we need to be able to solve? From D. Shattuck

18. We have, We got the Solution the Hard Way Solution in complex numbers We have everything we need to be able to solve. This is a complex equation with two unknowns. Therefore, we can set the real parts equal, and the imaginary parts equal, and get two equations, with two unknowns, and solve. Alternatively, we can set the magnitudes equal, and the phases equal, and get two equations, with two unknowns, and solve. The solution must be it in time domain Return to #12 From D. Shattuck

19. Solving the Same Circuit Using Phasors The first step is to transform the problem into the phasor domain. Transformed circuit – phasor domain Original circuit – time domain  where Im and q are the values we need to obtain. Our phasors are the following complex numbers

20. Now, we examine this circuit, combining the two impedancesin series as we would for resistances, we can write in one step, Solution with Phasors = the Easy Way cont. Impedance Ohm’s Law

21. Original circuit The Phasor Solution Using phasors phase also here Compare the magnitude of each side. Then the phase of each side. We get

22. The Sinusoidal Steady-State Solution Do the inverse phasor transformation to obtain time domain signal. This is the same solution obtained before in the time domain. Phasor Domain solutions are MUCH EASIER and SHORTER than those in the Time Domain.