Oluwayomi Adamo Department of Electrical Engineering

1. EENG 2610: Circuit AnalysisClass 14: Sinusoidal Forcing Functions, Phasors, Impedance and Admittance Oluwayomi Adamo Department of Electrical Engineering College of Engineering, University of North Texas

2. AC Steady-State Analysis • Sinusoidal forcing function (f(t) is forcing function) • The natural response xc(t) is a characteristics of the circuit network and it is independent of the forcing function. • The forced response xp(t) depends on the type of forcing function. • Why study sinusoidal forcing function? • This is the dominant waveform in electric power industry. • Any periodic signal can be represented by a sum of sinusoids (you will learn it in Fourier analysis) • We will only concentrate on the steady-state forced response of networks with sinusoidal forcing functions. • We will ignore the initial conditions and the transient or natural response

3. Sinusoids • Definition • XM is the amplitude • ω is radian or angular frequency (unit: radian/second) • θ is phase angle (unit: radian), • T=2π/ω is period (unit: second) • f=1/T is frequency (unit: Hertz),

4. In-phase and out-of-phase • Any point on the waveform XMsin(ωt+θ) occurs θ radians earlier in time than the corresponding point on the waveform XMsin(ωt). • We say XMsin(ωt+θ) leads XMsin(ωt) by θ radians, or • XMsin(ωt) lags XMsin(ωt+θ) by θ radians. • In the more general situation, if • Then, • x1(t) leads x2(t) by (θ1 - θ2) radians, or, • x2(t) lags x1(t) by (θ1 - θ2) radians. • If θ1 = θ2 ,the waveforms are identical and the functions are said in phase; otherwise, it is said out of phase.

5. Important Trigonometric Identities Polar form Rectangular form

6. Sinusoidal and Complex Forcing Functions • Forcing function and circuit response • If we apply a constant forcing function (i.e., source function) to a network, the steady state circuit response is also a constant. • If we apply a sinusoidal forcing function to a linear network, the steady state circuit response will also be sinusoidal. • Sinusoidal source function • If the input source is v(t)=Asin(ωt+θ), then the output will be in the same sinusoidal form. For example, i(t)=Bsin(ωt+φ). • That means: if the input source is sinusoidal function, we know the form of the output response, and therefore the solution involves simply determining the values of the two parameters B and φ.

7. Learning Example algebraic problem Determining the steady state solution can be accomplished with only algebraic tools!

8. FURTHER ANALYSIS OF THE SOLUTION

9. Phasors • If the forcing function for a linear network is of the form v(t)=VMejωt, • Then every steady-state voltage or current in the network will have the same form and the same frequency ω; for example, a current will be of the form i(t)=IMej(ωt+φ). • In our circuit analysis, we can drop the factor ejωt, since it is common to every term in the describing equations. Phasors are defined as: Sinusoidal signal: The magnitude of phasors are positive !

10. Phasor Analysis (or Frequency Domain Analysis) • The circuit analysis after dropping ejωt term is called phasor analysis or frequency domain analysis. • By phasor analysis, we have transformed a set of differential equations with sinusoidal forcing functions in the time domain into a set of algebraic equations containing complex numbers in the frequency domain. • The phasors are then simply transformed back to the time domain to yield the solution of the original set of differential equations. • Phasor representation: Time Domain Frequency Domain

11. Phasor Relationships for Circuit Elements • We will establish the phasor relationships between voltage and current for the three passive elements R, L, C. Phasor diagram

12. Voltage leads current by 90°

13. Current leads voltage by 90°

14. Definition of Impedance (unit: ohms): • Impedance is defined as the ratio of the phasor voltage V to the phasor current I at the two terminals of the element related to one another by the passive sign convention: • It’s important to note that: • Resistance R and reactance X are real function of the frequency of the forcing function ω, thus Z(ω) is frequency dependent. • Impedance Z is a complex number; however, it is not a phasor, since phasors denote sinusoidal functions.

15. Passive element impedance: Equivalent impedance if impedances are connected in series: KVL and KCL are both valid in frequency domain Equivalent impedance if impedances are connected in parallel: Two terminal input admittance: In general, R and G are not reciprocals of one another. The purely resistive case is an exception.

16. Example 8.9: Determine the equivalent impedance of the network. Then compute i(t) for f = 60 Hz and f = 400 Hz.

17. SPECIAL APPLICATION: IMPEDANCES CAN BE COMBINED USING THE SAME RULES DEVELOPED FOR RESISTORS LEARNING EXAMPLE

18. SERIES-PARALLEL REDUCTIONS LEARNING EXAMPLE

19. AC Steady-State Analysis • For relatively simple circuits (e.g., those with single source), use: • Ohm’s law for AC analysis, i.e., V=IZ • The rules for combining impedance Z (or admittance Y) • KCL and KVL • Current division and voltage division • For more complicated circuits with multiple sources, use: • Nodal analysis • Loop or mesh analysis • Superposition • Source exchange • Thevenin’s and Norton’s theorem • Software tools: MATLAB, PSPICE, …

20. LEARNING EXAMPLE COMPUTE ALL THE VOLTAGES AND CURRENTS

21. Steady-State Power Analysis • Here we study powers in AC circuits: • Instantaneous power • Average power • Maximum power transfer, • Power factor, • Complex power. • Device power ratings: • Typically, electrical and electronic devices have peak power or maximum instantaneous power ratings that cannot be exceeded without damaging the devices.

22. Instantaneous Power Steady-state voltage and current: Instantaneous power: With passive sign convention independent of time a function of time

23. LEARNING EXAMPLE constant Twice the frequency

24. Average Power Since p(t) is a periodic function of time, the average power: For passive sign convention. In the equation, t0is arbitrary, T=2π/ωis the period of the voltage or current. For purely resistive circuit (i.e., Z = R+j0 ): For purely reactive circuit (i.e., Z = 0 + jX): That’s why reactive elements are called lossless elements

25. If voltage and current are in phase LEARNING EXAMPLE Determine the average power absorbed by each resistor, the total average power absorbed and the average power supplied by the source Inductors and capacitors do not absorb power in the average Verification

26. Maximum Average Power Transfer Average power at the load: For maximum average power transfer: If the load is purely resistive (i.e., XL = 0): For maximum average power transfer:

27. Effective or RMS Values • The effective value of a periodic current (or voltage) is defined as a constant or DC value, which would deliver the same average power to a resistor R. • The 120 V AC electrical outlets in our home is the rms value of the voltage: • It is common practice to specify the voltage rating of AC electrical devices (such as light bulb) in terms of the rms voltage. Effective (or rms) value of a periodic current: The average power delivered to a resistor by DC effective current: The average power delivered to a resistor by a periodic current: On using the rms values for the sinusoidal voltage and current, the average power: The power absorbed by a resistor:

28. Compute the rms value of the voltage waveform LEARNING EXAMPLE