1. ELECTRIC CIRCUIT ANALYSIS - I
Chapter 14 – Basic Elements and Phasors
Lecture 16
by Moeen Ghiyas
16/04/2017

2. Chapter 14 – Basic Elements and Phasors
TODAY’S lesson

3. Today’s Lesson Contents
Examples….. Response Of Basic R, L, And C Elements To A Sinusoidal Voltage Or Current
Frequency Response Of The Basic Elements
Ideal Vis-à-vis Real Elements (Effects Of Frequency, Temperature And Current)
Average Power and Power Factor

4. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
EXAMPLE - Voltage across a resistor is v = 25 sin(377t + 60°) Find sinusoidal expression for the current if resistor is 10 Ω. Sketch the curves for v and i.
Solution: From voltage expression and ohm’s law
For resistive network, v and i are in phase, thus

5. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
EXAMPLE - Current through a 0.1-H coil is i = 7 sin(377t - 70°) Find the sinusoidal expression for the voltage across the coil. Sketch the v and i curves
For a coil, v leads i by 900, thus

6. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
EXAMPLE - Voltage across a 1-μF capacitor is v = 30 sin 400t. What is the sinusoidal expression for current? Sketch v and i.
Solution:
For capacitive network, i leads v by 900, thus

7. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
EXAMPLE - For the following pair of voltage and current, determine the element and the value of C, L, or R.
Solution:
Since v and i are in phase, the element is a resistor, and

8. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
EXAMPLE - For the following pair of voltage and current, determine the element and the value of C, L, or R.
Solution:
Since v leads i by 90°, the element is an inductor, and

9. RESPONSE OF BASIC R, L, AND C ELEMENTS TO A SINUSOIDAL VOLTAGE OR CURRENT
EXAMPLE - For the following pair of voltage and current, determine the element and the value of C, L, or R.
Solution:
Since v and i are in phase, the element is a resistor, and

10. Dc, High-, and Low-Frequency Effects on L and C
Inductors
Capacitors

11. Dc, High-, and Low-Frequency Effects on L and C
Inductors and Capacitors

12. Phase Angle Measurements between the Applied Voltage and Source Current
16/04/2017

13. FREQUENCY RESPONSE OF BASIC ELEMENTS Resistance
We know by now that:
resistance generally remains constant with change in frequency,
the inductive reactance increases with increase in frequency
the capacitive reactance decreases with increase in frequency
But in the real world each resistive element has stray capacitance levels and lead inductance that are sensitive to the applied frequency, which are usually so small that their real effect is not noticed until the megahertz range

14. FREQUENCY RESPONSE OF BASIC ELEMENTS Resistance

15. FREQUENCY RESPONSE OF BASIC ELEMENTS Resistance
Frequency, though has an impact on the resistance, but for frequency range of interest, we will assume that the resistance level of a resistor is independent of frequency as also shown in fig for up to 15 MHz.

16. FREQUENCY RESPONSE OF BASIC ELEMENTS Inductor
The equation of inductive reactance
is directly related to the straight-line equation
with a slope (m) of 2πL and a y-intercept (b) of zero
The larger the inductance, the greater the slope (m = 2πL) for the same frequency range

17. FREQUENCY RESPONSE OF BASIC ELEMENTS Capacitor
The equation of capacitive reactance
Can be written as
which matches the basic format of a hyperbola,
With
y = XC,
x = f,
and constant k = 1/(2πC).
Note that an increase in capacitance causes the reactance to drop off more rapidly with frequency

18. FREQUENCY RESPONSE OF BASIC ELEMENTS Capacitor
Therefore, as the applied frequency increases (up-to range of our interest);
the resistance of a resistor remains constant,
the reactance of an inductor increases linearly, and
the reactance of a capacitor decreases nonlinearly

19. FREQUENCY RESPONSE OF BASIC ELEMENTS
EXAMPLE - At what frequency will an inductor of 5 mH have the same reactance as a capacitor of 0.1 μF?
Solution:

20. Ideal vis-à-vis Real Elements (Effects of Frequency, Temperature and Current)
A true equivalent for an inductor in Fig
The series resistance Rs represents
the copper losses (resistance of thin copper wire turned);
the eddy current losses (losses due to small circular currents in the core when an ac voltage is applied);
and the hysteresis losses (core losses created by the rapidly reversing field in the core).
The capacitance Cp is the stray capacitance that exists between the windings of inductor.

21. Ideal vis-à-vis Real Elements (Effects of Frequency, Temperature and Current)
Dropping level of CP will begin to have a shorting effect across the windings of inductor at high frequencies.
Inductors lose their ideal characteristics and begin to act as capacitive elements with increasing losses at very high frequencies.

22. Ideal vis-à-vis Real Elements (Effects of Frequency, Temperature and Current)
The equivalent model for a capacitor
The resistance Rs, resistivity of the dielectric and the case resistance, will determine leakage current during the discharge cycle.
The resistance Rp reflects the energy lost as the atoms continually realign themselves in the dielectric due to the applied alternating ac voltage.

23. Ideal vis-à-vis Real Elements (Effects of Frequency, Temperature and Current)
The inductance Ls includes the inductance of the capacitor leads and any inductive effects introduced by the design of the capacitor.
The inductance of the leads is about μH per centimetre (or 0.2 μH for a capacitor with two 2-cm leads) — a level that can be important at high frequencies.

24. Ideal vis-à-vis Real Elements (Effects of Frequency, Temperature and Current)
As frequency increases, reactance Xs becomes larger, eventually the reactance of the coil equals that of capacitor (a resonant condition – Ch 20). Further increase in frequency will simply result in Xs being greater than XC, and the element will behave like an inductor.

25. Ideal vis-à-vis Real Elements (Effects of Frequency, Temperature and Current)
Frequency of application and expected temperature range of operation define the type of capacitor (or inductor) that would be used:
Electrolytic capacitors are limited to frequencies up to 10 kHz, while ceramic or mica handle beyond 10 MHz
Most capacitors tend to loose their capacitance values both at lower (sharp decline) and high temperatures than room temperatures but their sensitivity rests with their type of construction.

26. AVERAGE POWER AND POWER FACTOR
We know for any load
v = Vm sin(ωt + θv)
i = Im sin(ωt + θi)
Then the power is defined by
Using the trigonometric identity
Thus, sine function becomes

27. AVERAGE POWER AND POWER FACTOR
Putting above values in
We have
The average value of 2nd term is zero over one cycle, producing no net transfer of energy in any one direction.
The first term is constant (not time dependent) is referred to as the average power or power delivered or dissipated by the load.

28. AVERAGE POWER AND POWER FACTOR

29. AVERAGE POWER AND POWER FACTOR
Since cos(–α) = cos α,
the magnitude of average power delivered is independent of whether v leads i or i leads v.
Thus, defining θ as equal to | θv – θi |, where | | indicates that only the magnitude is important and the sign is immaterial, we have average power or power delivered or dissipated as

30. AVERAGE POWER AND POWER FACTOR
The above eq for average power can also be written as
But we know Vrms and Irms values as
Thus average power in terms of vrms and irms becomes,
16/04/2017

31. AVERAGE POWER AND POWER FACTOR
For resistive load,
We know v and i are in phase, then |θv - θi| = θ = 0°,
And cos 0° = 1, so that
becomes or

32. AVERAGE POWER AND POWER FACTOR
For inductive load ( or network),
We know v leads i, then |θv - θi| = θ = 90°,
And cos 90° = 0, so that
Becomes
Thus, the average power or power dissipated by the ideal inductor (no associated resistance) is zero watts.

33. AVERAGE POWER AND POWER FACTOR
For capacitive load ( or network),
We know v lags i, then |θv - θi| = |–θ| = 90°,
And cos 90° = 0, so that
Becomes
Thus, the average power or power dissipated by the ideal capacitor is also zero watts.

34. AVERAGE POWER AND POWER FACTOR
In the equation,
the factor that has significant control over the delivered power level is cos θ.
No matter how large the voltage or current, if cos θ = 0, the power is zero; if cos θ = 1, the power delivered is a maximum.
Since it has such control, the expression was given the name power factor and is defined by
For situations where the load is a combination of resistive and reactive elements, the power factor will vary between 0 and 1

35. AVERAGE POWER AND POWER FACTOR
In terms of the average power, we know power factor is
The terms leading and lagging are often written in conjunction with power factor and defined by the current through load.
If the current leads voltage across a load, the load has a leading power factor. If the current lags voltage across the load, the load has a lagging power factor.
In other words, capacitive networks have leading power factors, and inductive networks have lagging power factors.

36. AVERAGE POWER AND POWER FACTOR
EXAMPLE - Determine the average power delivered to network having the following input voltage and current:
v = 150 sin(ωt – 70°) and i = 3 sin(ωt – 50°)
Solution

37. AVERAGE POWER AND POWER FACTOR
EXAMPLE - Determine the power factors of the following loads, and indicate whether they are leading or lagging:
Solution:

38. AVERAGE POWER AND POWER FACTOR
EXAMPLE - Determine the power factors of the following loads, and indicate whether they are leading or lagging:
Solution:

39. Summary / Conclusion
Examples….. Response Of Basic R, L, And C Elements To A Sinusoidal Voltage Or Current
Frequency Response Of The Basic Elements
Ideal Vis-à-vis Real Elements (Effects Of Frequency, Temperature And Current)
Average Power and Power Factor

40. 16/04/2017