Chapter 13 The Basic Elements and Phasors

Objectives Be able to add and subtract sinusoidal voltages or currents Use phasor format to add and subtract sinusoidal waveforms Become familiar with complex numbers and the rectangular and polar formats Perform all basic mathematical operations using complex numbers Using phasor notation be able to apply Kirchhoff’s voltage and current law to ac networks

Objectives Understand the response of resistors, inductors and capacitors to an ac signal Apply phasor notation to the analysis of the basic elements Become aware of how the resistance of a resistor or reactance of a capacitor or inductor will change with frequency

Adding and Subtracting Sinusoidal Waveforms For ac circuits, the currents and voltages will be sinusoidal (vary with time) and may be out of phase –The methods used in this chapter will be only for sinusoidal waveforms that are the same frequency When adding or subtracting sinusoidal waveforms of the same frequency the resulting waveform will also be sinusoidal with the same frequency

Adding and Subtracting Sinusoidal Waveforms Vectors are a “snapshot” of the rotating vectors at t = 0 s or  = 0° The radius vectors are called phasors when applied to electric circuits –the peak value is the sum –  T is the phase angle

Adding and Subtracting Sinusoidal Waveforms Finding the sum or difference of two sinusoidal waveforms requires putting each in the phasor format and performing the required vector algebra –the angle associated with each vector is the phase angle associated with the standard format for a sinusoidal waveform

Complex Numbers A complex number defines a point in a two- dimensional plane established by two axes at 90  to one another –the horizontal axis is the real axis –the vertical axis is the imaginary axis Two forms are used to represent a complex number: rectangular and polar –each can define the vector drawn from the origin to a point in the two-dimensional plane

Complex Numbers The rectangular form is: –the boldface notation is for any number with magnitude and direction –the italic notation is for magnitude only The polar form is: –Z indicates magnitude only –  is measured counterclockwise from the positive real axis

Complex Numbers Conversion between forms: –Rectangular to Polar: –Polar to Rectangular:

Mathematical Operations with Complex Numbers Complex numbers lend themselves readily to the basic mathematical operations of addition, subtraction, multiplication and division The symbol j associated with imaginary numbers is by definition:

Complex Numbers The complex conjugate of a complex number can be found by: –changing the sign of the imaginary part in rectangular form –using a negative sign of the angle in the polar form

Complex Numbers Addition or subtraction of complex numbers will normally be performed in rectangular form –addition or subtraction of two complex numbers requires that the real and imaginary parts be worked on independently addition or subtraction cannot be performed in polar form unless the complex numbers have the same angle  or unless they differ only by multiples of 180 

Complex Numbers Multiplication (division) can be performed using either rectangular or polar form, although it is usually much easier to perform these operations in the polar form –in polar form the magnitudes are multiplied (divided) and the angles are added (subtracted) algebraically

Applying Kirchhoff’s Laws using Phasor Notation Kirchhoff’s laws can be applied to any sinusoidal waveform with any phase angle –apply each law in the same way it would be applied to dc circuits –work with the phasor notation for each waveform to determine the desired solution

Resistors and the AC Response At every instant of time, other than when i R = 0A (or v R = 0V), power is being delivered to the resistor irrespective of the direction of the current through (or polarity of the voltage across) the resistor For resistive elements, the applied voltage and the resulting current are in-phase The impedance of a resistive element is: The frequency or angular velocity is not a part of the phasor notation

Inductors and the AC Response The opposition of an inductor to the flow of current is directly related to the inductance of the inductor and the frequency (rate of change) of the current The opposition of an inductor is called its reactance Reactance is quite different from resistance in that all the electrical energy transferred to an ideal inductor is not dissipated but simply stored in the form of a magnetic field

Inductors and the AC Response For a pure inductor the peak values of the current and voltage are related by an Ohm’s law relationship: The voltage across an inductor leads the current through the inductor by 90 

Inductors and the AC Response Using phasor notation, the impedance of a coil will be defined by: Vector representation of the impedance of an inductor ensures that the phase angle associated with the voltage or current is correct –this format does not define a sinusoidal function

Capacitors and the AC Response The larger the capacitance of a capacitor, the smaller the opposition to the flow of current For a capacitor, the higher the applied frequency the less the opposition factor, termed reactance The reactance of a capacitor is quite different from resistance in that all the electrical energy transferred to an ideal capacitor is not dissipated but simply stored in the form of an electric field

Capacitors and the AC Response For a capacitor the peak values of current and voltage are related by an Ohm’s law relationship: The current of a capacitor leads the voltage across the capacitor by 90 

Capacitors and the AC Response Using phasor notation, the impedance of a capacitor will be defined by: Vector representation of the impedance of a capacitor ensures that the phase angle associated with the voltage or current is correct –this format does not define a sinusoidal function

Power and the Basic Elements ac power delivered to resistive elements has the same format as for dc circuits: the peak value of the sinusoidal function defining the power curve is the product of the effective or rms values of the voltage and current

Power and the Basic Elements Note that the entire power curve is above the horizontal axis, indicating that all power delivered is dissipated by the resistive element The area under the curve is the energy dissipated by the resistive element –for one full period of the applied voltage or current:

Power and the Basic Elements The power curve has a sinusoidal pattern at twice the frequency of the applied voltage or current –for every cycle of the voltage or current there are two cycles of the power curve (delivered to the inductor)

Power and the Basic Elements Due to the 90  phase shift, there are regions where either the voltage or current will be negative, resulting in a negative product for the power level The power curve has equal areas above and below the axis for one full period of the applied signal –over one full cycle the energy absorbed is equal to that returned - no net dissipation For inductors, a quantity called reactive power has been defined

Power and the Basic Elements Since the ideal capacitor and inductor are purely reactive elements the results obtained for the capacitor will be a close match to those for the inductor

Frequency Response of the Basic Elements Every commercial element available today will not respond in the ideal fashion for the full range of possible frequencies We will consider the response of ideal elements For the ideal resistor, frequency will have no effect on the impedance level

Frequency Response of the Basic Elements At a frequency of 0 Hz an inductor takes on the characteristics of a short-circuit At very high frequencies the characteristics of an inductor approach those of an open- circuit –the higher the inductance value, the quicker it will approach the open-circuit equivalence

Frequency Response of the Basic Elements At, or near, 0 Hz the characteristics of a capacitor approach those of an open-circuit At very high frequencies, a capacitor takes on the characteristics of a short circuit –reactance drops very rapidly as the frequency increases

Summary Sinusoidal waveforms can be added graphically or by representing each by a radius vector and adding the vectors using vector algebra The addition or subtraction of two sinusoidal waveforms of the same frequency will result in a sinusoidal waveform of the same frequency Addition and subtraction are performed in the rectangular form unless the phase angles are the same or 180  out of phase, in which case we can use the polar form

Summary Multiplication and division can be performed in both the rectangular and polar forms, but usually we use the polar form The symbol j is equal to the square root of -1 with j 2 equal to -1 Using the phasor notation, two sinusoidal waveforms can be added or their difference found using phasor (complex) algebra

Summary When applied to electric circuits the magnitude of a phasor quantity is the effective or rms value of the voltage or current. The angle is the phase angle of the voltage or current The vector form for the resistance of a resistor or the impedance of an inductor or capacitor does not represent a sinusoidal function The power delivered to a resistive element is dissipated in the form of heat

Summary The average or real power delivered to a purely inductive or capacitive element is zero, although there is a reactive power level associated with each that is found using equations having the same format as those used for the resistive element Ideally the resistance of a resistor is independent of the frequency applied The impedance of an ideal inductive element will increase in a straight line fashion as the frequency increases

Summary The impedance of an ideal capacitor will be very high at low frequencies and drop off dramatically as the frequency increases At very low frequencies an inductor can be replaced by a short-circuit equivalent and a capacitor by an open-circuit equivalent. At very high frequencies the reverse is true