EEE2001 Network Theory Module 1 Sinusoidal Steady State Analysis Lecture 1 Dr. S. Hemamalini Professor School of Electrical Engineering VIT Chennai Contents • Review of Phasors • For circuits with independent and dependent sinusoidal sources – Nodal Analysis – Mesh Analysis – Thevenin’s Theorem – Norton’s Theorem – Maximum Power Transfer Theorem – Superposition Theorem DC Vs AC • Thomas Alva Edison Vs Nikola Tesla • DC sources have fixed polarities and constant magnitudes and produce currents with constant value and unchanging direction. • AC sources alternate in polarity and vary in magnitude and produce currents that vary in magnitude and alternate in direction. Voltage and current versus time for dc Periodic waveform Sinusoidal AC Voltage & Current Symbol for an AC Voltage Source Generating AC Voltages Frequency and Period Time Scales Cycle scaled in time Frequency The number of cycles per second of a waveform is defined as its frequency. 1 Hz = 1 cycle per second The length of time required to generate one cycle depends on the velocity of rotation. • • • • • • Power line frequencies : 50 Hz Audible sound frequencies : 20 Hz to 20 kHz Standard AM radio band: 550 kHz to 1.6 MHz FMband : 88 MHz to 108 MHz TV transmissions : 54-MHz to 890-MHz Optical and X-ray frequencies : Above 300 GHz Frequency and Period The period, T, of a waveform, is the duration of one cycle. It is the inverse of frequency. Period may be measured between any two corresponding points ??? a. What is the period of a 50-Hz voltage? b. What is the period of a 1-MHz current? c. Determine the period and frequency of the waveform. Instantaneous value Instantaneous value : The value of the voltage at any point on the waveform AC voltage and current reference conventions Amplitude and Peak-to-Peak Value The amplitude of a sine wave is the distance from its average to its peak. Peak-to-peak voltage is measured between minimum and maximum peaks. Peak-to-peak voltages are denoted Ep-p or Vp-p Peak-to-peak currents are denoted as Ip-p Peak Value The peak value of a voltage or current is its maximum value with respect to zero. Sine wave rides on top of a dc value The Basic Sine Wave Equation The voltage produced by the generator is Voltage at any point on the sine wave may be found by multiplying Em times the sine of the angle at that point. Angular Velocity • The rate at which the generator coil rotates is called its angular velocity. • denoted by the Greek letter ω (omega). • Expressed in radians per second Cycle length scaled in degrees and radians 2π radians = 360° Sinusoidal Voltages and Currents as Functions of Time A 100-Hz sinusoidal voltage source has an amplitude of 150 volts. Write the equation for e as a function of time. Voltages and Currents with Phase Shifts If a sine wave does not pass through zero at t 0 s it has a phase shift. Phasors Phase Difference – the angular displacement between different waveforms of the same frequency. Sinusoids A sinusoid is a signal that has the form of the sine or cosine function. Consider the sinusoidal voltage v(t) = Vm sin ωt where Vm = the amplitude of the sinusoid ω = the angular frequency in radians/s ωt = the argument of the sinusoid A periodic function is one that satisfies f (t) = f (t + nT), for all t and for all integers n. Sinusoids Two sinusoids with different phases v(t) = Vm sin(ωt + φ) A sinusoid can be expressed in either sine or cosine form. v2 leads v1 by φ or that v1 lags v2 by φ Given the sinusoid 5 sin(4πt − 60◦), calculate its amplitude, phase, angular frequency, period, and frequency. Calculate the phase angle between v1 = −10 cos(ωt + 50◦) and v2 =12 sin(ωt − 10◦). State which sinusoid is leading. Phasors A phasor is a complex number that represents the amplitude and phase of a sinusoid. Phasors Phasors Given a sinusoid v(t) = Vm cos(ωt + φ) V is thus the phasor representation of the sinusoid v(t) A phasor is a complex representation of the magnitude and phase of a sinusoid. A phasor may be regarded as a mathematical equivalent of a sinusoid with the time dependence dropped. Phasors Time-domain vs Frequency domain The differences between v(t) and V: • v(t) is instantaneous or time-domain representation • v(t) is time dependent. • v(t) is always real with no complex term • V is the frequency or phasor-domain representation. • V is not time dependent • V is generally complex Note: Phasor analysis applies only when frequency is constant; when it is applied to two or more sinusoid signals only if they have the same frequency. PHASOR RELATIONSHIPS FOR RESISTOR (a) time domain, (b) frequency domain Phasor diagram for the resistor For a purely resistive element, the voltage across and the current through the element are in phase, with their peak values related by Ohm’s law. PHASOR RELATIONSHIPS FOR INDUCTOR Phasor diagram for the Inductor (a) time domain, (b) frequency domain PHASOR RELATIONSHIPS FOR CAPACITOR (a) time domain, (b) frequency domain Phasor diagram for the capacitor PHASOR RELATIONSHIPS FOR CIRCUIT ELEMENTS Summary of voltage-current relationships If the source current leads the applied voltage, the network is predominantly capacitive. If the applied voltage leads the source current, it is predominantly inductive. IMPEDANCE Complex Impedance in Phasor Notation VR RI R VC Z C I C 1 1 1 ZC j 90 j C C C VL Z L I L Z L j L L 90 IMPEDANCE • The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in ohms (Ω). • Impedance Z is a complex number in units of Ohms. • The impedance represents the opposition which the circuit exhibits to the flow of sinusoidal current. • The impedance may be expressed in rectangular form as Z = R + jX • Re(Z) = R & Im (Z)= X are called resistance and reactance, respectively. • Although impedance is complex, it’s not a phasor. In other words, it cannot be transformed into a sinusoidal function in the time domain. Impedances The impedance may also be expressed in polar form as Behaviour of inductor and capacitor at different frequencies ADMITTANCE The admittance Y is the reciprocal of impedance, measured in siemens (S).
