Electric Circuits (EELE 2312) Chapter 7 Sinusoidal Steady-State Analysis Basil Hamed
Introduction Thus far, we have focused on circuits with constant sources; in this chapter we are now ready to consider circuits energized by time-varying voltage or current sources. In particular, we are interested in sources in which the value of the voltage or current varies sinusoidally. Basil Hamed
Introduction Sinusoidal sources and their effect on circuit behavior form an important area of study for several reasons First, the generation, transmission, distribution, and consumption of electric energy occur under essentially sinusoidal steady-state conditions. Second, an understanding of sinusoidal behavior makes it possible to predict the behavior of circuits with nonsinusoidal sources. Third, steady-state sinusoidal behavior often simplifies the design of electrical systems. Basil Hamed
7.1 The Sinusoidal Source A sinusoidal voltage source (independent or dependent) produces a voltage that varies sinusoidally with time. A sinusoidal current source (independent or dependent) produces a current that varies sinusoidally with time. Basil Hamed
7.1 The Sinusoidal Source Basil Hamed
Example 7.1 Basil Hamed
Example 7.2 Basil Hamed
Example 7.3 We can translate the sine function to the cosine function by subtracting 90o (π/2 rad) from the argument of the sine function. Verify this translation by showing that sin(ωt+θ)=cos(ωt+θ-90o) Use the result in (a) to express sin(ωt+30o) as a cosine function Basil Hamed
Example 7.4 Basil Hamed
Example 7.4 Basil Hamed
7.2 The Sinusoidal Response Basil Hamed
7.2 The Sinusoidal Response Basil Hamed
7.3 The Phasor The Phasor Transform Basil Hamed
7.3 The Phasor Inverse Phasor Transform Basil Hamed
Example 7.5 Basil Hamed
Example 7.5 Basil Hamed
Example 7.5 Basil Hamed
7.4 Passive Circuit Elements in f-Domain V-I Relationship for a Resistor Basil Hamed
7.4 Passive Circuit Elements in f-Domain V-I Relationship for a Resistor Basil Hamed
Passive Circuit Elements in f-Domain V-I Relationship for an Inductor Basil Hamed
Passive Circuit Elements in f-Domain V-I Relationship for an Inductor Basil Hamed
7.4 Passive Circuit Elements in f-Domain V-I Relationship for a Capacitor Basil Hamed
Basil Hamed
Impedance and Reactance Basil Hamed
7.5 Kirchhoff’s Laws Kirchhoff’s Voltage Law in Frequency Domain
7.5 Kirchhoff’s Laws Kirchhoff’s Current Law in Frequency Domain
7.6 Circuit Simplifications Combining Impedances in Series
Example 7.6 A 90Ω resistor, a 32 mH inductor, and a 5 μF capacitor are connected in series across the terminals of a sinusoidal voltage source. The steady-state expression for the source voltage υs is 750 cos(5000t+30o) V. Construct the frequency-domain equivalent circuit Calculate the steady-state current i by the phasor method Basil Hamed
Example 7.6 Basil Hamed
7.6 Circuit Simplifications Combining Impedances in Parallel Basil Hamed
7.6 Circuit Simplifications Combining Impedances in Parallel Basil Hamed
7.6 Circuit Simplifications Combining Impedances in Parallel Basil Hamed
Example 7.7 The sinusoidal current source in the circuit produces the current is=8cos200000t A. Construct the frequency-domain equivalent circuit Find the steady-state expression for υ, i1, i2, and i3 Basil Hamed
Example 7.7 Basil Hamed
Example 7.7 Basil Hamed
7.6 Circuit Simplifications Source Transformation
7.6 Circuit Simplifications Thevenin Equivalent Circuit
7.6 Circuit Simplifications Norton Equivalent Circuit
Example 7.8 Use the concept of source transformation to find the phasor voltage Vo in the circuit.
Example 7.8 Basil Hamed
Example 7.8 Basil Hamed
Example 7.9 Find the Thevenin equivalent circuit with respect to terminals a,b for the circuit.
Example 7.9
Example 7.9 Basil Hamed
Example 7.9 Basil Hamed
7.7 The Node-Voltage Method Example 7.10 Use the node-voltage method to find the branch currents Ia, Ib, and Ic in the circuit Basil Hamed
Example 7.10 Basil Hamed
7.8 The Mesh-Current Method Example 7.11 Use the mesh-current method to find the voltages V1, V2, and V3 in the circuit Basil Hamed
7.8 The Mesh-Current Method Example 7.11 Basil Hamed
7.9 Instantaneous, Average, and reactive Power The Instantaneous Power Basil Hamed
7.9 Instantaneous, Average, and reactive Power The Instantaneous Power Basil Hamed
Active and Reactive Power Basil Hamed
Power for Purely Resistive Circuit Basil Hamed
Power for Purely Resistive Circuit Basil Hamed
Power for Purely Inductive Circuit Basil Hamed
Power for Purely Inductive Circuit Basil Hamed
Power for Purely Capacitive Circuit Basil Hamed
Power for Purely Capacitive Circuit Basil Hamed
The Power Factor
Example 7.12 Calculate the average power and the reactive power at the terminals of the network shown given that: υ=100cos(ωt+15o) i=4sin(ωt-15o) State whether the network is absorbing or delivering average power. State whether the network is absorbing or delivering magnetizing vars. Basil Hamed
box delivers average power Example 7.12 box delivers average power box absorbs magnetizing vars Basil Hamed
Appliance Ratings
7.10 The RMS Value & Power Calculation Basil Hamed
7.10 The RMS Value & Power Calculation Basil Hamed
7.10 The RMS Value & Power Calculation Basil Hamed
Example 7.13 A sinusoidal voltage having a maximum amplitude of 625 V is applied to the terminals of a 50 Ω resistor. Find the average power delivered to the resistor. Repeat (a) by first finding the current in the resistor
7.11 Complex Power & Power Calculations
7.11 Complex Power & Power Calculations
7.11 Complex Power & Power Calculations
Example 7.14 An electric load operates at 240 V rms. The load absorbs an average power of 8 kW at 0.8 lagging power factor. Calculate the complex power of the load Calculate the impedance of the load
Example 7.14
Power Calculations
Power Calculations
Alternate Forms for Complex Power
Alternate Forms for Complex Power
Example 7.15 In the circuit shown, a load having an impedance of 39+j26 Ω is fed from a voltage source through a line having an impedance of 1+j4 Ω. The effective, or rms, value of the source voltage is 250 V. Calculate the load current IL and voltage VL Calculate the average and reactive power delivered to the load Calculate the average and reactive power delivered to the line Calculate the average and reactive power supplied by the source
Example 7.15
Example 7.16 The two loads in the circuit shown can be considered as follows: load 1 absorbs an average power of 8 kW at a leading power factor of 0.8 and load 2 absorbs 20 kVA at 0.6 lagging power factor. Determine the power factor of the two load in parallel Determine the apparent power required to supply the loads, the magnitude of the current, Is, and the average power loss in the transmission line Given that the frequency of the source is 60 Hz, compute the value of the capacitor that would correct the power factor to 1 if placed in parallel with the two loads. Recompute the values in (b) for the load with the corrected power factor
Example 7.17 Calculate the total average and reactive power delivered to each impedance Calculate the average and reactive powers associated with each source Verify that the average power delivered equals the average power absorbed, and that the magnetizing reactive power delivered equals the magnetizing reactive power absorbed
End Of Chapter Seven