Lecture 14 SteadyState DC RLC Circuits Lecture Objectives The lecture today will focus on Introduction to steadystate DC analysis of RLC Circuits Circuits with only DC sources amp without switches ID: 1028400
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1. EEL 3004 Electrical Networks Lecture #14Steady-State (DC) RLC Circuits
2. Lecture ObjectivesThe lecture today will focus on:Introduction to steady-state (DC) analysis of RLC Circuits Circuits with only DC sources & without switches.Examples
3. Like the resistor, the capacitor exhibits an algebraic relationship between its branch voltage and the stored charge on its plate.Time varying voltage applied across the capacitor will induce current flow in the capacitor caused by charge variations through the charging process.Constant voltage (DC) applied across the capacitor will result in zero capacitor current.If the voltage across the capacitor changes its value instantly i.e. dt=0, then an infinite current is required. hence,Capacitor ObservationsRECALL
4. When time varying current is applied across the inductor, it induces voltage across the inductor winding caused by magnetic field variation.When the inductor current is a fixed DC value, there is no voltage induced across it.If the inductor current changes its value instantly i.e. dt=0, then the resultant induced voltage will be infinity. hence, Inductor ObservationsLike the resistor, the inductor exhibits an algebraic relationship between its terminal current and the induced magnetic flux in the core.RECALL
5. If the circuit consists only of direct current and voltage sources (dc), then the capacitor behaves as an open circuit, and the inductor behaves as a short circuit.RLC circuits with DC sources and NO Switching(Steady-state)Under DC conditions – the circuit has been undisturbed for a long time
6. ExampleUnder DC conditions, find:The current passing through the inductor,The energy stored in the inductorVariableValuesUnitVDC2VIDC20mAR11.2kΩR21.6kΩR32.4kΩR43.4kΩL56mH
7. The voltage across the inductor is considered zero under dc conditions:
8. The voltage across the inductor is considered zero under dc conditions:Simplifying the circuit yields
9. Where Req is given by:
10. Where Req is given by:Applying KCL at node V1 yields:
11. Where Req is given by:Applying KCL at node V1 yields:Solving for V1 yields:
12. Now IL can be calculated by voltage division:
13. Now IL can be calculated by voltage division:The energy stored in the inductor is given by:
14. ExampleFor the RLC circuit under DC conditions, find:The inductor currents and capacitor voltages,The energy stored in each inductor and capacitor.VariableValuesUnitV8.6VI4.6mAR11.2kΩR21.6kΩR32.4kΩL1120mHL2360mHL3320mHC1240FC2340F
15. The inductor acts as a short circuit, the capacitor like an open circuit under dc conditions: = 2.1 V
16. The energy stored in the inductors is given by:
17. The energy stored in the capacitors is given by:
18. ExampleDesign the load resistor RL so that in steady-state the energy stored in the capacitor and inductor are equal.Find the voltage VL.Find the power dissipated in RL.VariableValuesUnitVS6VR200ΩL250mHC25F
19. In steady state, the circuit becomes:The current in the inductor is given by:
20. In steady state, the circuit becomes:The current in the inductor is given by:Energy stored in the inductor is given by:
21. In steady state, the circuit becomes:The current in the inductor is given by:Energy stored in the inductor is given by:The voltage in the capacitor is given by:
22. In steady state, the circuit becomes:The current in the inductor is given by:Energy stored in the inductor is given by:The voltage in the capacitor is given by:Energy stored in the capacitor is given by:
23. In steady state, the circuit becomes:Equalizing the energy stored yields:
24. In steady state, the circuit becomes:Equalizing the energy stored yields:Solving for the load resistance:
25. In steady state, the circuit becomes:The voltage VL is easily found as:
26. In steady state, the circuit becomes:The voltage VL is easily found as:The power dissipated in RL is given by: