Last time : energy storage elements capacitor. Charge on plates Energy stored in the form of electric field Passive sign convention Vlt Voltage drop across real capacitor can not change abruptly because instant voltage change means instant change of accumulated charge and this, in turn, requires infinite current. 1
Last time : capacitor charge/discharge. Charging capacitor by practical DC source Energy gets stored in the capacitor Discharging capacitor Energy previously stored in the capacitor gets dissipated in resistor 2
Energy can be stored in circuit element in the form of magnetic field. Ampere s law magnetic field created by electric current in vacuum magnetic flux density Coil of wire can be used to store energy in the form of magnetic field Biot Savart law: 3
Magnetic flux density generated by current in the coil of wire. Magnetic field in center generated by Magnetic field in center generated by full coil of wire Biot Savart law: For circuit itelement containing i N coils and in media with magnetic permeability In every case depends on geometry but always: 4
Inductance. Closed path for Ampere s law calculation Number of turns inside the closed path Total number of turns in solenoid Length of solenoid Magnetic flux density inside solenoid core: Magnetic flux inside solenoid core: Inductance (self inductance) relates the magnetic flux to the current that created it. Depends on geometry of the circuit element. 5
Linear inductor. Energy stored in inductor Recall that in parallel plate capacitor with plate area A and spacing d:, where volume energy density Volume energy density of magnetic field: Energy stored in solenoid of length l and cross section of the core A: In solenoid we just considered: 6
Voltage drop across ideal linear inductor. When DC current is flowing through ideal inductor the voltage drop across it is zero. What happens if the current is changing with time? Magnetic flux: Magnetic flux change due to change of current: Faraday s law electromotive ti force: Recall EMF in battery: In inductor: Change of current in inductor generates the voltage drop. 7
Passive sign convention Voltage and current in inductor. Voltage drop appears because the induced EMF force opposes the change of current Assume: moved charges in the direction opposite to the direction of current change. 8
Inductor power and energy. Initial current Current in inductor implies presence of magnetic field, hence, current is associated with energy and energy can not be changed abruptly without infinite power. Hence, current through practical inductor can not be changed abruptly since it would imply infinite voltage. 9
1. Simple ideal case: Let s put the energy into inductor. 2. More realistic case with practical power supply: Time constant of the RL circuit 10
Increase of current through inductor. 11
Increase of current through inductor by practical voltage source. Obviously, everything is just the same since we used Thevenin form of the practical source and it is equivalent to Norton one used on slide 11. 12
Decrease of current through inductor removal of energy. 1. Simple ideal case: See ignition coil in cars 2. Realistic case: 13
Series connection of inductors Physical sense: 14
Parallel connection of inductors Physical sense current divider. 15
Example 1. Assume DC steady state 16
Example 2. Current trough inductor in DC steady state Energy stored in inductor 17
Example 2 cont. Energy stored in inductor: Energy dissipated in the circuit during transition from initial DC steady state to final DC steady state condition: 18
Example 2 cont. KVL: 19
Example 2 cont. KVL: 20
Example 2 cont. Find B from initial conditions: 21
Example 2 cont. 22
Superposition. 23
Superposition cont. 24
Superposition cont. 25
Superposition cont. 26
Superposition cont. 27
Example. DC steady state Energy stored in inductor will get dissipated in resistor. Eventually all voltages and currents will become zero since circuit will contain no sources and there is resistor that does not store but dissipates energy. To find the transient values of voltage and currents we will need to solve integrodifferential equation not an easy task, in general. 28