Inductance and Capacitance Response of First Order RL and RC Part 7

Inductance and Capacitance Energy can be stored in both magnetic and electric fields. Inductors and capacitors are capable of storing energy. The behavior of inductors is based on magnetic fields where the source of magnetic field is current. If the current is varying with time, the magnetic field is varying with time which induces a voltage in the conductor. The behavior of capacitors is based on electric fields where the source of electric field is voltage. If the voltage is varying with time, the electric field is varying with time which produces current in the space.

Inductor Inductance is symbolized by “L” Inductance is measured in Henrys (H)

Practical Inductors

How Inductors Work? An inductor stores energy in the form of magnetic field When a current starts flowing in the coil, the coil wants to build up a magnetic field. While the field is building, the coil resists the flow of current. Once the field is built, current can flow normally through the wire.

Capacitor Capacitance is symbolized by “C” Capacitance is measured in Farads (F)

Practical Capacitors

How Capacitors Work? Capacitor is like a battery A capacitor stores electrical energy in the form of electric charge When a capacitor is connected to a battery, charge starts accumulating on its plates Once the capacitor is charged, it has the same voltage as the battery that was connected to it

Series-Parallel Combinations of Inductance and Capacitance

Response of First Order RL and RC Natural response Step response

Response of First Order RL and RC Why called First order? Natural response: Happens when the inductor or capacitor is suddenly disconnected from its dc source. Step response: Happens when a dc voltage or current source is applied to an inductor or a capacitor.

Natural Response of an RL circuit If the current is constant, the voltage across the inductor is zero. Thus the inductor behaves as a short circuit in the presence of a constant, or dc, current. Assume a constant current source. Assume that the switch has been closed for long time. So, no current passes through Ro or R and all source current Is appears in the inductive branch. We find the natural response after the switch has been opened at t=0 ( i(0-) = Is).

Natural Response of an RL circuit For t > 0, the circuit reduces to the one shown in the figure. We need to find the voltage and current at the terminal of the resistor after the switch has been opened.

Time Constant

Natural Response of an RL circuit Calculating the natural response of an RL circuit can be summarized as follows: Find the initial current, Io, through the inductor. Find the time constant of the circuit, τ = L/R. Use equation Ioe-t/τ, to generate i(t) from Io and τ

Example 7.1 Find d) power dissipated in the 10 Ohm resistor

Example 7.1 The switch has been closed for a long time prior to t=0   After opening the switch The switch has been closed for a long time prior to t=0 The voltage across the inductor must be zero at t=0- Therefore the initial current in the inductor is 20 A      

                 

Problem 7.4

Natural Response of an RC circuit If the voltage is constant, the current across the capacitor is zero. Thus the capacitor behaves as an open circuit in the presence of a constant, or dc, voltage. Assume a constant voltage source. Assume that the switch has been in position a for long time. Then the circuit made up of Vg, R1 and C reaches a dc steady-state circuit where the capacitor is an open circuit. The voltage across the capacitor at (t = 0-)is Vg.

Natural Response of an RC circuit Because there can be no instantaneous change in the capacitor voltage, the initial voltage across the capacitor is V(0+) = Vg

Natural Response of an RC circuit Calculating the natural response of an RC circuit can be summarized as follows: Find the initial voltage, Vo , through the capacitor. Find the time constant of the circuit, τ = RC. Use equation Voe-t/τ, to generate v(t) from Vo and τ

Example 7.3 Find d) power dissipated in the 60K Ω resistor        

Assessment Problem 7.3/page 246

Energy Stored in Inductor and Capacitor  

Step Response of an RL circuit Finding the currents and voltages generated when either dc voltage or current sources are suddenly applied. The initial current through the inductor is zero. The switch closes at t = 0. The voltage across the inductor is 0 for t < 0 and Vs for t > 0.

Step Response of an RL circuit After the switch has been closed, From the graph it is clear that Vs/R is also the final value of current i.e., i(∞) = Vs/R τ= L/R

Example 7.5

Assessment Problem 7.5

Step Response of an RC circuit Finding the currents and voltages generated when either dc voltage or current sources are suddenly applied. The switch closes at t = 0. The initial current through the capacitor is zero for t < 0 and Is for t >0. After the switch has been closed, IsR is also the final value of voltage, i.e., vC(∞) = IsR

               

Example 7.6/page 252

Problem 7.55

A general solution for natural and step responses

Procedure

Example