1. Inductance and Capacitance
2. 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:
1) Find the initial current, Io, through the inductor.
2) Find the time constant of the circuit, τ = L/R.
3) 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
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 steadystate 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:
1) Find the initial voltage, Vo , through the capacitor.
2) Find the time constant of the circuit, τ = RC.
3) 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