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Electric Circuits
Chapter 7
First-Order Circuits
Dr Jalal Al Roumy
Israa University
2019/2020
Introduction:
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A first-order circuit is characterized by a first-order
differential equation.
Examples: RC & RL circuits.
Natural Response:
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A circuit response is the manner in which the circuit reacts
to an excitation.
The natural response of a circuit refers to the behavior (in
terms of voltages and currents) of the circuit itself, with no
external sources of excitation.
The Source-Free RC Circuit:
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The natural response depends on the nature of the circuit
alone, with no external sources.
In fact, the RC circuit has a response only because of the
energy initially stored in the capacitor.
Time Constant:
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The time constant of a circuit is the time required for the
response to decay to a factor of 1∕e or 36.8% of its initial
value.
Approximate Full Discharging Time:
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The voltage v(t) is less than 1% of V0 after 5τ. Thus, it is
customary to assume that the capacitor is fully discharged
(or charged) after five time constants.
Response for Different Time Constants:
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The smaller the time constant, the more rapidly the voltage
decreases and the faster the response.
Power & Energy in RC circuits:
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Exercise 7.1:
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Exercise 7.2:
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The Source-Free RL Circuit:
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The RL circuit has a response only because of the energy
initially stored in the inductor.
Time Constant:
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The time constant of a circuit is the time required for the
response to decay to a factor of 1∕e or 36.8% of its initial
value.
Time Constant:
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The smaller the time constant of a circuit, the faster the rate
of decay of the response.
The larger the time constant, the slower the rate of decay
of the response.
The response decays to less than 1% of its initial value
(i.e., reaches steady state) after five time constants.
Power & Energy in RL circuits:
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Exercise 7.3:
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Exercise 7.4:
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Example 7.5:
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Singularity Functions:
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Singularity functions are functions that either are
discontinuous or have discontinuous derivatives.
Singularity functions (also called switching functions) are
very useful in circuit analysis since they serve as good
approximations to the switching signals that arise in
circuits with switching operations.
Examples:
impulse function, step function, and ramp function.
The Unit Step Function:
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The unit step function u(t) is 0 for negative values of t
and 1 for positive values of t.
Representing a Current Source:
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We use the step function to represent an abrupt change in
voltage or current, like the changes that occur in the circuits
of control systems and digital computers.
For a current source:
Representing a Voltage Source:
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For a voltage source:
The Unit Impulse Function:
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The unit impulse function δ(t) is zero everywhere except
at t = 0, where it is undefined.
The Unit Ramp Function:
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The unit ramp function is zero for negative values of t
and has a unit slope for positive values of t.
Exercise 7.6:
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Example 7.8:
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Step Response of an RC Circuit:
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The step response of a circuit is its behavior when the
excitation is the step function, which may be a voltage or
a current source.
Initially Charged Capacitor:
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Initially Uncharged Capacitor:
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Step Response of an RC Circuit:
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Step Response of an RC Circuit:
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The transient response is the circuit’s temporary response
that will die out with time.
The steady-state response is the behavior of the circuit a
long time after an external excitation is applied.
Example 7.10:
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Step Response of an RL Circuit:
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The step response of a circuit is its behavior when the
excitation is the step function, which may be a voltage or
a current source.
Initially Charged Inductor:
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Initially Uncharged Inductor:
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Step Response of an RL Circuit:
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The transient response is the circuit’s temporary response
that will die out with time.
The steady-state response is the behavior of the circuit a
long time after an external excitation is applied.
Example 7.13:
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HW #6:
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7.1, 7.2, 7.7, 7.10, 7.12, 7.22, 7.26, 7.28, 7.44 & 7.54.
Assignment is due to 16/12/2019.
