Spring - 2022
Lecture - 3
Chapter 2:
First-Order Circuits
Prepared By
Dr. Salma Subh
salma_s_mohammed@cic-cairo.com
Outlines
1.
2.
3.
4.
The Source-Free RC Circuit
The Source-Free RL Circuit
Step Response of an RC Circuit
Step Response of an RL Circuit
2
Introduction
•
Three passive elements (resistors, capacitors, and inductors) and one active element (the
op-amp)
•
In this chapter, we shall examine two types of simple circuits: a circuit comprising a
resistor and capacitor and a circuit comprising a resistor and an inductor. These are called
RC and RL circuits, respectively.
•
The only difference is that applying Kirchhoff’s laws to purely resistive circuits results
in algebraic equations while applying the laws to RC and RL circuits produces
differential equations, which are more difficult to solve than algebraic equations.
•
The differential equations resulting from analyzing RC and RL circuits are of the first
order. Hence, the circuits are collectively known as first-order circuits.
3
There are two ways to excite the circuits:
•
Source-free circuits, we assume that energy is initially stored in the capacitive or
inductive element.
•
By independent DC sources.
•
The two types of first-order circuits and the two ways of exciting them add up to the four
possible situations we will study in this chapter.
4
The Source-free RC Circuits
Find the voltage across the capacitor v(t) , circuits response
𝑣 𝑜 = 𝑣𝑜
1 2
𝐶𝑣
2 𝑜
𝑖𝑐 + 𝑖𝑅 = 0
𝑤 𝑜 =
𝑑𝑣 𝑣
𝐶
+ =0
𝑑𝑡
𝑅
𝑑𝑣
𝑣
+
=0
𝑑𝑡
𝐶𝑅
𝑑𝑣
𝑑𝑡
=−
𝑣
𝐶𝑅
5
The Source-free RC Circuits
1
1
න 𝑑𝑣 = −
න 𝑑𝑡
𝑣
𝑅𝐶
𝑡
+ ln 𝐴
𝐶𝑅
𝑡
ln 𝑣 − ln 𝐴 = −
𝐶𝑅
𝑣
𝑡
𝑙𝑛 = −
𝐴
𝐶𝑅
ln 𝑣 = −
𝑣(𝑡) = 𝐴𝑒
−
𝑡
𝐶𝑅
From the initial conditions v(0) = A = 𝑣𝑜 , Hence
𝑣(𝑡) = 𝑣𝑜 𝑒
𝑡
− 𝐶𝑅
6
RC Circuits Response ..
𝑣(𝑡) = 𝑣𝑜 𝑒
−
𝑡
𝐶𝑅
Rth
7
To Solve RC Circuits..
8
Multisim..
9
Example 1 (page 281):
Let Vc(0) = 15 V, Find Vc, Vx and Ix
for t > 0
10
Example 1 (page 281):
The switch in the circuit in Fig.
has been closed for a long
time, and it is opened at t > 0,
Find v(t) for
t >=0
Calculate the initial energy
stored in the capacitor.
11
Example 2 (page 282):
The switch in the circuit in Fig. has been closed
for a long time, and it is opened at t=0 Find v(t)
for t >0 . Calculate the initial energy stored in
the capacitor.
12
The Source-free RL Circuits
𝑖 𝑜 = 𝑖𝑜
1 2
𝐿𝑖
2 𝑜
𝑉𝑙 + 𝑉𝑅 = 0
𝑤 𝑜 =
𝑑𝑖
+ 𝑅𝑖 = 0
𝑑𝑡
𝑑𝑖 𝑅
+ 𝑖=0
𝑑𝑡 𝐿
𝑑𝑖
𝑅 𝑑𝑡
=−
𝑖
𝐿
𝐿
13
The Source-free RL Circuits
න
𝑑𝑖
𝑅 𝑑𝑡
=− න
𝑖
𝐿
ln i = −
Rt
L
+ ln A
ln 𝑖 − ln 𝐴 = −
𝑙𝑛
𝑅𝑡
𝐿
𝑖
𝑅𝑡
=−
𝐴
𝐿
𝑖(𝑡) = 𝐴𝑒
−
𝑅𝑡
𝐿
From the initial conditions i(0) = A = 𝑖𝑜 , Hence
𝑖(𝑡) = 𝑖𝑜 𝑒
−
𝑅𝑡
𝐿
14
RL Circuits Response ..
𝑖(𝑡) = 𝑖𝑜 𝑒
−
𝑅𝑡
𝐿
15
To Solve RL Circuits with no external
source ..
16
Example 3:
Assuming that i(o) = 10 A, calculate i(t)
and ix(t) in the circuit of Fig.
17
Example 4:
The switch in the circuit of Fig. 7.16 has
been open for a long time. At t= 0 the
switch is closed . Calculate i(t) fort > 0
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Step Response of an RC Circuit
𝑉 0 = 𝑉𝑜
𝑑𝑣 𝑉 − 𝑉𝑠 𝑢(𝑡)
𝐶
+
=0
𝑑𝑡
𝑅
𝑑𝑣 𝑉 𝑉𝑠 𝑢(𝑡)
𝐶
+ =
𝑑𝑡 𝑅
𝑅
𝑑𝑣
𝑉
𝑉𝑠
+
=
𝑓𝑜𝑟 𝑡 > 0
𝑑𝑡 𝐶𝑅 𝐶𝑅
𝑑𝑣
𝑉 − 𝑉𝑠
=−
𝑑𝑡
𝐶𝑅
19
𝑑𝑣
𝑑𝑡
=−
𝑉 − 𝑉𝑠
𝐶𝑅
𝑣(𝑡)
𝑡
𝑑𝑣
𝑑𝑡
න
= න−
𝑉 − 𝑉𝑠
𝐶𝑅
𝑉0
0
ln 𝑉 − 𝑉𝑠 = −
𝑇
𝐶𝑅
ln 𝑉(𝑡) − 𝑉𝑠 − ln 𝑉𝑜 − 𝑉𝑠 = −
ln
𝑡
𝐶𝑅
𝑉 𝑡 − 𝑉𝑠
𝑡
=−
𝑉𝑜 − 𝑉𝑠
𝐶𝑅
𝑡
𝑉 𝑡 − 𝑉𝑠
−
𝐶𝑅
=𝑒
𝑉𝑜 − 𝑉𝑠
𝑤ℎ𝑒𝑟𝑒 𝑅𝐶 = 𝓉
𝑉 𝑡 − 𝑉𝑠 = (𝑉𝑜 −
𝑡
−
𝓉
𝑉𝑠 ) 𝑒
𝑡
𝑉 𝑡 = 𝑉𝑠 + (𝑉𝑜 − 𝑉𝑠 ) 𝑒 −𝓉
20
To solve RC circuits with source
21
Example 5 (page 277):
The switch in Fig. has been in position A for a long time. At t = 0 the
switch moves to B. Determine v(t) for t > 0 and calculate its value at t
=1 and t = 4 s.
22
Example 6:
Find v(t) for t> 0 in the circuit of fig. Assume the switch has been open for
long time and is closed at t= 0 calculate v(t) value at t =0.5.
23
Step Response of an RL Circuit
Prove !!
24
Example 7:
Find i(t) for t> 0 in the circuit of fig. Assume the switch has been close for
long time and is opened at t= 0 calculate i(t).
Check your answer ! ☺
25
Example 9:
First-Order Op Amp Circuits
For the op-amp circuit in fig. find vo for
t > 0 given that v(o) = 3 V let Rf = 80 K
ohm, R1 = 20 K ohm and C= 5 u F
26