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EENG 2610: Circuit AnalysisClass 11: Capacitor and Inductor Combinations RC Operational Amplifier Circuits
Oluwayomi AdamoDepartment of Electrical EngineeringCollege of Engineering, University of North Texas
Series Capacitors
Since the same current flows in each of the series capacitors, each capacitor gains the same charge in the same time period:
The voltage across each capacitor will depend on this charge and the capacitance of the element.
N
N
j jS CCCCC
11111
211
iii vCq
Ni qqqq 21
)()(1
)()(1
)()(1
)()()()(
0
10
1
10
21
0
0
0
tvdttiC
tvdttiC
tvdttiC
tvtvtvtv
t
tS
N
jj
t
t
N
j j
N
jj
t
tj
N
Example 6.12: Determine the equivalent capacitance, the initial voltage for the circuit, total energy stored in the circuit.
Example 6.13: Two previously uncharged capacitors are connected in series and then charged with a 12 V source. One capacitor is 30 μF and the other is unknown. If the voltage across the 30 μF capacitor is 8 V, find the capacitance of the unknown capacitor.
V 12 μF 30
V 8
1C
Parallel Capacitors
NP CCCC 21
dt
tdvC
dt
tdvC
dt
tdvC
dt
tdvC
dt
tdvC
titititi
P
N
jj
N
N
)(
)()(
)()()(
)()()()(
1
21
21
Example E6.7: Determine equivalent capacitance at terminals A-B.
A
B
Series Inductors
N
N
jjS LLLLL
211
dt
tdiL
dt
tdiL
dt
tdiL
tvtvtvtv
S
N
jj
N
jj
N
)(
)(
)(
)()()()(
1
1
21
Parallel Inductors
N
N
j jP LLLLL
11111
211
)()(1
)()()1
(
)()(1
)()()()(
0
10
1
10
21
0
0
0
tidxxvL
tidxxvL
tidxxvL
titititi
t
tP
N
jj
t
t
N
j j
N
jj
t
tj
N
RC Operational Amplifier Circuits - Differentiator
iR
vvvv
dt
dC o
211 )(
dt
tdvCRtvo
)()( 1
12
General rules for op-amp circuit analysis Use the ideal op-amp model conditions: Write nodal equations at the op-amp input terminals
.0 , iivv
RC Operational Amplifier Circuits - Integrator
General rules for op-amp circuit analysis Use the ideal op-amp model conditions: Write nodal equations at the op-amp input terminals
.0 , iivv
Example 6.17:The waveform below is applied to the input of the differentiator circuit. Determine the waveform at the output of the op-amp.
μF 2
k 1
Example 6.18:The waveform below is applied to the input of the integrator circuit. Determine the waveform at the output of the op-amp if the capacitor is initially discharged.
k 5
μF 2.0
