Analysis using Laplace Transform (non unit-step) | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

 Page 1


LaPlace Transform in Circuit Analysis 
Objectives: 
•Calculate the Laplace transform of common functions 
using the definition and the Laplace transform tables 
•Laplace-transform a circuit, including components with 
non-zero initial conditions. 
•Analyze a circuit in the s-domain 
•Check your s-domain answers using the initial value 
theorem (IVT) and final value theorem (FVT) 
•Inverse Laplace-transform the result to get the time-
domain solutions; be able to identify the forced and 
natural response components of the time-domain solution. 
(Note – this material is covered in Chapter 12 and Sections 
13.1 – 13.3) 
Page 2


LaPlace Transform in Circuit Analysis 
Objectives: 
•Calculate the Laplace transform of common functions 
using the definition and the Laplace transform tables 
•Laplace-transform a circuit, including components with 
non-zero initial conditions. 
•Analyze a circuit in the s-domain 
•Check your s-domain answers using the initial value 
theorem (IVT) and final value theorem (FVT) 
•Inverse Laplace-transform the result to get the time-
domain solutions; be able to identify the forced and 
natural response components of the time-domain solution. 
(Note – this material is covered in Chapter 12 and Sections 
13.1 – 13.3) 
LaPlace Transform in Circuit Analysis 
What types of circuits can we analyze? 
•Circuits with any number and type of DC sources and 
any number of resistors. 
•First-order (RL and RC) circuits with no source and with a 
DC source. 
•Second-order (series and parallel RLC) circuits with no 
source and with a DC source. 
•Circuits with sinusoidal sources and any number of 
resistors, inductors, capacitors (and a transformer or op 
amp), but can generate only the steady-state response. 
Page 3


LaPlace Transform in Circuit Analysis 
Objectives: 
•Calculate the Laplace transform of common functions 
using the definition and the Laplace transform tables 
•Laplace-transform a circuit, including components with 
non-zero initial conditions. 
•Analyze a circuit in the s-domain 
•Check your s-domain answers using the initial value 
theorem (IVT) and final value theorem (FVT) 
•Inverse Laplace-transform the result to get the time-
domain solutions; be able to identify the forced and 
natural response components of the time-domain solution. 
(Note – this material is covered in Chapter 12 and Sections 
13.1 – 13.3) 
LaPlace Transform in Circuit Analysis 
What types of circuits can we analyze? 
•Circuits with any number and type of DC sources and 
any number of resistors. 
•First-order (RL and RC) circuits with no source and with a 
DC source. 
•Second-order (series and parallel RLC) circuits with no 
source and with a DC source. 
•Circuits with sinusoidal sources and any number of 
resistors, inductors, capacitors (and a transformer or op 
amp), but can generate only the steady-state response. 
LaPlace Transform in Circuit Analysis 
What types of circuits will Laplace methods allow us to 
analyze? 
•Circuits with any type of source (so long as the function 
describing the source has a Laplace transform), resistors, 
inductors, capacitors, transformers, and/or op amps; the 
Laplace methods produce the complete response! 
Page 4


LaPlace Transform in Circuit Analysis 
Objectives: 
•Calculate the Laplace transform of common functions 
using the definition and the Laplace transform tables 
•Laplace-transform a circuit, including components with 
non-zero initial conditions. 
•Analyze a circuit in the s-domain 
•Check your s-domain answers using the initial value 
theorem (IVT) and final value theorem (FVT) 
•Inverse Laplace-transform the result to get the time-
domain solutions; be able to identify the forced and 
natural response components of the time-domain solution. 
(Note – this material is covered in Chapter 12 and Sections 
13.1 – 13.3) 
LaPlace Transform in Circuit Analysis 
What types of circuits can we analyze? 
•Circuits with any number and type of DC sources and 
any number of resistors. 
•First-order (RL and RC) circuits with no source and with a 
DC source. 
•Second-order (series and parallel RLC) circuits with no 
source and with a DC source. 
•Circuits with sinusoidal sources and any number of 
resistors, inductors, capacitors (and a transformer or op 
amp), but can generate only the steady-state response. 
LaPlace Transform in Circuit Analysis 
What types of circuits will Laplace methods allow us to 
analyze? 
•Circuits with any type of source (so long as the function 
describing the source has a Laplace transform), resistors, 
inductors, capacitors, transformers, and/or op amps; the 
Laplace methods produce the complete response! 
LaPlace Transform in Circuit Analysis 
Definition of the Laplace transform: 
 
 
 
Note that there are limitations on the types of functions for 
which a Laplace transform exists, but those functions are 
“pathological”, and not generally of interest to engineers! 
?
?
?
? ?
0
) ( ) ( )} ( { dt e t f s F t f
st
L
Page 5


LaPlace Transform in Circuit Analysis 
Objectives: 
•Calculate the Laplace transform of common functions 
using the definition and the Laplace transform tables 
•Laplace-transform a circuit, including components with 
non-zero initial conditions. 
•Analyze a circuit in the s-domain 
•Check your s-domain answers using the initial value 
theorem (IVT) and final value theorem (FVT) 
•Inverse Laplace-transform the result to get the time-
domain solutions; be able to identify the forced and 
natural response components of the time-domain solution. 
(Note – this material is covered in Chapter 12 and Sections 
13.1 – 13.3) 
LaPlace Transform in Circuit Analysis 
What types of circuits can we analyze? 
•Circuits with any number and type of DC sources and 
any number of resistors. 
•First-order (RL and RC) circuits with no source and with a 
DC source. 
•Second-order (series and parallel RLC) circuits with no 
source and with a DC source. 
•Circuits with sinusoidal sources and any number of 
resistors, inductors, capacitors (and a transformer or op 
amp), but can generate only the steady-state response. 
LaPlace Transform in Circuit Analysis 
What types of circuits will Laplace methods allow us to 
analyze? 
•Circuits with any type of source (so long as the function 
describing the source has a Laplace transform), resistors, 
inductors, capacitors, transformers, and/or op amps; the 
Laplace methods produce the complete response! 
LaPlace Transform in Circuit Analysis 
Definition of the Laplace transform: 
 
 
 
Note that there are limitations on the types of functions for 
which a Laplace transform exists, but those functions are 
“pathological”, and not generally of interest to engineers! 
?
?
?
? ?
0
) ( ) ( )} ( { dt e t f s F t f
st
L
LaPlace Transform in Circuit Analysis 
Aside – formally define the “step function”, which is often 
modeled in a circuit by a voltage source in series with a 
switch. 
 
 
 
 
 
 
 
When K = 1, f(t) = u(t), which we call the unit step function 
0 ,
0 , 0 ) (
? ?
? ?
t K
t t f
K 
t 
f(t) = Ku(t)