Lecture 38: Recap of Laplace Transform and its Applications
12 Recap of Laplace Transform and its Applications
In this lecture, we review the second part of the course, consisting of:
- The Laplace Transform
- Linear Time-Invariant (LTI) Systems
- Application of the Laplace Transform to LTI Differential Systems
- Time and Frequency Analysis of BIBO stable, continuous-time LTI systems
- Application of Laplace Transform Techniques to Electric Circuit Analysis
12.1 The Laplace Transform
The Laplace transform is a generalization of the Fourier transform.
Advantages of the Laplace transform over the Fourier transform:
The Fourier transform was defined only for stable systems or signals that taper off at infinity (signals of finite energy or absolutely integrable.) On the other hand, the Laplace transform of an unbounded signal or of an unstable impulse response is defined. The Laplace transform can also be used to analyze differential LTI systems with nonzero initial conditions.
The Laplace transform of is defined as follows:
Important note: The ROC is an integral part of a Laplace transform. It must be specified.
The inverse Laplace transform is in general given by
This contour integral is rarely used because we are mostly dealing with linear systems and standard signals whose Laplace transforms are found in tables of Laplace transform pairs.
Poles and zeros:
The zeros of the numerator are called the zeros of the Laplace transform. If is a zero, then . The zeros of the denominator are called the poles of the Laplace transform. If is a pole, then
Pole-zero plot:
Example
12.1.1 Convergence of the Two-Sided Laplace Transform
Convergence of the integral depends on the real part of the complex Laplace variable. Thus the region of convergence in the complex plane or "s"-plane is either the whole plane, a vertical half-plane, a vertical strip, or nothing.
12.1.2 Some Important Properties of the ROC
· If is of finite duration and is absolutely integrable, then the ROC is the entire s-plane.
· The ROC for a right-sided signal is a right half-plane (special case: "causal" signal).
· The ROC for a left-sided signal is a left half-plane.
· If is two-sided and if the line is in the ROC, then the ROC consists of a strip in the -plane that includes the line .
12.1.3 Important Properties of the Two-Sided Laplace Transform
Time-Shifting:
Time-Scaling: ,
Convolution Property: .
Differentiation in the Time Domain: ,
Differentiation in the Frequency Domain: ,
Integration in the Time Domain:,
The Initial and Final Value Theorems:
Initial-value theorem: ,
Final-value theorem: .
12.1.4 The Unilateral Laplace Transform
The one-sided or unilateral Laplace transform of is defined as follows:
Note that two signals that differ for but are equal for will have the same one-sided Laplace transform. Also note that the unilateral Laplace transform of is identical to the (two-sided) Laplace transform of .
The differentiation property of the unilateral Laplace transform is useful to analyze the response of causal differential systems to nonzero initial conditions.
If , then ,
12.2 Analysis and Characterization of LTI Systems using the Laplace Transform
The Laplace transform of the output of an LTI system with impulse response is:
.
Also recall that the frequency response of the system is .
The Laplace transform of the impulse response is called the transfer function.
12.2.1 Causality
Recall that for for a causal system, and thus is right-sided. Therefore,
The ROC associated with the transfer function of a causal system is a right half-plane.
The converse is not true. BUT, if we know that the transfer function is rational, then it suffices to check that the ROC is the right half-plane to the right of the rightmost pole in the s-plane to conclude that the system is causal.
12.2.2 Stability
· An LTI system is stable if and only if the ROC of its transfer function contains the -axis.
· A causal system with rational transfer function is stable if and only if all of its poles are in the left-half of the s-plane (i.e., all of the poles have negative real parts.)
12.3 Application of the Laplace Transform to LTI Differential Systems
Consider the general form of an LTI differential system:
.
Transfer function :
.
Note that we haven't specified an ROC yet. If the differential system is causal, then the ROC is the right half-plane to the right of the rightmost pole in the s-plane. The impulse response is then uniquely defined.
Causality
An LTI differential system is causal if and only if the ROC of its transfer function is an open right half-plane located to the right of the rightmost pole.
Stability
A causal LTI differential system is stable if and only if the poles of its transfer function lie in the open left half-plane.
Routh's Criterion for Stability
Consider the transfer function
Assume that . Compute the Routh array:
The fourth row is computed in the same manner from the two rows immediately above it, and so on until the last row labeled is reached.
Theorem (Routh):
The system is stable if and only if all the entries in the first column are positive. If there are sign changes in this column, then there are as many RHP (unstable) poles are there are sign changes.
12.3.1 Realization of a Transfer Function
Cascade: Write as the multiplication of first-order systems
Parallel: Expand in partial fractions
Direct Form: A direct form can be obtained by breaking up a general transfer function into two subsystems as follows
The input-output system equation of the first subsystem is
,
and for the second subsystem we have
.
The direct form realization is then (for a second-order system):
12.3.2 Transient and Steady-State Analysis Using the Laplace Transform
For a causal, stable LTI system, a partial fraction expansion of the transfer function allows us to determine which terms correspond to transients (the terms with the system poles) and which correspond to the steady-state response (terms with the input poles).
Example: Consider the step response
The steady-state response corresponds to the last term , which in the time-domain is . The other two terms correspond to the transient response .
Step response: We can use the final value theorem to determine the steady-state component of a step response. In general, this component is a step function . The "gain" is given by
Response to a sinusoid or a periodic exponential:
For, the steady-state response is
For a sinusoidal input, say ,
Response to a periodic signal:
For , the steady-state response is
.
where is the Fourier series coefficient of the periodic output signal.
12.4 Time and Frequency Analysis of BIBO Stable, Continuous-Time LTI Systems
12.4.1 Relation of Poles and Zeros of the Transfer Function to the Frequency Response
The vector representation of each first-order pole and zero factor in a transfer function can help us visualize its contribution to the overall frequency response of the system.
Example: A stable third-order system with transfer function
has the frequency response
.
In the s-plane, the vector-valued functions of originating at the poles and zeros are depicted below.
The phase at frequency is given by the sum of the angles of the vectors originating at zeros minus the sum of the angles of the vectors originating at poles.
The magnitude at frequency is given by the product of the lengths of the vectors originating at zeros divided by the product of the lengths of the vectors originating at poles.
12.4.2 Bode Plot
A Bode plot is the combination of a magnitude and phase plots using log scales for the magnitude and the frequency, and a linear scales (radians or degrees) for the phase. Only positive frequencies are normally considered. As stated above, the Bode plot is quite useful since the overall frequency response of cascaded systems is simply the graphical addition of the Bode plots of the individual systems. In particular, this property is used to hand sketch a Bode plot of a rational transfer function in pole-zero form by considering each first-order factor corresponding to a pole or a zero to be an individual system with its own Bode plot.
First-Order Example: Consider the first-order system with transfer function which has the frequency response .
It is convenient to write it as the product of a gain and a first-order transfer function with unity gain at dc:
.
The break frequency is 2 radians/s. The Bode magnitude plot is
The Bode phase plot is the graph of
.
The Bode phase plot is shown below.
12.4.3 Frequency Response of First-Order Lag, Lead and Second-Order Lead-Lag Systems
First-order lag ,
where , is the time constant and is either called the natural frequency, the cutoff frequency, or the break frequency, depending on the application. Bode plot:
First-order lead , where but .
A second-order lead-lag is simply a cascade of a lead and a lag.
12.4.4 Frequency Response of Second-Order Systems
Causal, stable LTI second-order systems
.
where is the damping ratio is the undamped natural frequency of the second-order system.
Case
In this case, the system is said to be overdamped. The step response doesn't exhibit any ringing. The two poles are real, negative and distinct: . The second-order system can be seen as a cascade of two standard first-order systems (lags).
Case
In this case, the system is said to be critically damped. The two poles are negative and real, but they are the same. We say that it’s a repeated pole; . In this situation, the second-order system can also be seen as a cascade of two first-order transfer functions having the same pole.
Case
In this case, the system is said to be underdamped. The step response exhibits some ringing, although it really becomes visible only for . The two poles are distinct, complex conjugates of each other: . The frequency response has a peak for . The maximum of the magnitude occurs at the resonant frequency which is close to for low damping ratios. At the resonant frequency, the magnitude of the peak resonance is given by
The Bode plot of
When the damping ratio is very low, a second-order filter becomes highly selective due to its high peak resonance at . The quality Q of the filter is defined as .
12.4.5 Step Response of Stable LTI Systems
Rise Time: time taken by the output to rise from 5% to 95% of its final value.
Overshoot: percentage of the final value of the output signal.
Settling time: The ±5% settling time is the time when the response gets to within 5% of its final value for all subsequent times.
12.4.6 Non-Minimum Phase and All-Pass Systems
Systems whose transfer functions have RHP zeros are said to be non-minimum phase.
Note that any non-minimum phase transfer function can be expressed as the product of a minimum-phase transfer function and an allpass transfer function. For example
12.5 Application of Laplace Transform Techniques to Electric Circuit Analysis
12.5.1 Nodal Analysis
Kirchhoff's Current Law (KCL) states that the sum of all currents entering a node is equal to zero. This law can be used to analyze a circuit by writing the current equations at the nodes (nodal analysis) and solving for the node voltages. Nodal analysis is usually applied when the unknowns are voltages. For a circuit with N nodes and N-1 unknown node voltages, one needs to solve N-1 nodal equations.
12.5.2 Mesh Analysis
Kirchhoff's Voltage Law (KVL) states that the sum of all voltages around a mesh (a loop that has no element within it) is equal to zero. This law can be used to analyze a circuit by writing the voltage equation for each mesh (mesh analysis) and solving for the mesh currents. Mesh analysis is usually applied when the unknowns are currents.
12.5.3 Transform Circuit for Nodal Analysis
For a resistive network, the currents are usually written as the difference between two voltages divided by the impedance (resistance). For a transform circuit, the same principle applies, although the impedance of each element is in general a function of the Laplace variable.
Using the unilateral Laplace transform, an inductor current can be written as:
and the corresponding circuit diagram replacing the inductor is
The capacitor voltage-current relationship in the Laplace domain is written as
and the corresponding circuit diagram replacing the capacitor is
12.5.4 Transform Circuit for Mesh Analysis
Mesh analysis can be used to analyze a circuit by applying KVL around the circuit meshes. For a resistive network, the voltages are usually written as the mesh current times the resistances. The same principle applies for a transform circuit, although in this case the impedance of each element is in general a function of the Laplace variable.
Using the unilateral Laplace transform, an inductor voltage can be written as:
.
The corresponding circuit diagram replacing the inductor is
The capacitor voltage-current relationship in the Laplace domain is written as
.
The corresponding circuit diagram replacing the capacitor is
12.5.5 Thévenin Equivalent Circuit
Consider the following linear single-port network containing basic circuit elements and independent and controlled sources.
Thévenin's theorem: a one-port network is equivalent to a voltage source in series with an impedance. The Thévenin equivalent circuit is shown below.
12.5.6 Norton Equivalent Circuit
Consider again our linear single-port network containing basic circuit elements and independent and controlled sources.
Norton's theorem: a one-port network is equivalent to a current source in parallel with an impedance.
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