Review: Laplace Transform, Linear Circuits and Bode Plots

Alessandro SpinelliPhone: (02 2399) 4001

alessandro.spinelli@polimi.ithome.deib.polimi.it/spinelli

Electronics 96032 Alessandro Spinelli

Slides are supplementary material and are NOT a

replacement for textbooks and/or lecture notes

Laplace transform

𝐹 𝑠 = න0

+∞

𝑓 𝑡 𝑒−𝑠𝑡𝑑𝑡 = ℒ(𝑓)

ℒ is a linear operator:ℒ(𝛼𝑓 𝑡 + 𝛽𝑔(𝑡)) = 𝛼ℒ(𝑓) + 𝛽ℒ(𝑔)

Note: in the following, all signals are defined only for 𝑡 ≥ 0, i.e., they are (implicitely) multiplied by the unit step function

Electronics 96032 Alessandro Spinelli

Properties

1. Time differentiation:

ℒ 𝑓′ 𝑡 = 𝑠𝐹 𝑠 − 𝑓(0)

– If we think in terms of distributional derivatives(e.g., Dirac delta as derivative of step function), then 𝑓(0) → 𝑓(0−) (= 0 in our case)

– If we think in terms of classical derivatives (e.g., zero as derivative of step function), then 𝑓(0) →𝑓(0+)

Electronics 96032 Alessandro Spinelli

Properties

2. “Frequency” differentiation:

−𝑑𝐹 𝑠

𝑑𝑠= ℒ(𝑡𝑓(𝑡))

– More generally

−1 𝑛𝑑𝑛𝐹 𝑠

𝑑𝑠𝑛= ℒ(𝑡𝑛𝑓(𝑡))

Electronics 96032 Alessandro Spinelli

Properties

3. Time integration:

ℒ න0

𝑡

𝑓 𝜏 𝑑𝜏 =𝐹 𝑠

𝑠

4. Frequency integration:

ℒ𝑓 𝑡

𝑡= න

𝑠

∞

𝐹 𝜎 𝑑𝜎

Electronics 96032 Alessandro Spinelli

Properties

5. Time shifting:

ℒ 𝑓 𝑡 − 𝑇 = 𝑒−𝑠𝑇𝐹(𝑠)

6. Frequency shifting:

ℒ 𝑒𝑎𝑡𝑓 𝑡 = 𝐹(𝑠 − 𝑎)

7. Convolution:ℒ 𝑓 𝑡 ∗ 𝑔(𝑡) = 𝐹 𝑠 𝐺(𝑠)

– Dual properties is more complicated, involvingintegral in complex domain

Electronics 96032 Alessandro Spinelli

Properties

8. Initial value theorem:𝑓 0+ = lim

𝑠→∞𝑠𝐹(𝑠)

9. Final value theorem:𝑓 +∞ = lim

𝑠→0𝑠𝐹(𝑠)

Both can be extended to compute the values of the derivatives of 𝑓 just by recalling #1.

Electronics 96032 Alessandro Spinelli

Elementary signals – 1

• Dirac delta function:

ℒ 𝛿 𝑡 = න0

+∞

𝛿 𝑡 𝑒−𝑠𝑡𝑑𝑡 = 1

• Heaviside step function (integral of 𝛿 𝑡 ):

ℒ 𝑢 𝑡 =1

𝑠• Ramp function (integral of 𝑢 𝑡 )

ℒ 𝑡 =1

𝑠2Electronics 96032 Alessandro Spinelli

Also obtainedfrom #2.

Elementary signals – 2

• Rectangle function:

ℒ rect 0, 𝑇 = ℒ 𝑢 𝑡 − 𝑢 𝑡 − 𝑇

=1 − 𝑒−𝑠𝑇

𝑠• Decreasing exponential function (from #6.):

ℒ 𝑒−𝑡/𝜏𝑢(𝑡) =1

𝑠 + 1/𝜏=

𝜏

1 + 𝑠𝜏

Electronics 96032 Alessandro Spinelli

Example

• Let 𝑓 𝑡 = 𝑒−𝑡/𝜏 ⇒ 𝐹 𝑠 = 𝜏/(1 + 𝑠𝜏)

• Then ℒ 𝑓′ 𝑡 = 𝑠𝐹 𝑠 = 𝑠𝜏/(1 + 𝑠𝜏)

• In fact, in a distribution frame:

𝑓′ 𝑡 = 𝛿 𝑡 −1

𝜏𝑒−𝑡/𝜏

ℒ 𝑓′ 𝑡 = 1 −1

𝜏

𝜏

1 + 𝑠𝜏=

𝑠𝜏

1 + 𝑠𝜏

Electronics 96032 Alessandro Spinelli

Elementary signals – 3

• Sine function

ℒ sin 𝜔𝑡 =𝜔

𝑠2 +𝜔2

• Cosine function (from #1.)

ℒ cos 𝜔𝑡 =𝑠

𝑠2 +𝜔2

Electronics 96032 Alessandro Spinelli

Application to LTI systems

Electronics 96032 Alessandro Spinelli

𝐻(𝑠)

ℎ(𝑡)𝑥(𝑡) 𝑦 𝑡 = 𝑥 𝑡 ∗ ℎ(𝑡)

𝑋(𝑠) 𝑌 𝑠 = 𝑋 𝑠 𝐻(𝑠)

ℒ ℒ ℒ−1

Resistors

Electronics 96032 Alessandro Spinelli

𝑖 𝑡

𝑣 𝑡

𝑣 𝑡 = 𝑅𝑖 𝑡

𝑉 𝑠 = 𝑅𝐼(𝑠)

𝑉(𝑠)

𝐼(𝑠)= 𝑅

Capacitors

𝑖 𝑡 = 𝐶𝑑𝑣(𝑡)

𝑑𝑡

𝐼 𝑠 = 𝑠𝐶𝑉(𝑠)

𝑉(𝑠)

𝐼(𝑠)=

1

𝑠𝐶= 𝑍𝐶

Electronics 96032 Alessandro Spinelli

𝑖 𝑡

𝑣 𝑡

Inductors

𝑣 𝑡 = 𝐿𝑑𝑖(𝑡)

𝑑𝑡

𝑉 𝑠 = 𝑠𝐿𝐼(𝑠)

𝑉(𝑠)

𝐼(𝑠)= 𝑠𝐿 = 𝑍𝐿

Electronics 96032 Alessandro Spinelli

𝑖 𝑡

𝑣 𝑡

RC network

Electronics 96032 Alessandro Spinelli

𝑣𝑖 𝑡 𝑣𝑜 𝑡

𝐶𝑅

𝑖 𝑡

ቐ

𝑣𝑖 = 𝑣𝑜 + 𝑅𝑖

𝑖 = 𝐶𝑑𝑣𝑜𝑑𝑡

⇒ 𝑅𝐶𝑑𝑣𝑜𝑑𝑡

+ 𝑣𝑜 = 𝑣𝑖

𝑉𝑜(𝑠)

𝑉𝑖(𝑠)=

1/𝑠𝐶

𝑅 + 1/𝑠𝐶=

1

1 + 𝑠𝐶𝑅

Output signals

• Delta function response, 𝑣𝑖 𝑡 = 𝐴𝛿(𝑡)

𝑉𝑖 𝑠 = 𝐴 ⇒ 𝑉𝑜 =𝐴

1 + 𝑠𝐶𝑅⇒ 𝑣𝑜 =

𝐴

𝜏𝑒−𝑡/𝜏

• Step function response, 𝑣𝑖 𝑡 = 𝐴𝑢(𝑡)

𝑉𝑖 𝑠 =𝐴

𝑠⇒ 𝑉𝑜 =

𝐴

1 + 𝑠𝜏

1

𝑠= 𝐴

1

𝑠−

𝜏

1 + 𝑠𝜏

⇒ 𝑣𝑜 = 𝐴 𝑢 𝑡 − 𝑒−𝑡/𝜏𝑢 𝑡 = 𝐴(1 − 𝑒−𝑡/𝜏)

Electronics 96032 Alessandro Spinelli

Ex: Lag network (step response)

Electronics 96032 Alessandro Spinelli

𝑉𝑖 𝑉𝑜𝐶

𝑅1

𝑅2

𝑉𝑜𝑉𝑖=

𝑅2 + 1/𝑠𝐶

𝑅1 + 𝑅2 + 1/𝑠𝐶=

1 + 𝑠𝐶𝑅21 + 𝑠𝐶(𝑅1 + 𝑅2)

𝑉𝑜 =1 + 𝑠𝐶𝑅2

1 + 𝑠𝐶(𝑅1 + 𝑅2)

𝐴

𝑠

𝑉𝑜 = 𝐴1

𝑠−

𝐶𝑅11 + 𝑠𝐶 𝑅1 + 𝑅2

⇒ 𝑣𝑜(𝑡) = 𝐴 1 −𝑅1

𝑅1 + 𝑅2𝑒−𝑡/𝜏

Alternative method

• Compute:

𝑣𝑜 0 = lim𝑠→∞

𝑠𝑉𝑜(𝑠) = 𝐴𝑅2

𝑅1 + 𝑅2𝑣𝑜 ∞ = lim

𝑠→0𝑠𝑉𝑜(𝑠) = 𝐴

• Connect extremes withan exponential curvehaving time constantdetermined by the singlepole

Electronics 96032 Alessandro Spinelli

𝑡

𝑣𝑜

One more approach

• Voltage across a capacitors cannot changeabruptly (unless you are applying a 𝛿-functioncurrent) 𝑣𝑐 0+ = 𝑣𝑐 0− = 0

– 𝑣0 0+ = 𝐴𝑅1/(𝑅1 + 𝑅2)

• For 𝑡 → ∞, the capacitor behaves as an open circuit

– 𝑣𝑜 ∞ = 𝐴

Electronics 96032 Alessandro Spinelli

LTI systems and sinusoidal inputs

• The output of a stable LTI system to a sinusoidalinput contains a transient and a steady-state term

• The steady-state term is also sinusoidal at the same frequency as the input, but with differentamplitude and phase:

where𝐵 = 𝑇 𝑗𝜔 𝜙 = ∡(𝑇 𝑗𝜔 )

Electronics 96032 Alessandro Spinelli

𝐴 sin(𝜔𝑡) 𝑇(𝑠) 𝐴𝐵 sin(𝜔𝑡 + 𝜙)

Bode plots

• Provide an efficient way to plot the frequencyresponse (magnitude and phase) of LTI systems

• We consider asymptotic Bode plots, i.e., piecewise linear approximations on a suitablescale:

– Magnitude: dB = 20 log10 | ∙ |

– Phase: linear scale

– Frequency: log scale

Electronics 96032 Alessandro Spinelli

Single real pole: magnitude

𝑇 𝑠 =1

1 + 𝑠𝜏⇒ 𝑇 𝑗𝜔 𝑑𝐵 = 20 log10

1

1 + 𝜔𝜏 2

= −20 log10 1 + 𝜔𝜏 2 ≈ ቊ0 𝜔𝜏 ≪ 1

−20 log10𝜔𝜏 𝜔𝜏 ≫ 1

Electronics 96032 Alessandro Spinelli

log10 𝑓

Slope = −20 dB/dec

1

2𝜋𝜏

Bode plot of a real zero

Single real pole: phase

Electronics 96032 Alessandro Spinelli

∡ 𝑇 𝑗𝜔 = ∡1

1 + 𝑗𝜔𝜏= −arctan(𝜔𝜏)

log10 𝑓

1

2𝜋𝜏∡ ⋅

Bode plot of a real zero

General case (real singularities)

• 𝑇 𝑠 can always be expressed as

𝑇 𝑠 = 𝐺1 + 𝑠𝜏𝑧1 1 + 𝑠𝜏𝑧2 …(1 + 𝑠𝜏𝑧𝑛)

1 + 𝑠𝜏𝑝1 1 + 𝑠𝜏𝑝2 …(1 + 𝑠𝜏𝑝𝑚)

• Thanks to log and arctan properties:

𝑇 𝑑𝐵 =

𝑖=1

𝑛

1 + 𝑗𝜔𝜏𝑧𝑖 𝑑𝐵 −

𝑗=1

𝑚

1 + 𝑗𝜔𝜏𝑝𝑗 𝑑𝐵

∡(𝑇) =

𝑖=1

𝑛

∡(1 + 𝑗𝜔𝜏𝑧𝑖) −

𝑗=1

𝑚

∡(1 + 𝑗𝜔𝜏𝑝𝑗)

Electronics 96032 Alessandro Spinelli

Ex: Lag network

𝑇(𝑠) =1 + 𝑠𝐶𝑅2

1 + 𝑠𝐶(𝑅1 + 𝑅2)

Electronics 96032 Alessandro Spinelli

log10 𝑓

1

2𝜋𝜏𝑧

1

2𝜋𝜏𝑝

𝜏𝑧

𝜏𝑝

1

𝑅2𝑅1 + 𝑅2

𝑇 𝑑𝐵

In practice…

• Magnitude:

– At every zero frequency, the slope is increased by 1 (20 dB/dec)

– At every pole frequency, the slope is reduced by 1(20 dB/dec)

• Phase:

– Composition is a bit more complicated, but a rough view can be obtained by abruptly adding± 90° at every zero/pole frequency

Electronics 96032 Alessandro Spinelli

Asymptotic values

𝑇(𝑠) =1 + 𝑠𝐶𝑅2

1 + 𝑠𝐶(𝑅1 + 𝑅2)

Electronics 96032 Alessandro Spinelli

𝑉𝑖 𝑉𝑜𝐶

𝑅1

𝑅2

log10 𝑓

log10 𝑓

𝑉𝑖 𝑉𝑜𝐶

𝑅1

𝑅21

2𝜋𝜏𝑧

1

2𝜋𝜏𝑝

𝜏𝑧

𝜏𝑝

1

𝑅2𝑅1 + 𝑅2

∡ 𝑇

𝑇 𝑑𝐵

Ex: BP filter

𝑇 𝑠 =𝐴𝑠𝜏0

(1 + 𝑠𝜏1)(1 + 𝑠𝜏2)𝜏1 ≫ 𝜏2

Electronics 96032 Alessandro Spinelli

log10 𝑓

log10 𝑓

1

2𝜋𝜏2

1

2𝜋𝜏1

90

∡ 𝑇

𝑇 𝑑𝐵

−90

Homework

1. Compute the step response of a CR filter

2. Design a simple network (made with 𝑅s and 𝐶s) with a zero and a pole having 𝑓𝑧 < 𝑓𝑝 (a

lead network)

3. Quote the Bode diagram of the previous slide

4. Work out the step response of the BP filter of the previous slide

Electronics 96032 Alessandro Spinelli