Linear System exercise

Exercise Package 2:

Systems and its properties: (Tip: Always use the components symbols, C, RS, KT, etc., in the derivation of transfer function and only plug in component values at the last step. Show your steps and tell me a complete story.)

1) Consider a 100mH inductor with v-i relationship in passive device labeling convention: a. Find transfer function H(s) with current flowing through the inductor as the input, i(t),

and voltage across the inductor as the output, v(t), (in the unit of Ohms). b. Find the same input-output relationship in the expression of differential equation. c. Find v1(t) with input i1(t)=2sin(100t) (mA) and v2(t) with input i2(t)=0.4cos(500t) (mA)

respectively. d. Show time invariant such that v(t)=v1(t−τ) as i(t)=i1(t−τ)=2sin(100t−0.9) (mA). e. Show linearity using superposition such that v(t)=v1(t)+v2(t) as i(t)=i1(t)+i2(t).

2) Given following, a practical integrator, circuit, where Rf=100KΩ, R1=9.1KΩ, RS=100Ω, C=0.1µF, and the OpAmp is an ideal operational amplifier:

a. Find the transfer function in between the output VO(t) and input VS(t), VO(t)=H(s){VS(t)}. b. Find the same input-output relationship in the expression of differential equation. c. Find VO1(t) (sinusoidal steady state response) with input VS1(t)=0.2sin(100t) (V) and VO2(t)

with input VS2(t)=0.4cos(5000t) (V) respectively. d. Show time invariant such that VO(t)= VO1(t−τ) as VS(t)= VS1(t−τ)=0.2sin(100t−0.9) (V). e. Show linearity using superposition such that VO(t)= VO1(t)+VO2(t) with VS(t)=VS1(t)+ VS2(t).

3) Here is a typical coupling network in electronics where coupling capacitor, selected, C=0.022µF, input impedance, Zi=5.7KΩ, and input source resistor, RS=520Ω:

a. Find the transfer function, H(s), Vout(t)=H(s){Vin(t)}. b. Find the same input-output relationship in the expression of differential equation. c. Find VOut(t) (sinusoidal steady state response) with input Vin1(t)=2sin(50t+0.4) (V) and

Vin2(t) with input Vin2(t)=4cos(10000t) (V) respectively. 4) Here is a typical bypass network in electronics where bypass capacitor, selected, C=10µF, and

the equivalent (Thevenin) resistor of circuit to be bypassed, Req=376Ω:

Vcc+

Vcc-

Vo Vs

Rf

R1Rs

C

Vin Vout

CRs

Zi

a. Find the transfer function, H(s), VS(t)=H(s){IS(t)} (note: the unit is ohm). b. Find the same input-output relationship in the expression of differential equation. c. Find VS1(t) (sinusoidal steady state response) with input Is1(t)=0.2cos(10t+0.3) (A) and

VS2(t) with input IS2(t)=0.5cos(10000t) (A) respectively. 5) The following circuit is an active filter (2nd order Butterworth low-pass filter), with the selected

values: R=10KΩ, C=8200pF, Rf=68KΩ, and R1=120KΩ.

a. Derive the transfer function, H(s), Vout(t)=H(s){Vin(t)}. (Tip: the selected R is much greater

than RS such that RS can be ignored in the derivation. Label extraordinary nodes and use node voltage method. OpAmp is considered ideal.)

b. Show that the canonical form of the transfer function: H(s)=𝐵𝐵(𝑠𝑠) 𝐴𝐴(𝑠𝑠)

= 𝐴𝐴𝑉𝑉 𝜔𝜔02

𝑠𝑠2+(3−𝐴𝐴𝑉𝑉)𝜔𝜔0𝑠𝑠+𝜔𝜔02

where AV=(1 + 𝑅𝑅𝑓𝑓 𝑅𝑅1

) and ω0=1/(RC). c. Find VO1(t) (sinusoidal steady state response) with input VS1(t)=0.2cos(100t) (V) and

VO2(t) with input VS2(t)=0.2cos(50000t+0.2) (V) respectively. d. Find the power associated with VO1(t) and VO2(t) respectively (comments). e. Find the same input-output relationship in the expression of differential equation.

6) Find the transfer function of a field-controlled (armature current is fixed) DC motor where the torque generated by the motor is proportional (linearized) to the field current, T(t)=KTif(t), where KT is motion-torque constant (Newton•meter/A). It shows in the figure that the combined rotor and load moment of inertia is J (Kg•meter2=Newton•meter•second2/radian) and ω(t) is the angular speed (radian/second). s{•} is the derivative operator. B is the viscous damping coefficient (Newton•meter•second/radian). θ(t) is angular position (radian).

a. Find the transfer function, Hω(s), ω(t)=H(s){Vin(t)}, (unit radian/(second•V)).

Vs C Req

Is

Vcc+

Vcc-

Vo Vs

Rf R1

Rs

C

C

RR

Vin

Rf

Lf M

Fixed armature current Ia

if J

Bω(t)

θ(t)

T(t)

Js{ω(t)}

b. Find the transfer function, Hθ(s), θ(t)=H(s){Vin(t)}, (unit radian/V). c. Typically Lf<<Rf in a real motor, Lf can be ignored to reduce the order of transfer

function. Find the transfer function, Hω(s), in the reduced order. d. With a specific motor Rf=2Ω, Lf=3.5µH, J=3.4*10−6 (kg•meter2), KT=0.03

(Newton•meter/A), and B=3.6*10−6 (Newton•meter•second/radian), find transfer functions, Hω(s), in both original and reduced order.

7) Find the transfer function of a armature-controlled (field current is fixed) DC motor where the torque generated by the motor is proportional (linearized) to the field current, T(t)=KTia(t), where KT is motion-torque constant (Newton•meter/A). The combined rotor and load moment of inertia is J (Kg•meter2=Newton•meter •second2/radian) and ω(t) is the angular speed (radian/second). s{•} is the derivative operator. B is the viscous damping coefficient (Newton•meter•second/radian). θ(t) is angular position (radian). Kb is the electromagnetic field constant and equal to KT for an ideal DC motor. Kbω(t) is back electromotive force (according to Faraday’s law of electromagnetic induction) in Volt.

a. Find the transfer function, Hω(s), ω(t)=H(s){Vin(t)}, (unit radian/(second•V)). b. Find the transfer function, Hθ(s), θ(t)=H(s){Vin(t)}, (unit radian/V). c. Typically La<<Ra in a real motor, La can be ignored to reduce the order of transfer

function. Find the transfer function, Hω(s), in the reduced order. d. With a specific motor Ra=4.2Ω, Lf=5µH, J=5.4*10−6 (kg•meter2), KT= Kb=0.036

(Newton•meter/A), and B=3*10−6 (Newton•meter•second/radian), find transfer functions, Hω(s), in both original and reduced order.

8) Determine the following system is LTI and if it is a LTI system find its transfer function:

a. 6 𝑑𝑑 3𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡3

+ 11 𝑑𝑑 2𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡2

+ 25 𝑑𝑑𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡

+ 6𝑦𝑦(𝑡𝑡) = 2 𝑑𝑑𝑑𝑑(𝑡𝑡) 𝑑𝑑𝑡𝑡

+ 𝑥𝑥(𝑡𝑡)

b. (𝑑𝑑𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡

)2 + 𝑦𝑦(𝑡𝑡) = 2𝑥𝑥(𝑡𝑡)

c. 3 𝑑𝑑𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡

+ 𝑡𝑡𝑦𝑦(𝑡𝑡) = 𝑥𝑥(𝑡𝑡)

d. 𝑑𝑑𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡

+ 2𝑦𝑦(𝑡𝑡) = 𝑥𝑥(𝑡𝑡) 𝑑𝑑𝑑𝑑(𝑡𝑡) 𝑑𝑑𝑡𝑡

9) Here is a circuit network for a lowpass filter where bypass capacitor, selected, Ci=1µF, and the

source resistor Rs=500Ω and the load resister RL=1KΩ:

a. Find the transfer function, H(s), Vout(t)=H(s){VS(t)}. b. Find the same input-output relationship in the expression of differential equation.

Vin

Ra La

M

Fixed field current If

ia J

Bω(t)

θ(t)

T(t)

Js{ω(t)}

Lf

+

Lf

_ Kbω(t)

Vs VoutCi

Rs

RL

c. Find Vout1(t) (sinusoidal steady state response) with input Vs1(t)=cos(100t) (V) and Vout2(t) with input VS2(t)=5cos(100000t) (V) respectively.