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 th ...

1. 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

2. 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

3. 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

4. node voltage method. OpAmp is considered ideal.)
b. Show that the canonical form of the transfer function:
H(s)=��(��)
��(��)
= ����
��02
��2+(3−����)��0��+��02
where AV=(1 +
����
��1
) 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

5. 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

6. 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

7. (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 ��
3��(��)
����3
+ 11 ��
2��(��)
����2

8. + 25 ����(��)
����
+ 6��(��) = 2 ����(��)
����
+ ��(��)
b. (����(��)
����
)2 + ��(��) = 2��(��)
c. 3 ����(��)
����
+ ����(��) = ��(��)
d. ����(��)
����
+ 2��(��) = ��(��) ����(��)
����
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

9. 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

10. Vs1(t)=cos(100t) (V) and Vout2(t)
with input VS2(t)=5cos(100000t) (V) respectively.