chp4
1
Active Circuit Analysis
chp4
2
Electrical System
• Composed of resistors, capacitors, inductors,
transistors, amplifiers, power supplies
– Passive circuits: respond to applied voltage or current and
do not have any amplifiers
– Active circuits: made of transistors and/or amplifiers,
require active power source to work
• Basic quantities
–
–
–
–
–
Charge q [coulomb] = 6.24x1018 electrons
Current i [ampere] = dq/dt
Voltage e [Volt] = dw/dq
Energy or Work w [joule]
Power p [watt] = e x i = dw/dt
chp4
3
Operational Amplifier
• Op-amp: integrated circuit that amplifies voltage
positive power supply
inverting input
non-inverting input
(reference, usu. grounded)
output
negative power supply
• Key properties
– High gain (> 106 volt/volt) -> ideal computation device
– Low output impedance (< 100 ohms) -> output voltage does not vary with
output current, so amplifier drives loads as ideal voltage source
– High input impedance (106 ohms) and low input voltage -> no current is
required by amplifier
– Idealizations: zero noise, infinite bandwidth
chp4
4
Operational Amplifier
•
Component equations:
Zf: feedback impedance
Zi: input impedance
•
Node equation:
Input is grounded and
differential power supply is used
•
Substitute component eqs. into node eq:
Zf/Zi is small compared to G
chp4
Basic Op-Amp Circuits
5
chp4
6
chp4
7
Example 8: Op-Amp Circuit
• Above is an op-amp circuit with impedances on the
plus and minus inputs, derive the output equation
e0 as a function of en and ep. The amplifier has
characteristic e0=G(eap-ean), where G >> 1.
• Show that if all impedances are resistive and equal
to R, then e0=ep-en.
chp4
8
Example 8: Op-Amp Circuit
iZf
iZn
iZp
iZg
chp4
9
Example 8: Op-Amp Circuit
iZf
iZn
iZp
iZg
chp4
10
Example 8: Op-Amp Circuit
chp4
11
Example 9: Pair-Share: Op-Amp Circuit
iRf
• Above is a an op-amp
circuit used to drive an
electromagnetic coil on a
servo valve. Write all the
modeling equations and
derive the transfer function
for iv as a function of input
voltage ei.
• Derive a state-space
representation for the
system.
iC
iRi
iv
chp4
12
Example 9: Pair-Share: Op-Amp Circuit
chp4
13
Example 9: Pair-Share: Op-Amp Circuit
chp4
14
Example 10: Full-Bridge Strain Gauge Circuit
chp4
15
Example 10: Full-Bridge Strain Gauge Circuit
=R2
chp4
16
Example 10: Full-Bridge Strain Gauge Circuit
chp4
17
Example 11: Pair-Share: Audio Amplifier Circuit w/ Light Bulb
chp4
18
Example 11: Pair-Share: Audio Amplifier Circuit w/ Light Bulb
chp4
19
Example 11: Pair-Share: Audio Amplifier Circuit w/ Light Bulb
chp4
20
Case Study:
A Speaker Model
chp4
21
Modeling a Speaker
• Read Chapter 4 Case Study Handout
• We are interested in developing a model
relating the output x to input v
– What is the order of the system?
– Use Newton’s law and Kirchhoff’s laws to develop
the transfer function between x and v
chp4
22
Modeling a Speaker
From Newton's law :
d 2x
dx
m 2  c  kx  K f i
dt
dt
From Kirchoff' s law :
di
dx
v  L  Ri  K e
dt
dt
V ( s )  K e sX ( s )
I (s) 
Ls  R
or
or ms 2 X ( s )  csX ( s )  kX ( s )  K f I ( s )
V ( s )  LsI ( s )  RI ( s )  K e sX ( s )
Substituting I(s) into the above equation,
Kf
X (s)

V ( s ) mLs3  (cL  mR) s 2  (kL  cR  K f K e ) s  kR
chp3
23
References
• Woods, R. L., and Lawrence, K., Modeling and Simulation of
Dynamic Systems, Prentice Hall, 1997.
• Close, C. M., Frederick, D. H., Newell, J. C., Modeling and
Analysis of Dynamic Systems, Third Edition, Wiley, 2002
• Palm, W. J., Modeling, Analysis, and Control of Dynamic
Systems