electrical circuits ii (ece233b)
TRANSCRIPT
Electrical Circuits II (ECE233b)
The University of Western Ontario Faculty of Engineering Science
Anestis Dounavis
Variable-Frequency Network Performance (Part 3)
ScalingOften the values of circuit parameters vary by orders of magnitudes
For example: Resistors units MCapacitors units pF 10-12
Inductors units nH 10-9
Time units ns 10-9
Frequency units GHz 109
In addition, scaling can sometimes make the results more presentable.
Since computers are finite precision machines, scaling circuit parameters can result in numerically more accurate results.
There are two ways to scale circuit parameters
Can scale: Magnitude (or Impedance) scalingFrequency scaling
106
Magnitude Scaling
Note magnitude scaling does not affect the frequency response
Lets verify for RLC series circuit
00
MM K
CLK
1CL
1
QRK
LKR
LQM
M00
Magnitude Scaling RKR M
LKL M
MKCC
Frequency ScalingNote resistors are frequency independent and are unaffected by this scaling
LjLj
The new inductor and capacitor values must have the same impedance at the scale frequency as the original circuit and must satisfy:
L C0
CjCj 11
where FK and KF is the frequency scaling factor
Frequency Scaling RR
FKLL
FKCC
Frequency independent
Note that frequency scaling affects the resonant frequency and bandwidth but not the Q.
Frequency Scaling
00 F
FF
K
KC
KL
1CL
1
QRL/KK
RLQ M0M0
(BW)KQ
WB F
0
Lets verify for RLC series circuit
Example 14An RLC network has the following parameters values: R = 10, L=1 H and C=2F. Determine the values of the circuit elements if the circuit is magnitude scaled by a factor of 100 and frequency scaled by factor of 10 000.
Filter NetworksTwo types: PASSIVE and ACTIVE circuits
Passive Filters – circuits composed of passive RLC elements
Types of Filters 1. Low pass filters: allows low frequencies to pass and rejects
high frequencies2. High pass filters: allows high frequencies to pass and rejects
low frequencies3. Band pass filters: allows some particular band of frequencies
to pass and rejects all frequencies outside the range
4. Band reject filters: rejects some particular band of frequencies and allows all other frequencies to pass
Low Pass Filters
R
C+V0
-
+-VI
Ideal Characteristic of Low Pass Filter
Magnitude
Simple Low Pass Filter
RCjsC1R(sC)
VV(s)G
I
0V
1
1)(
1
RC1
0
Magnitude1
21
Actual characteristico resonant frequency
break frequencyhalf power frequency
Low Pass Filters
0
1
1
jV
V(s)GI
0V
R
C+V0
-
+-VI
RC1
0
Bode plot approximationActual response
-20dB
0dB
00
-900
-450
2/12
0
1
1
)M(
0
1tan
)(
High Pass FilterIdeal Characteristic of High Pass Filter
Magnitude
Simple High Pass Filter
RCjRCj
sC1RR
VV(s)G
I
0V
1)(
RC1
0 Magnitude 1
21
Actual characteristic
RC
+-VI
+ Vo -
RC
+-VI
+ Vo -High Pass Filters
0
0
1
jVV(s)G
I
0V
j
RC1
0
Bode plot approximationActual response
-20dB
0dB
900
00
450
2/12
0
0
1
)M(
0
10 tan90
)(
Band Pass Filter
LC
R
+VO
-
+VI
-
Ideal Characteristic of Band Pass Filter
LO
Magnitude
HI
Simple Band Pass Filter
)(sC1sLRR
VV(s)G
I
0V
C)(1LjRR
Band Pass Filter
C)(1LjRR
VV)(jG
I
0V
LC
R
+VO
-
+VI
-Magnitude
Actual characteristic
LO HI
1
21
222 1)LC(RC)(RC)M(
Band Pass FilterMagnitude
1
21
Actual characteristic
222 1)LC(RC)(RC)M(
LO HI
The center frequency 0
LC1
0 rad/s01LC2
Lower cut off frequency Upper cut off frequency 1-LCRC 2 1-LCRC 2
24(R/L)(R/L)
LO
20
24(R/L)(R/L)
LO
20
BandwidthLR
Q BW 0 LOHI
LO
Magnitude
HI
Band Reject Filter (Notch Filter)Ideal Characteristic of Band Reject Filter
LC R
+ VO -
+VI
-
Simple Reject Pass Filter
)()(
sC1sLRsC1sL
VV(s)G
I
0V
C)(1LjR
C)(1Lj
LO HI
Magnitude1
21Actual characteristic
Example 15
LC R
+VR
-
+VI=1|0 0
-
+ VL -+ VC -
Given the following circuit parameter values: L=159mH, C=159mF and R=10W. Demonstrate that this circuit can be used to produce a low-pass, high-pass, or band-pass filter.
Active FiltersDrawbacks of Passive Filters
1. Inability to generate a network with a gain greater than one since passive elements cannot add energy to signals
2. Inductors are generally expensive and occupy to much space
1. Active Filters are able to add energy to signals 2. Can construct inductors using resistors, capacitors and
operational amplifiers (Op-amps).
Advantages of Active Filters
Active filters
-
+
Z1
Z2
+VI
-
+VO
-
Inverting operational amplifier
Z1-+
+VI
-I
1
2 VZZ +
VO
-
+
-
Z1Z2
+
VI
-
+VO
-
Noninverting operational amplifier
+-
+VI
-
I1
2 VZZ )1( +
VO
-
Filter characteristics are determined by the choice of Z1 and Z2.
Example 16Find the voltage gain Vo/VI for the following circuit
R2
-+
+VI
-
+VO
-
CR1
Example 17 (Difference Amplifier)R2
-++
VI
-
+VO
-
C1
R1
R3
C2
Find the voltage gain Vo/VI for the following circuit
Example 18Find the input impedance for the following circuit
R2
-+
+V-
CR1
Zin
Inductor ReplacementAntoniou Inductance Circuit
- +
-+
R1 R2
R3 C R4Zin
I
+V-
sL/RRRsCRIVZ 2431in
where 2431 /RRRCRL
Example 19
R5
- +
-+
+ VO -
+VI
-
R2
C1R1
R3
R4 C2
Find the transfer function Vo/VI for the circuit shown below and state what type of filter this transfer function represents
Example 20The network shown is a circuit model for a single stage tuned transistor amplifier. Find the transfer function Vo/VA, and the value of C so that the center frequency is 91.1 MHz.
R=25k L=1H C+
Vo-
0.004Vx+-VA
+Vx-
Example 21Design 20db attenuation at 22.05 KHz for the following two circuits R
C+V0
-
+-VI
a) Single pole low pass filter
+
-
R
C+-VI
R
C+V0
-b) Two-stage buffered low pass filter