Download Loop and Nodal Analysis and Op Amps

Circuits II
EE221
Unit 6
Instructor: Kevin D. Donohue
Active Filters, Connections of Filters, and
Midterm Project
Load Effects


If the output of the filter is not properly buffered, then for
different loads, the frequency characteristics of the filter will
change.
Example: Both low-pass filters below have fc = 1 kHz and GDC = 1
(or 0 dB).
1 k
1 k
+
vi(t)


1
2
F
+
vo(t)
-
vi(t)
1
2
-
F
10 k
+
vo(t)
-
Find: fc and GDC when a 100  load is placed across the output vo(t).
Result: fc = 11 kHz and GDC = 1/11 (or -21 dB) for the passive
circuit, while fc and GDC remain unchanged for the active circuit.
Filter Order


The order of a filter is the order of its transfer function
(highest power of s in the denominator). If the filter
order is increased, a sharper transition between the
stopband and passband of the filter is possible.
Example: For the two low-pass filters, determine circuit
parameters such that fc is 100 Hz, and GDC is 3  2  1.586
Plot transfer function magnitudes to observe the
transition near fc.
C
R
vi(t)
C
Rf
+
vo(t)
-
R
R
vi(t)
C
(K-1)Rf
R1
Rf
+
vo(t)
-
Transfer Function Results
First Order
1
Rf
R1
Hˆ ( s ) 
s
1
 1 


RC


GDC = 1+Rf / R1
fc = 1 / (2RC)
Second Order
Hˆ ( s) 
K
s
s2
1  (3  K )

1

  1 2

 

RC

  RC 
For GDC = K = 3  2  1.586 , fc = 1 / (2RC)
Formula not valid for any other value of K.
Value of K was contrived so the cutoff
would come out this way. For a general K
value fc =  / (2RC) , where


 7  6K  K   4
2 2
 7  6K  K 2 
2
Plot Comparison of Filter Order Effect
first order (-), second order (---)
1.4
1.2
3dB cut-off
gain
1
0.8
0.6
0.4
0.2
w = [0:1024]*2*pi;
p = j*w;
0
100
200
300
400
500
600
700
800
Hz
h1 = (3-sqrt(2)) ./ (1 + (p / (2*pi*100)));
h2 = (3-sqrt(2)) ./ (1 + (sqrt(2)*p / (2*pi*100)) + ( p / (2*pi*100)) .^2);
plot(w/(2*pi),abs(h1),'b-', w/(2*pi), abs(h2), 'b:')
title('first order (-), second order (---)'); xlabel('Hz'); ylabel('gain')
900
1000
SPICE Transfer Function Analysis


The simulation option for “.AC Frequency Sweep” with plot
transfer function for simulated circuit. The frequency range
(in Hz) must be selected along with plot parameters such as
log or linear scales. Both the phase and magnitude can be
plotted if requested. The input can be a voltage or current
source with amplitude of 1 and phase 0.
Selection of the frequency range is critical. If a range is
selected in an asymptotic region (missing the dynamic details
of the transfer function) the plot will be misleading. The
range can be determined by looking at large scale plots and
making adjustments. Selecting a large range on a log scale
may make it easier to identify the frequency range where
change is happening, and then a smaller range can be selected
to better show the details of the plot.
Spice Example:


Design the second order LPF so
that it has a cutoff at 3.5 kHz
with a gain of 3  2  1.586
Verify the design with a SPICE
R
simulation.
The key design equations
become: K  3  2
3500  (2 ) 
vi(t)
1
RC
So let C = 0.01F and Rf = 10k.
This implies R = 4.55k and
(K-1)Rf = 5.86k
C
R
C
(K-1)Rf
Rf
+
vo(t)
-
SPICE Results (Plot range 10 to 10 MHz)

The magnitude plot in dB. Crosshair marker near 3 dB
cutoff (What is happening near 100kHz)?
TFLPF2ex-Small Signal AC-6-Graph
100.000
10.000
1.000k
10.000k
100.000k
0.0
-20.000
-40.000
FREQ
3.447k
87.325
D(PH_DEG(v (IVm1)))
DB(v (IVm1))
1.116
D(FREQ)
0.0
Frequency (Hz)
1.000Meg
10.000Meg
SPICE Results (Plot range 10 to 10 MHz)

The phase plot in degrees. Crosshair marker near 3 dB
cutoff (What is happening near 100kHz)?
Frequency (Hz)
TFLPF2ex-Small Signal AC-6-Graph
10.000
100.000
1.000k
10.000k
100.000k
1.000Meg
-100.000
-200.000
-300.000
FREQ
3.548k
D(D B(v (IVm1)))
-1.000
PH_DEG(v (IVm1))
-91.852
D(FREQ)
0.0
10.000Meg