Circuit Structure And Diagram

Follow the guidelines (MAKE SURE TO USE MULTILISM) and the answers should be filled in to the cw booklet and for the oscillators use the numbers that I will attach in the image below

S5.1 Circuit Diagram
S5.2
Output impedance of the function generator, RS = 50Ω
Input impedance of the oscilloscope, RL = 1 MΩ
S5.3
Cut off frequency, fc = 10 kHz
S5.4
The cut-off frequency of the filter in question is 10 kHz and is given by the equation
f c= 1
2 π Req C
Req takes into account both the source and load resistance. If the value of capacitance (C) is
chosen to be C=0.01 μF, the resistance is approximately
Req = 1
2 π f c C ≅ 1.591 k Ω
Thus,
Capacitor, C =0.33 F
Resistance, R ≅ 1.591 kΩ

S5.5
Find an attached file of the RC circuit as built in multism
S5.6
The preferred standard manufactured values for the capacitor and the resistor are:
Capacitor, C = (10 + 000 +/- 5%) pF = 0.01∓5 % μ F
Resistance, R ≅ 16 x 102 ∓5 % Ω=1.6 ∓5 %k Ω
S5.7
Frequency,f
(kHz)
I/P
Volts
Vi
O/P Volts
Vo
Gain G 20 log10 ( V O
V i ) =GdB Phase
∅ O
0.1 0.2 0.2 1 0 -0.85
0.5 0.8 0.8 1 0 -2.97
0.8 1 1 1 0 -4.74
1 1.1 1.1 1 0 -5.94
2 1.5 1.5 1 0 -11.7
4 1.8 1.66 0.9222 -0.70329 -22.5
6 2.2 1.87 0.85 -1.41162 -31.9
8 2.4 1.84 0.7667 -2.30787 -39.6
10 3 2.08 0.6933 -3.18116 -46
20 3.2 1.39 0.4344 -7.2427 -64.2
40 4 0.94 0.235 -12.5786 -76.4
50 4.5 0.85 0.1889 -14.4759 -79
90 5 0.54 0.108 -19.3315 -83.9

S5.8 & S5.9

S5.10 Graph Measurements:
S5.10.1 The cut-off frequency is 10 kHz (as indicated on the graphical plots)
f c
actual=10 kHz
S5.10.2 The per decade slope of the gain magnitude is -20dB/decade (as indicated on the
graphical plots)
Sactual
decade =−20 dB /decade
S5.10.3 The per octave slope of the gain magnitude is -6dB/octave (as indicated on the graphical
plots)
Sactual
octave =−6 dB /octave
S5.10.4 The pass band phase is between 0 to -45o (as indicated on the graphical plots). In
radians,
−1
4 π ≤ θactual
pass ≤ 0
S5.10.5 The stop band phase is between -45o to -90o (as indicated on the graphical plots). In
radians,
−π
2 ≤θactual
stop ≤− 1
4 π
S5.10.6 The phase in radians at the cut-off frequency is obtained from the graph as -45o. In
radians,
θactual
cutoff =−45
180 π =−1
4 π rads
S5.10.7 The cut-off frequency obtained from the graph is 10 kHz and that in design specification
is 10 kHz. Therefore the fractional difference is zero i.e.
Ef =0

S5.11 Derivation of the transfer function for the filter circuit
From the equivalent circuit diagram in S.5.1, the transfer function is derived as below
The output voltage is obtained by voltage divider rule as,
V out =
( RL
sC
RL + 1
sC )
(Rs + R+
RL
sC
RL+ 1
sC ) V ¿
Simplifying the expression,
V out =
( RL
1+sC RL )
( Rs + R+ RL
1+ sC RL ) V ¿
On re-arranging, the transfer function for the above filter reduces to,
V out
V ¿
= RL
Rs + R+RL+ sC RL( Rs + R)
But the output impedance of the function generator, RS = 50Ω and the input impedance of the
oscilloscope, RL = 1 MΩ. Substituting these values in the above expression, the transfer function
becomes,
V out
V ¿
= (1 x 106)
50+ R+(1 x 106 )+(1 x 106)sC (50+ R)
The calculated values of the capacitor (C) and the resistor (R) at the cut-off frequency are 0.01
μ F and 1.6 k Ω respectively. Replacing these values in the transfer function above, the equation
reduces to
H ( s )= V out
V ¿
= 106
1001650+ 16.5 s

The above transfer function is in the s-domain. To convert it the complex frequency domain, s is
replaced with jω. The result is
H ( jω ) = V out
V ¿
= 106
1001650+ j 16.5 ω
S5.12 Derivation of the two port representation of the circuit
The two port network is defined in terms of Z-parameters. For the given RC network, the Z-
parameters are obtained as follows:
Z11= V 1
I1 |(I ¿¿ 2=0)=Rs + R+ 1
jωC ¿
Z21= V 2
I 1 |( I¿¿ 2=0)= 1
jωC ¿
Z22= V 2
I2 |(I ¿¿ 1=0)= 1
jωC ¿
Z12= V 1
I 2 |(I ¿¿ 1=0)= 1
jωC ¿
For C=0.33 F∧R=1.591 kΩ , Rs=50 Ω
Z11=1.591 x 103 +50+ 1
j 2 π x 10 x 0.33 x 103 =1641− j 4.82 x 10−5 Ω
Z12= 1
j 2 π x 10 x 0.33 x 103 =− j 4.82 x 10−5 Ω
Z22= 1
j 2 π x 10 x 0.33 x 103 =− j 4.82 x 10−5 Ω
Z21= 1
j 2 π x 10 x 0.33 x 103 =− j 4.82 x 10−5 Ω
In matrix form, it is
[ Z ]= [ 1641− j 4.82 x 10−5 − j 4.82 x 10−5
− j 4.82 x 10−5 − j 4.82 x 10−5 ]

6. Passive LC Low Pass Filter
S6.1
S6.2
Cut-off frequency, fc = 15 kHz
S6.3
Component values of the components are obtained from the defining equations of the filter. The
cut-off frequency is given by the relation,
ωc= 2
√LC rad s−1
In hertz, the frequency is

f c= 1
π √ LC
If a capacitor value of 0.1 μ F is chosen, the inductor value is
L= ( 1
π f c √C )2
≅ 4.503 mH
The matching impedances Rs and RL are therefore obtained as
Rs=RL= √ L
C ≅ 212.21 Ω
Inductor, L = 4.503 mH
Capacitor, C = 0.1 μ F
S6.4
Find an attached file of the LC circuit as built in multism
S6.5
The standard preferred manufacturer’s values are;
L = (45 + 00 +/- 1%) μ H = 4.5 mH
C = (10 + 0000 +/- 5%) pF = 0.1∓ 5 % μ F

S6.6
Table of results
Frequency,f
(kHz)
I/P
Volts
Vi
O/P
Volts
Vo
Gain G 20 log10 ( V O
V i ) =GdB Phase
∅ O
0.1 0.2 0.1 0.5 -6.02 -1.31
0.5 0.8 0.4 0.5 -6.02 -3.83
1.0 1 0.5 0.5 -6.02 -7.7
1.5 1.1 0.55 0.5 -6.02 -11.5
2 1.5 0.75 0.5 -6.02 -15.4
4 1.8 0.9 0.5 -6.02 -31
8 2.5 1.24 0.496 -6.1 -64.8
10 2.7 1.3 0.48 -6.4 -84.4
20 3.1 0.6 0.193 -14.3 -173
30 3.2 0.2 0.062 -24.2 -209
40 3.8 0.1 0.026 -31.6 -226
50 4.5 0.06 0.013 -37.4 -235

90 13 0.03 0.0023 -52.7 -251

S6.7 & 6.8

S6.9 Graph Measurements:
S6.9.1 The cut-off frequency is 10 kHz (as indicated on the graphical plots)
f c
actual=16 kHz
S6.9.2 The per decade slope of the gain magnitude is -20dB/decade (as indicated on the
graphical plots)
Sactual
decade =−20 dB /decade
S6.9.3 The per octave slope of the gain magnitude is -6dB/octave (as indicated on the graphical
plots)
Sactual
octave =−6 dB /octave
S6.9.4 The pass band phase is between 0 to -135o i.e. angles greater than -135 degrees (as
indicated on the graphical plots). In radians,
−3
4 π ≤θactual
pass ≤0
S6.9.5 The stop band phase is between -135o to -360o i.e. angles less than 135 degrees (as
indicated on the graphical plots)
−2 π ≤ θactual
stop ≤− 3
4 π
S6.9.6 The phase in radians at the cut-off frequency is obtained from the graph as
θactual
cutoff =−135
180 π=−3
4 π rads
S6.9.7 The cut-off frequency obtained from the graph is 16 kHz and that in design specification
is 15 kHz. Therefore the fractional difference is
¿ |16−15|
15 ×100 ¿ 6.6667 %
Ef =6.6667 %

S6.10
Let the voltage across the capacitor be V1
By voltage divider rule, this voltage V1 is given as
V 1=
1
s C ∨¿ ( RL+s L'
)
1
s C ∨¿ ( RL+ s L' ) +Rs +s L'
V i
¿
( RL+ s L'
s C ( RL+ s L'+ 1
s C ) )
RL+s L'
s C ( RL +s L' + 1
s C ) + Rs + s L'
V i
Simplifying, we have
V 1= RL+s L'
RL+ s L'+ s C (RL+s L'+ 1
s C ) ( Rs +s L' )
V i
But the output voltage, by voltage divider rule, is
V o = RL
RL+ s L' V 1
¿ ( RL
RL+ s L' ) ( RL +s L'
RL +s L' + s C ( RL +s L' + 1
s C ) ( Rs+ s L'
) ) V i
¿ RL
RL+ s L' +s C ( RL+ s L' + 1
s C ) ( Rs+s L' )
V i

The transfer function of the circuit is defined as the ratio of the output voltage to the input
voltage i.e.
H ( s )= V o
V i
= RL
RL +s L' + s C (RL+s L' + 1
s C ) ( Rs+s L' )
Simplifying the above expression, the transfer function reduces to
H ( s )= V o
V i
= RL
L'2 Cs3 + ( RL L' C+ Rs L' C ) s2+ ( RL Rs C +2 L' ) s+ ( RL+ Rs )
Where,
L' = L
2
Replacing the variables with their equivalent values we have,
H ( s )= 212.21
5.069 x 10−13 s3+ 9.556 x 10−8 s2 +0.00901 s +424.42

S6.11
The two port network is defined in terms of Z-parameters. For the given LC network, the Z-
parameters are obtained as follows:
Z11= V 1
I 1 |(I ¿¿ 2=0)=Rs + jωL
2 + 1
jωC =Rs + j ( ωL
2 − 1
ωC ) ¿
Z21= V 2
I 1 |( I¿¿ 2=0)= 1
jωC ¿
Z22= V 2
I 2 |(I¿¿ 1=0)= jωL
2 + 1
jωC = j ( ωL
2 − 1
ωC ) ¿
Z21= V 1
I 2 |(I ¿¿ 1=0)= 1
jωC ¿
For C=0.1 μF∧L=4.503 mH , Rs=212.21 Ω
Z11=212.21+ j ( 2 π x 15 x 4.503
2 − 1
2 π x 15 x 0.1 x 10−3 )=212.21+ j 106.096 Ω
Z21= 1
j 2 π x 15 x 0.1 x 10−3 =− j106.103 Ω
Z22= j ( 2 π ×15 × 4.503
2 − 1
2 π x 15 x 0.1 x 10−3 )= j 106.096 Ω
Z12= 1
j 2 π x 15 x 0.1 x 10−3 =− j106.103 Ω
[ Z ]= [212.21+ j 106.096 − j106.103
− j106.103 j106.096 ] Ω

S7.1
The 6.667% fractional error in the cut-off frequency of the LC low pass filter could be as a result
of;
1. Round-off of the component values to the nearest equivalent manufactured values.
Design equations occasionally yield values of resistors, inductors and capacitors which
are not standard (Brown, 2019). This necessitates for round off of these values to the
nearest manufactured equivalents. Hence, this may contribute to erroneous results (Huang
et al., 2015)
2. Approximations in readings obtained from the manual plotted responses on the semi-log
paper (Uysal & Watkins, 2015)
S7.2
It is observed that the highest power of s in the denominators of transfer functions derived for the
RC low pass filter is one thus it is a first order low pass filter. Its first order nature is because
the circuitry consists of only two passive elements i.e. a capacitor and resistors. The oscilloscope
provides load resistance (RL) while the function generator offers resistance (Rs). Combined
effects of all these two resistances with the resistor making up the filter unit make up the overall
resistance of the circuit.
However, for LC low pass filter, the highest power of s in the characteristic equation is three
making it a three pole system or a third order system. This is because the circuitry is made up
of all the three passive components i.e. there exists a capacitor, an inductor and resistors. The
resistors are modeled from the function generator and oscilloscope and they are used for
impedance matching (Nassir, 2017)

The LC low pass filter gives a phase of -135o at the cut off frequency because of the inductor
involved while that of the RC low pass filter is -45o because only a capacitor is involved.
References
Brown, J. M. (2019). Chapter 2: Classical results of network synthesis. Passive Network
Synthesis: An Approach to Classification, 5-18.
Huang, C., Chen, C., Zhou, M. J., & Wu, J. H. (2015). An active-RC reconfigurable lowpass-
polyphase biquad filter for wireless receiver. Analog Integrated Circuits and Signal
Processing, 84(1), 97-105.
Nassir, H. S. (2017). Basic analysis of resistive circuits. Electric Circuits and Signals, 83-112.
Uysal, S., & Watkins, J. (2015). Novel microstrip low-pass/high-pass directional couplers/filters.
IEEE Proceedings on Southeastcon.