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Passive components and circuits - CCP
Lecture 4
1
Content
 Basics
Circuit characteristics and parameters
Logarithmical representation
 Passive circuit elements
Resistor
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Circuit characteristics
The transmittances of the electronic circuits are
described by the linear or non-linear functions (for linear
and non-linear circuits).
 The graphical representation of those functions is called
electrical characteristic. In the case of linear circuits, they
have linear segments representation.



By the graphical representation of a mathematical function which
approximate the circuit operation  theoretical characteristics .
(If those functions are approximated for simplification, they are
called ideal characteristics).
By experimental measurements  experimental characteristics.
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Errors in determination of electrical
characteristics

In the case of theoretical characteristics:



The incapacity of the mathematical model to consider all factors
which act over the circuit (sometimes, a simple model is used).
Approximations in solving of mathematical model (sometimes, the
mathematical model is too difficult to be solved).
In the case of experimental measurements:


The incapacity of total separation of interested quantities from
others (noise).
Errors introduced by the measurements instruments and
measurements method .
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Family of characteristics



In most cases, an electric quantity is not dependent of a
single variable (electric or non-electric variable) .
The dependency of a characteristic by the second (or even
third) variable is represented as a family of characteristics
in plane (or in space) .
In electronics circuit behavior, one of most important
non-electric variable is the temperature.
v  v(i, p) 
v  v(i ) p  cst. one characteristic
v  v(i ) p  p1, p 2,... family of characteristics
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Family of characteristics - example
a) Family of linear characteristics
b) Family of non-linear characteristics
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Circuit parameters

The coordinates of some points from the electric
characteristic are called circuit parameters.

The parameters are chosen so that they emphasize the
significant points on the characteristics (maximum and
minimum, modulation points, etc.).

If those points are referred to all the characteristics from a
family, suggesting this way operation limits, the
parameters are called limits parameters or limits values.
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Electronics data books

They represent a collection of characteristics and
parameters, made by the manufacturers, which describe
the electronics components behavior.

Usually, the data book characteristics and parameters
have standardized signification => the signification is the
same for all manufacturers.
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Classification of circuit characteristics and
parameters

Generally, the following characteristics and
parameters regarding operation regimes, can be
found in all data books:





Static (or DC) characteristics/parameters
Dynamics (or AC ) characteristics/parameters
Transient regime characteristics/parameters
Environment characteristics/parameters
Dissipated power characteristics/parameters
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DC Operation Regime

In this operation regime, the electric quantities are
not time dependent (during the observation period).

These parameters reflect significant points of the
characteristics or absolute limit values that cannot
be exceeded.
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Alternating current (AC) operation regime
• The electrical stimulus applied
to the circuit (component) are
usually sinusoidal.
• In this case, the ratios (the
transmittances) of output to
input signals are called Gains
(Amplifications) if they are
greater than the unit value or
Attenuations if they are smaller
than the unit value.
• For an Amplifier Block the
power amplification (power
gain) Ap is defined as the ratio
of output (load) power to input
power.
vo
Av :
non  dimensiona l
vi
Ai :
io
non  dimensiona l
ii
Av / i
vo
: []
ii
Ai / v
io
: [S]
vi
Ap :
PO
non  dimensiona l
PI
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AC operation – frequency representation



In alternative current, the transmittances are dependant by the
signal frequency f, or the signal pulsation,  =2f.
The frequency dependency of transmittances is represented by
the frequency characteristics.
For frequency representation, the sinusoidal quantities are
described like vectors:
v(t )  2V cos(t   )  V ( j )  V  e jt  j
http://mathworld.wolfram.com/Phasor.html
http://www.clarkson.edu/~svoboda/eta/phasors/MatchPhasors10.html
http://www.physics.udel.edu/~watson/phys208/phasor-animation.html
http://www-ccrma.stanford.edu/~jos/filters/Phasor_Notation.html
http://www.usna.edu/MathDept/CDP/ComplexNum/Module_5/PhasorForm.htm
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Transfer function


The ratio between two electric quantities represented by
vectors is called transfer function.
The transfer function is a complex quantity, characterized
by the modulus and phase. Consequently, the frequency
representation has two components:


Modulus-frequency characteristic (the amplitude ratio)
Phase-frequency characteristic
H ( j ) 
vo ( j )
 Re[ H ( j )]  jIm[ H ( j )]
v i ( j )
H ( j )  Re 2 [ H ( j )]  Im 2 [ H ( j )]
 H ( jt )  arctg
Im[ H ( j )]
Re[ H ( j )]
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The transient regime behavior

The transient regime is an operation regime at a signal
variation. The signal variation can be:



From a static value to another static value;
From a frequency value to another frequency value;
In data-books this regime is described by time values
such as: rising time, falling time, propagation time, etc.
For example, if a power supply in a circuit is switched on
there may be a surge, possibly with oscillations, before a
steady flow of current is established. Circuits exhibit transients
when they contain components that can store energy, such as
capacitors and inductors.
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Environment effect over the circuits
 The environment acts over the electrical circuits through
different factors. In the majority of situations, those factors
have a disturbing effect on the circuit.
 The main environment factor affecting the electronic
circuits is the temperature. Changes in the temperature
affect the internal physical processes of the component
(dimensions, chaotic thermal motion), changing its
electrical characteristics.
 The temperature coefficients reflect the variation of
different parameters:
p(T 1)  p(T 2)
 
T1  T 2
p
T
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The dissipated power
 The electrical phenomena taking place in electronic
devices and circuits are constantly affected by Joule effect
(heat dissipation). The heat accumulation in the circuit
structure will increase its temperature.
 Therefore, in data books are presented parameters and
characteristics that restrict the dissipated power in the
circuit under particular environment conditions.
 Not all the parameters specifying limit values are
connected with the dissipated power; there are also other
destructive phenomena.
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D.C. dissipated power
 Usually, the power is dissipated by a circuit
regardless of the functioning regime: d.c., a.c.,
transient regime.
 Applying a constant voltage VR to a
resistance R in d.c., the current will be:
v
R
A
VR
IR 
R
B
iR
R
 The power dissipated by the resistance will be:
VR2
P  VR  I R 
 R  I R2
R
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A.C. dissipated power
 Applying a sinusoidal voltage on a resistance:
vR (t )  Vmax sin( t )
v
R
A
B
 The current through the resistance will be:
vR (t ) Vmax
iR (t ) 

sin( t )
R
R
iR
R
 The instantaneous power dissipated by the resistance is:
2
Vmax
p(t )  vR (t )  iR (t ) 
sin 2 (t )
R
 The average power dissipated:
T
Pmed
T
T
2
2
Vef
1
1 Vmax
1 1
2
2
2
2
  p(t )dt  
sin (t )dt 
V
sin
(

t
)
dt


R

I
max
ef
T0
T0 R
R T 0
R
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Tolerances of electrical parameters

In data books, the parameters values indicated by the
manufacturers are target values (nominal values).

Due to a different number of factors (technological factors,
reduced costs etc) the real values of the parameters are
near to the rated (target) values. By the selective
measurements, the manufacturers offers only those
components which have the parameters in the some
specific limits around the rated value.

The maximum accepted difference between real and rated
values is called tolerance.
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The tolerance expression



The tolerance can be evaluate pnom  [ pmin , pmax ]
as absolute tolerance,
specifying the minimum and
maximum values of a
p  pnom
t p   max
parameter.
pnom
The percentage tolerance
reflects the maximum
difference from a rated value. p  [ pnom (1  t p ), pnom (1  t p )]
Knowing the percentage
tolerance makes it easy to
determine the absolute
tolerance.
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Representation to a logarithmic scale

By logarithmic scale representation, the x variable
representation is replaced by the lgx (or lnx)
representation.

The logarithmic representation can be made only for
positive values of a variable. In order to achieve this
condition, the modulus representation of a variable is
used.


By logarithm, the 0 value of axes became -.
The old smaller than unity values became negative, and
greater than unity values became positive.
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Advantages of logarithmic technique
 Allows a compression of representation domain.
 Allows the obtainment of a linear characteristic.
 Convert the multiplying/dividing operations in
added/subtracting operation  these operations can
be graphical performed.
a(b  c)
lg
 lg a  lg(b  c)  lg( d  e)
(d  e)
22 /37
Linear representation - example


The representation
of the following
complex functions.
1
A1v 
1  j  f 10 6
j  f 10 6
A2 v 
1  j  f 10 6
Representation of
a 100Hz-100MHz
frequency domain.
23 /37
Logarithmic representation- example

The representation
of the same values
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Characterization of electrical quantities by
logarithmic ratio


The transfer ratio represents the
logarithms of a non-dimensional
ratio (regarding the input and
output).
Av [dB]  20 lg Av
Ai [dB]  20 lg Ai
Ap [dB]  10 lg Ap
The transfer ratios are used to
characterize the system transfer
properties (ex: amplification, line
attenuation etc).
25 /37
The double logarithmic representation example



The previous complex functions are used;
The vertical axes is represented in dB (logarithmic scale);
It can be observed the linear representation of these two
functions.
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Bode diagrams representation



The Bode diagram method assumes the replacement of
the double logarithmic representation with asymptotes
and tangents on the graphics.
We obtain a graphical representation only with straight
lines.
This type of representation allows an easier additional
graphical operation.
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Bode diagrams representation - example

In this figure we added the Bode diagrams for the
previous two functions:

Green for |A1v|;
Red for |A2v|;
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Additional operation for Bode diagrams example

In the second figure
we shown the
amplification (with
black- at the
logarithmic scale,
and with blue- by
the Bode diagrams)
Av [dB]  20 lg  A1v  A2v  
 20 lg  A1v   20 lg  A2v  
 A1v [dB]  A2v [dB]
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Example 1 – Using Bode diagrams
A1v 
A2 v 
1
1  j  f 104
j  f 108
1  j  f 108
30 /37
Example 2 – Using Bode diagrams
A1v 
A2 v 
1
1  j  f 108
j  f 104
1  j  f 104
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Signals levels
VdB  20 lg


Absolute signal level
report the system signal
values to a fixed
reference value.
The relative level
signal report the
analyzed signal to an
unknown value signal.
Vx
V0
[dBμV ]
 Voltage level-the reference
value is V0=1V
I dB  20 lg
Ix
I0
[dBμA ]
Current level-the reference
value is I0=1A
PdB  10 lg
Px
P0
[dB pW ]
Power level- the
reference value is P0=1pW
32 /37
Absolute levels in dB


Observation 1 - knowing the
absolute level makes it very easy to
reconstruct the signal value:
Vvolt  10
VdB
20
1μV
Observation 2- if the resistance Rx, on which the signal is
measured, is equal with the resistance R0, on which the
reference signal is measured, then the dB value of the power
level is equal with the voltage and current levels
Vx2
2
 Vx 
Px
Rx
Vx


PdB  10 lg  10 lg 2  10 lg    20 lg  VdB ; Rx  R0
V0
P0
V0
 V0 
R0
33 /37
Absolute levels in dB

Observation 3 – if the power level in dB and the
resistance value are known, the absolute voltage and
current levels can be calculated in the following way:
VdB  PdB  10lg( R /1) respectively
IdB  PdB  10lg( R /1)

Example – the following levels in dB have the corresponding
values on linear scale:
3dB21/2
6dB2;
20dB10;
120dB106
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Absolute levels in Np


If the decimal logarithms used
for dB representation are
replaced by natural logarithm,
the levels will be evaluated in
Nepers (Np).
The relation between Np and
dB and vice-versa are:
1Np8,686dB, 1dB0.115Np
Vx
VNp  ln [ Np]
V0
Ix
I Np  ln [ Np]
I0
1 Px
PNp  ln [ Np]
2 P0
35 /37
Operations with signal levels - example

On a 50 resistor a VdB=120 dBV level is measured. What is the
absolute power level? And the level of current through the resistor?

Method 1
Vvolt  10
VdB
20
1μV  10
120
20

Method 2
10 6  1V
V2 1
P
  20mW
R 50
V 1
I    20mA
I 50
20 10 3
9
PdB  10 lg

10
lg
20

10
 103dB
12
10
20 10 3
3
I dB  20 lg

20
lg
20

10
 86dB
6
10




PdB  VdB  10 lg( R / 1) 
50
 120  17  103dB
1
 PdB  10 lg( R / 1) 
 120  10 lg
I dB
50
 103  10 lg
 103  17  86dB
1
36 /37
Passive circuits elements

Resistance as circuit element – Homework


Ohm’s Law
Power dissipated on a resistor





In direct current
In alternative current
Series and parallel connections
How resistance can limit the current?
How resistance can limit the voltage?
37 /37