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Insturmentation
Hayt et. al.
Chapters 1 - 5
Amplifiers
Kirchhoff's Voltage Law:
- based on the principle of conservation of energy
- the algebraic sum of the changes of voltage (potential differences) encounte
transversal of a circuit loop must be zero
- the potential difference of a circuit element used is the voltage change which
would experience if passed through the element under consideration in the dir
transversal chosen.
- for any g



 Vi
=0
V1 -V2 - V
Circuit Concepts
Kirchhoff's Voltage Law:
Simple example:
Given V1, V2, V3 and R1, R2, R3 and R4
Find i?
V1 - i R1 - V2 - i R2 - i R3 - i R4 + V3 = 0
V  V2  V3
i 1
R eq
Req = R1 + R2 + R3 + R4
Circuit Concepts
Kirchhoff's Voltage Law:
Example:
Circuit Concepts
- Vab the potential difference experienced by moving from point b to point a, can be
determined by moving from b to a along any circuit path and adding algebraically
the potential differences encountered across each circuit element along the path
Simple Example:
- moving from b to a through R we get:
Vb + iR = Va
Vab = Va – Vb
= iR =

R
Rr
-choosing the path through the
voltage source and r :
Vb +  - ir = Va
Vab =  – ir =  R  r
R
The actual voltage from the battery Vab is only equal to its ;
- if r = 0 (i.e. if it is an ideal voltage source) or
- if R = ∞ (i.e. no load is connected to it).
Circuit Concepts
Potential Difference:
Second Example:
Properties of Circuit Components:
Independent Voltage Sources:
Circuit Concepts
Kirchhoff's Current Law:
- based on the law of conservation of charge
- no source or sink of current exists at a node.
- the current entering any node must be equal to the current leaving that node
- a current which enters a node is positive a current leaving a node in negative
- at any junction (node) the algebraic sum of the currents must be zero
 in = 0
i1 - i2 + i3 + i4 - i5 = 0
Circuit Concepts
Kirchhoff's Current Law:
Simple Example :
Given i1, i2, i3 and R1, R2, R3
Find V?
i1 
V
V
V
 i2 

 i3  0
R1
R 2 R3
V = Req (i1 + i2 - i3)
1
1
1
1



R eq R1 R 2 R 3
Circuit Concepts
Kirchhoff's Current Law:
Example:
Circuit Concepts
Equivalence of Sources:
- In many circuit analysis situations it is important to convert power sources from one type (voltage or
current ) to another (current or voltage)
- to convert (replace) the voltage source to (with) an equivalent current source
- equivalence is defined as having identical terminal characteristics.
Properties of Circuit Components:
Independent Voltage Sources:
Properties of Circuit Components:
Independent Current Sources:
Circuit Concepts
Equivalence of Sources:
- for the sources to be equivalent, for any specific current,
they must produce the same voltage across their terminals (and vice versa).
- voltage and current sources are linear devices
therefore match outputs for two extreme cases
- consider an open circuit condition (load impedance infinite)
Use KVL
Use KCL
Circuit Concepts
Equivalence of Sources:
-
now consider a short circuit condition (load impedance zero):
Use KVL
Use KCL
Circuit Concepts
Equivalence of Sources:
- these two considerations lead to:
Io =
Eo
R series
=
Rparallel = Rseries =
Vs
Rs
Rs
and similarly to:
Eo = Io Rparallel
=
Rseries = Rparallel =
Rs is
Rs
Circuit Concepts
Equivalent Circuit Components:
Equivalent Resistance:
Resistors in series:
Resistors in parallel:
for two resistors:
Req = R1R2 / (R1 + R2 )
Circuit Concepts
Equivalent Circuit Components:
Equivalent Capacitance:
Capacitors in series:
1/Ceq = 1/C1 + 1/C2 + ... + 1/Cn
for two capacitors:
Ceq = C1C2 / (C1 + C2 )
Capacitors in parallel:
Ceq = C1 + C2 + ... + Cn
Circuit Concepts
Equivalent Circuit Components:
Equivalent Inductance:
Inductors in series:
Leq = L1 + L2 + ... + Ln
Inductors in parallel:
1/Leq = 1/L1 + 1/L2 + ... + 1/Ln
for two inductors:
Leq = L1L2 / (L1 + L2 )
Circuit Concepts
Thevinen and Norton Equivalent Circuits:
- when only external behaviour is important
- equivalent circuits reduce complex circuits to simple complimentary forms
- Thevenin's theorem (for resistive linear circuits):
any circuit can be replaced by an equivalent ideal voltage source
and an equivalent series resistance
- Norton's theorem (for resistive linear circuits):
any circuit can be replaced by an equivalent ideal current source and
an equivalent parallel resistance
Circuit Concepts
Thevinen and Norton Equivalent Circuits:
- duality of Thevenin and Norton predicted by the equivalence of sources developed earlier
- vt is referred to as the Thevenin voltage
- in is referred to as the Norton current
- Req is called the equivalent resistance
- in = vt / Req
vt = in Req
- internal information lost
- used to theoretically replace static circuit sections during analysis and design
Circuit Concepts
Thevinen and Norton Equivalent Circuits:
For circuits containing independent sources only:
- find voc and isc
vt = voc
in = isc
Req = voc / isc
- Req is also the equivalent circuit resistance looking into the reference
terminal with all the independent sources removed
Circuit Concepts
Thevinen and Norton Equivalent Circuits:
Example:
Properties of Circuit Components:
Thevinen and Norton Equivalent Circuits:
6 kW
→
40 V
Circuit Concepts
Thevinen and Norton Equivalent Circuits:
For circuits containing dependent sources :
- find voc and isc
- remove independent sources, then
- apply an external voltage Vx and measure the current ix it supplies
to the circuit
- Req = Vx / ix
Circuit Concepts
Voltage Divider:
- very useful for circuit analysis simplification
Vi
Ro

Vi
Vo = R o i  R o
R o  R1
R o  R1
- R1 and Ro can be the equivalent series resistance
of any combination of resistors
Ro
Ro
R1
Vo 

Ro
R o  R1
1
R1
- goes to 1 as
Ro
increases
R1
Circuit Concepts
Voltage Divider:
Example:
Circuit Concepts
Current Divider:
- very useful for circuit analysis simplification
R R
i i R eq
Vo
R1
1 o 1 
io 

 ii
ii
R  R Ro
Ro
Ro
R

R
1
o
1
o
- R1 and Rocan be the equivalent parallel resistance
of any combination of resistors
Io
R1
1


R
Ii
R o  R1
1 o
R1
- goes to 0 as
Ro
increases
R1
Circuit Concepts
Current Divider:
Example:
Circuit Concepts
Circuit Analysis Simplification:
- using equivalent component values and
- Kirchoff's voltage and current laws and
- using voltage and current division concepts
- circuit analysis can be greatly simplified
- intuitive step by step method
Example:
Circuit Concepts
Circuit Analysis Simplification:
Circuit Concepts
Circuit Analysis Simplification:
Circuit Concepts
Circuit Analysis Simplification:
Circuit Concepts
Superposition:
- in a linear circuit, any voltage or current circuit response can be determined
by considering each source separately
- algebraically add the individual responses
- sources not being considered are removed from the circuit
- to remove ideal voltage source replace it with a short circuit
- to remove an ideal current source replace it with an open circuit
- assumed internal source resistances remain in the circuit at all times
- circuit analysis simplification using superposition
Example:
Use superposition to find the Thevinen and Norton equivalent circuits.
Circuit Concepts
Superposition:
Example:
Voc’ = 30(12/18) = 20 V
Circuit Concepts
Superposition:
Example:
Voc’’ = 40 V
Circuit Concepts
Superposition:
Example:
Voc’’’ = -7x15 = -105
Circuit Concepts
Superposition:
Example:
Vt = Voc’ + Voc’’ + Voc’’’ = 20+40-105 = -45
Req = (6//12)+11+5 = 4+16 = 20
Circuit Concepts
Transfer Functions:
- the ratio of a circuit’s output value to a circuit’s input value
is termed a circuit transfer function
- the ratio of output to input voltage
is a voltage transfer function
TFv 
Vo
V
 2
Vi
V1
Io
I
 2
Ii
I1
- a current transfer function is
TFI 
- a power transfer function is
P
VI
VI
TFP  o  o o  2 2
Pi
Vi I i
V1I1
- useful for considering circuit sections as block elements
- assumes no loading effects
- function of frequency
Circuit Concepts
Input Resistance:
- resistance looking into the input terminals of a circuit section
- important in determining the loading effects on previous section
- should be as large (small) as possible for voltage (current) transfer
I→
+
V
Rin
Assume RL = ∞
Rin = V/I
Circuit Concepts
Output Resistance:
- resistance looking into the output terminals of a circuit section
- important in determining the loading effects on next section
- should be as small (large) as possible for voltage (current) transfer
←I
+
V
Assume RS = 0
RO = V/I
Circuit Concepts
Loading Effects:
- how do previous and next circuit sections affect
the actual transfer characteristics of a specific circuit section?
- dependent on ratios of output to input and input to output resistances
respectively
Consider the voltage transfer characteristics of the following circuits:
Ideal Case
V
Circuit Concepts
Loading Effects:
Ideal Case
V
Source Resistance
V
Circuit Concepts
Loading Effects:
Ideal Case
V
Load Resistance
V
Circuit Concepts
Loading Effects:
Ideal Case
V
Source and Load Resistance
V
Circuit Concepts
Measuring Currents and Voltages (Ammeters and Voltmeters ):
Ammeters are used to measure the current in a
particular part of a circuit.
Ammeters must be inserted into the circuit so that the
current can be measured.
Ammeter resistance RA should be as small as possible.
(Ideal RA = 0. Why?)
Voltmeters measure potential differences
between two points in a circuit.
Voltmeters are connected to the circuit at the two points of
interest without breaking the circuit.
Voltmeter resistance RV should be as large as possible.
(Ideal RV =  Why?)
Circuit Concepts
Ammeter and Voltmeter Construction:
Circuit Concepts