Hayt et. al. Chapters 1 - 5
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Transcript of Hayt et. al. Chapters 1 - 5
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Insturmentation Amplifiers
Kirchhoff's Voltage Law:
- based on the principle of conservation of energy
- the algebraic sum of the changes of voltage (potential differences) encountered in a complete transversal of a circuit loop must be zero
- the potential difference of a circuit element used is the voltage change which a positive charge would experience if passed through the element under consideration in the direction of transversal chosen.
- for any given loop:
Vi = 0
V1 -V2 - V3 + V4 -V5 = 0
Hayt et. al.Chapters 1 - 5
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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
iV V V
Req
1 2 3
Req = R1 + R2 + R3 + R4
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Circuit Concepts
Kirchhoff's Voltage Law:
Example:
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- 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 =
Circuit Concepts
R
R r
R
R r
-choosing the path through the voltage source and r :
Vb + - ir = Va Vab = – ir =
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).
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Circuit Concepts
Potential Difference:
Second Example:
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Properties of Circuit Components:
Independent Voltage Sources:
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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
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Circuit Concepts
Kirchhoff's Current Law:
S i m p l e E x a m p l e : G i v e n i 1 , i 2 , i 3 a n d R 1 , R 2 , R 3 F i n d V ?
iV
Ri
V
R
V
Ri 01
12
2 33
V = R e q ( i 1 + i 2 - i 3 )
1 1 1 1
1 2 3R R R Re q
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Circuit Concepts
Example:
Kirchhoff's Current Law:
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Circuit Concepts
Equivalence of Sources:
- In many circuit analysis situations it is important to convert power sources from one type (voltage orcurrent ) 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.
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Properties of Circuit Components:
Independent Voltage Sources:
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Properties of Circuit Components:
Independent Current Sources:
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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
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Circuit Concepts
Equivalence of Sources:
- now consider a short circuit condition (load impedance zero):
Use KVL Use KCL
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Circuit Concepts
Equivalence of Sources:
-these two considerations lead to:
Io =o
series
ER
= s
s
VR
Rparallel = Rseries = Rs
and similarly to:
Eo = Io Rparallel = Rs is
Rseries = Rparallel= Rs
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Circuit Concepts
Equivalent Circuit Components:
Equivalent Resistance:
Resistors in series:
Resistors in parallel:
for two resistors:
Req = R1R2 / (R1 + R2 )
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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
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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 )
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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 sourceand 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
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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
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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
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Circuit Concepts
Thevinen and Norton Equivalent Circuits:
Example:
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Properties of Circuit Components:
Thevinen and Norton Equivalent Circuits:
40 V
6 k
→
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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
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Circuit Concepts
Voltage Divider:
- very useful for circuit analysis simplification
Vo = R i RV
R R
R
R RVo o
i
o
o
oi
1 1
- R1 and Ro can be the equivalent series resistance of any combination of resistors
VR
R R
R
RR
R
oo
o
o
o
1
1
11
- goes to 1 as R
Ro
1 increases
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Circuit Concepts
Voltage Divider:
Example:
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Circuit Concepts
C u r r e n t D i v i d e r :
- v e r y u s e f u l f o r c i r c u i t a n a l y s i s s i m p l i f i c a t i o n
iV
R
i R
Ri
R R
R R RR
R Rio
o
o
i e q
oi
o
oo o
i
1
1
1
1
1
- R 1 a n d R o c a n b e t h e e q u i v a l e n t p a r a l l e l r e s i s t a n c eo f a n y c o m b i n a t i o n o f r e s i s t o r s
I
I
R
R R R
R
o
i o o
1
1
1
1
1
- goes to 0 as R
Ro
1 increases
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Circuit Concepts
Current Divider:
Example:
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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:
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Circuit Concepts
Circuit Analysis Simplification:
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Circuit Concepts
Circuit Analysis Simplification:
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Circuit Concepts
Circuit Analysis Simplification:
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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.
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Circuit Concepts
Superposition:
Example:
Voc’ = 30(12/18) = 20 V
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Circuit Concepts
Superposition:
Example:
Voc’’ = 40 V
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Circuit Concepts
Superposition:
Example:
Voc’’’ = -7x15 = -105
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Circuit Concepts
Superposition:
Example:
Vt = Voc’ + Voc’’ + Voc’’’ = 20+40-105 = -45
Req = (6//12)+11+5 = 4+16 = 20
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Circuit Concepts
T r a n s f e r F u n c t i o n s :
- t h e r a t i o o f a c i r c u i t ’ s o u t p u t v a l u e t o a c i r c u i t ’ s i n p u t v a l u ei s t e r m e d a c i r c u i t t r a n s f e r f u n c t i o n
- t h e r a t i o o f o u t p u t t o i n p u t v o l t a g ei s a v o l t a g e t r a n s f e r f u n c t i o n
T FV
V
V
Vvo
i 2
1
- a c u r r e n t t r a n s f e r f u n c t i o n i s T FI
I
I
IIo
i 2
1
- a p o w e r t r a n s f e r f u n c t i o n i s T FPP
V IV I
V IV IP
o
i
o o
i i
2 2
1 1
- u s e f u l f o r c o n s i d e r i n g c i r c u i t s e c t i o n s a s b l o c k e l e m e n t s
- a s s u m e s n o l o a d i n g e f f e c t s - f u n c t i o n o f f r e q u e n c y
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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
Assume RL = ∞
I →
V
+
-
Rin = V/I
Rin
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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
Assume RS = 0
←I
V
+
-
RO = V/I
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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
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Circuit Concepts
Loading Effects:
Source Resistance
Ideal Case
V
V
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Circuit Concepts
Loading Effects:
Load Resistance
Ideal Case
V
V
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Circuit Concepts
Loading Effects:
Source and Load Resistance
Ideal Case
V
V
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Circuit Concepts
M easuring Currents and Voltages (Am m eters and Voltm eters ):
Ammeters are used to measure the current in aparticular part of a circuit.
Ammeters must be inserted into the circuit so that thecurrent can be measured.
Ammeter resistance R A should be as small as possible.(Ideal R A = 0. W hy?)
Voltmeters measure potential differencesbetween two points in a circuit.
Voltmeters are connected to the circuit at the two points ofinterest without breaking the circuit.
Voltmeter resistance R V should be as large as possible. (Ideal R V = W hy?)
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Circuit Concepts
Ammeter and Voltmeter Construction:
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Circuit Concepts