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

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

Circuit Concepts

Kirchhoff's Voltage Law:

Example:

- 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).

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:

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

Circuit Concepts

Example:

Kirchhoff's Current Law:

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.

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 =o

series

ER

= s

s

VR

Rparallel = Rseries = Rs

and similarly to:

Eo = Io Rparallel = Rs is

Rseries = Rparallel= 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 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

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:

40 V

6 k

→

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

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

Circuit Concepts

Voltage Divider:

Example:

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

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

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

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

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

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:

Source Resistance

Ideal Case

V

V

Circuit Concepts

Loading Effects:

Load Resistance

Ideal Case

V

V

Circuit Concepts

Loading Effects:

Source and Load Resistance

Ideal Case

V

V

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?)

Circuit Concepts

Ammeter and Voltmeter Construction:

Circuit Concepts