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3. MEASUREMENT CONVERTERS OF ELECTRICAL QUANTITIES measuring amplifiers: demands on measuring amplifiers, negative feedback, ideal operational amplifier, basic circuits of measuring amplifiers using operational amplifiers (OAs) measurement of low voltages and currents using OAs, estimating uncertainty of measurement (including influence of input voltage offset and input bias), rectifiers (converters of the rectified mean value): active – non-controlled, controlled rectifiers - principle, properties, measurement of voltage phasor

38EMB – P3

1

MEASUREMENT CONVERTERS OF ELECTRICAL QUANTITIES – Measuring amplifiers Demands: a) Defined gain, input and output impedance: 1.

A = U2/U1 voltage amplifier

Zin → ∞ or defined Zout → 0 (voltage source)

2.

A = I2/U1 voltage-controlled current source

Zin → ∞ or defined Zout → ∞ (current source)

3.

A = U2/I1 current-to-voltage converter

Zin → 0 or defined Zout → 0 (voltage source)

4.

A = I2/I1 current amplifier

Zin → 0 or defined Zout → ∞ (current source)

b) DC amplifier: Minimum input voltage offset and its drift (∆Ucc, ∆t, ∆T) c) AC coupled amplifier: Constant gain in defined frequency band, minimum phase shift. 38EMB – P3

2

Gain stabilization using negative feedback

U1-kU2

U1

A→∞

U2

kU2

U2 A = U 2 = A (U 1 − kU 2 ) ⇒ ACL = U 1 1 + kA A→∞ ⇒

Us

U out AU = , AU → ∞ ⇒ U D = 0 UD

k≤1

ACL =

1 k

for

Ideal operational amplifier (OA) I1+

+

UD

U1+

AUUD

I1U1-

38EMB – P3

-

Uout Us

I 1+ = I 1− = 0

3

Noninverting amplifier

U R1 R2

+ U1

U2

UR1

R1

U2 U1 I1 = = − I2 = − R2 R1

I2

R1

U2 R2 = − U1 R1

R2

-

38EMB – P3

U2 R2 = 1+ U1 R1

Rin → ∞ , Rout = 0

Inverting amplifier

U1

R1 = U1 = U 2 R1 + R2

I1

+

U2

U1 R in = = R1 , R out = 0 I1 4

Current-to-voltage converter (stems from inverting amplifier for R1=0) I2

I1 = − I 2

I1

U2 = − R2 I1

R

+

U2

Rin = 0 , Rout = 0

Voltage-controlled current source - VCCS (non-inverting – stems from noninverting amplifier) +

U 1 = I 2 R1

I2

-

RZ

U1 R1

U1 I2 = − R1 Rin → ∞ , R out → ∞

38EMB – P3

5

VCCS (inverting – stems from inverting amplifier) I2 I1

R -

RZ

+

U1

U1 I 2 = − I1 = − R1 Rin = R1 , Rout → ∞

Disadvantage of both current sources- necessity of “floating load“

Real (actual) operational amplifier

I 1+ ≠ I 1− ≠ 0

(non-zero input bias currents)

AU ≈ 10 4 ÷ 10 7

(finite voltage gain)

UD = 0 ⇔ / U 2 = 0 (non-zero input voltage offset)

38EMB – P3

6

MEASUREMENT OF LOW VOLTAGES AND CURRENTS USING OA Estimation of uncertainty of voltage measurement UX – inverting amplifier a) ideal OA

R2

I1N

R1

UX

+

U2

UD0

R1 UX = − U2 R2 2

uU X ( id )

Standard uncertainty:

2

 −U2   − R1   − U 2 R1   =  u R1  +  uU 2  +  u R2  2  R2   R2   R2 

2

where uU2 id standard uncertainty of the measured voltage U2

u R = ∆Rmax

3=

b) actual OA

Standard uncertainty:

38EMB – P3

δ Rmax

R , δR max tolerance of R1 or R2 in % 100 3

 R1 R1  U X = − U 2 m I1N R1 ± U D 0 1 +  R2  R2   I R   U (1 + R1 R2 )  +  1N 1  +  D 0  3  3    2

uU X ( OZ ) = uU2 X ( id )

2

7

Estimating uncertainty of measurement of voltage UX – non-inverting amplifier a) ideal OA

UD0 UX

+

R2

R1 UX = U2 R1 + R2

R1

I1N

U2

2

2

 U 2 R2   R1   U 2 R1       + + uU X ( id ) =  u u u U2  2 R2  2 R1     ( R1 + R2 )   R1 + R2   ( R1 + R2 ) 

Standard uncertainty

2

here uU2 is standard uncertainty of measurement of voltage U2

u R = ∆Rmax

3=

b) actual OA

Standard uncertainty: 38EMB – P3

δ Rmax

100 3

UX =

R,

δRmax - tolerance of R1 or R2 in %

 R1  R1  ± U D 0 U 2 m I1N R2  R1 + R2  R1 + R2 

 I R R / (R1 + R2 )   U D 0  +  1N 1 2   + 3   3   2

uU X ( OZ ) = uU2 X ( id )

2

8

Estimating uncertainty of measurement of current IX – I → U converter a) ideal OA

IX

I1N

R

Standard uncertainty:

+

IX = −U 2 R   −1   U 2 =  uU 2  + uR  2   R   R  2

u I X ( id )

U2

2

here uU2 is standard uncertainty of voltage U2

u R = ∆Rmax

3=

δ Rmax 100 3

R,

δRmax - tolerance of R in % b) actual OA

I X = − U 2 R m I1 N Standard uncertainty:

I  uU X ( OZ ) = u I2X ( id ) +  1 N   3

38EMB – P3

2

9

RECTIFIERS (CONVERTERS OF THE RECTIFIED MEAN VALUE) Non-controlled rectifiers (active) +

i2 i2

u1

i2

R1

R1

+

u1

non-inverting

u1 inverting

transfer characteristic

u1RM u1RMS = i2 RM ⇒ R1 = R1 1,11 ⋅ i2 RM note: Both circuits require „floating load“. If grounded load has to be used, differential amplifier or rectifier using two OAs should be used. 38EMB – P3 10

Controlled rectifiers

RL

u1

R

u2

R

u1

UC

UC

CC

Uc – rectangular pulse train with magnitude 1 ∞

1 sin (k ω t ) ∑ π k ´1 k u1 (t ) = U m sin (ω t − ϕ )

uř (t ) =

4

pro k liché

u2 (t ) = u1 (t ) ⋅ uř (t ) 38EMB – P3

11

+

u2

CC

    4 1   u2 (t ) = U m cos ω t − ϕ − ω t  − cos ω t − ϕ + ω t  +  14243  43  π 2   142 2ω t −ϕ −ϕ     

uC ϕ

u1

u2

u20

    4 1 ∞     +Um cos ω t − ϕ − kω t − cos ω t − ϕ + kω t  ∑  142 4 43 4  4 43 4  π 2 k =3   142  (1+k )ω t −ϕ    (1−k )ω t −ϕ  2 u2 (t ) = U m cos ϕ + AC components = u 20 + AC comp.

π

Notes to derivation

2

π

U m = U RM

1 sin α sin β = [cos (α − β ) − cos (α + β )] 2

38EMB – P3 12

Measuring voltage phasor using controlled rectifier-VECTORVOLTMETER After filtering out the AC components using a LP filter the DC component u2,0 at the c.r. output is proportional to the real part of the phasor. After shifting the control voltage by 900 (π/2), this DC component u2,90 corresponds to imaginary part of the measured phasor. u0 UX

ur

(u 90 )

CR

U2

FILTER

ux

uR,0 (uR,90 )

SC

=

0

o

90

o

SC

u2,0

uR,90

U m cos ϕ =

2 2

π

ReUx

u90 u2,90

Ux

38EMB – P3 13

u 2 , 90 = =

ur

ImUx

π

U ef cos ϕ

o

u0

ϕ

2

uR,0

Ur 90

u2,0 =

ϕ

2

π

U m sin ϕ =

2 2

π

U ef sin ϕ