BIOPOTENTIAL AMPLIFIERS - Design and Development of Medical Electronic Instrumentation - page 35

circuit is presented in Figure 1.30 a . For the purpose of analysis, input
voltage V in is assumed to be constant during the sampling period. It is further assumed that
the switches are thrown back and forth continuously with a clock period T according to the
timing diagram of Figure 1.30 b and that their connections shift instantly with no overlap.
During the

2 ), the circuit is equivalent to
Figure 1.30 c , and C S is charged instantaneously to V Φ odd

fi first odd-phase interval ( n 1) t / T ( n

( n 1):

in

V Φ odd V Φ odd

V C S ( t ) [( n 1) T ] ( n 1)

in in

While the odd-phase output voltage V Φ odd is equal to the voltage of V C H ,

out

V Φ odd V Φ odd

V C H ( t ) [( n 1) T ] ( n 1)

out out

2 )

During the even-phase interval ( n n that follows, the circuit is equivalent to

that of Figure 1.30 d , and charges are redistributed between C S and C H , which results in a
new output voltage. The analysis of the charge transaction is simpli

t / T

ed by assuming the
alternative equivalent circuit of Figure 1.30 e with uncharged capacitors. By applying the
initial voltages of the capacitors represented by the sources as step functions edged at
t

fi

( n 1) T , the new output voltage is

C

C S C

1
2

H
C H

S

V Φ even V Φ odd V Φ odd n ( n 1) C S ( n 1)

out in out

C H

2 ) , capacitor C H remains undisturbed, and
thus the output voltage is represented by the expression

During the odd-phase interval n t / T ( n

1
2

A general expression representing the odd-phase output voltage may now be written

V Φ odd V Φ even ( n ) n

out out

C C

S H

C H

V Φ odd V Φ odd V Φ odd

( n ) C S ( n 1) C S ( n 1)

out in out

C H

Applying the z -transform to this equation, we obtain

C

z
C

1

C S

V Φ odd V Φ odd

H

V Φ odd

( z ) C S ( z ) C ( z )

out in out

C H

S H

The odd-phase discrete-frequency-domain transfer function may then be solved directly:

Φ

o

o
d

d
d

d

z 1
1

V (

(

z
z

)
)

1
H /C S

H Φ in odd;Φ out odd ( z )

out

V 1

C

H /
H

C
/C

Φ

C

S

z 1

1

in

C

S

By replacing z by e j ω T and using Euler's formula, the time-domain representation of this
equation can be written as

Φ

o

o
d

d
d

T d j ω

V
V Φ

e

out

H Φ in odd;Φ out odd ( e j ω T )

j ω T

e

in

1
(1 C H /C S ) cos ω T C H /C S j (1 C H /C S ) sin ω T

where is the frequency of an applied sinusoidal signal and T is the clock period. From

this equation it is possible to determine the magnitude response and phase shift of the
switched-capacitor instrumentation block. In addition, a capacitance ratio C H /C S that

ω