a problem in equality of switch_capacitor with resistance

[akbarza]

hi
my problem is in proving that resistance can be replaced by switch_capacitor.
in attached file, in page 46 in fig b:
suppose from last step the voltage of cs is VB.
Now when s1 is connected, at end of this s1-connected step, we will have I1.t1=cs.(VA-VB)
IN next step S2 connected with noting that in this step initial voltage of capacitor is VA, then we will have in end of S2-connected: I2.t2=cs(VB-VA)
with t1=t2=T/2, then result: I1-I2=cs.(VA-VB)/T/2
MY PROBLEM IS THIS 2?
Can you explain to me where i am wrong? is this 2 extra?
thanks

Moderator action: removed attachment because it is clearly marked as copyrighted and must not be distributed without permission

 

[wwfeldman]

look at this

Switched capacitor - Wikipedia

en.wikipedia.org

the only real difference between the switched capacitor as resistor in your question
and that in the wiki article is the use of frequency (f) or period (T)

i think the issue is the definition of T
you appear to have broken T into t1 and t2, the times for each switch
since t1 = t2 = T/2, the 2 was introduced by using the time for a switch instead of the period

 

[LvW]

hi
my problem is in proving that resistance can be replaced by switch_capacitor.

Just one comment to the above referenced wiki-contribution: The first diagram (figure) is misleading.
This circuitry with one capacitor and two switches is not able to mimic a resistor. This is because there can be no current through the circuit because it is open at the output. Both nodes - input and output - need a fixed and defined potential.
Hence, a behaviour like a resistor requires at the output node either (a) a voltage source or (b) a very large capacitor or (c) ground or (d) virtual ground (most popular method; inverting opamp node).