Flip Flops, and equivalent circuits

In summary, the two circuits presented are not equivalent as they exhibit different state behaviors and output values for the same input sequence. The second circuit, which is asked to be simplified using only NAND and NOR gates, may not have an equivalent circuit using only those two types of gates.
  • #1
sandy.bridge
798
1

Homework Statement


Hey guys,
Can someone please explain how the following circuits that I attached are equivalent? I'm not seeing that they are when I develop equations for the output Z.

For the first one I get Z=Q1+Q2=(Q1'+Q2')+(XQ2')'=Q1'+Q2'+X'+Q2

For the second image I get Z=Q2'+Q1=(XQ2')'+Q1=X'+Q2+Q2+Q1'

so I must be messing up somewhere and not catching it.
Any suggestions?
Thanks!
 

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  • #2
sandy.bridge said:

Homework Statement


Hey guys,
Can someone please explain how the following circuits that I attached are equivalent? I'm not seeing that they are when I develop equations for the output Z.

For the first one I get Z=Q1+Q2=(Q1'+Q2')+(XQ2')'=Q1'+Q2'+X'+Q2

For the second image I get Z=Q2'+Q1=(XQ2')'+Q1=X'+Q2+Q2+Q1'

so I must be messing up somewhere and not catching it.
Any suggestions?
Thanks!

The AND/NAND and the 2nd flop do not look right...
 
  • #3
Are you referring to the image or the equations that I have given? If it's the equations, the first one?
 
  • #4
The image -- the connections do not look equivalent (but I could be wrong).
 
  • #5
Where did you get the image from?
 
  • #6
Second image I copied directly out of the textbook. The first image is the solution. It is asked for one to determine an equivalent circuit using merely NAND and NOR gates, and no inverters. The top image is answer.
 
  • #7
Would you agree that the equations I presented to represent the figures are indeed correct? Perhaps the solution is wrong?
 
  • #8
sandy.bridge said:
Second image I copied directly out of the textbook. The first image is the solution. It is asked for one to determine an equivalent circuit using merely NAND and NOR gates, and no inverters. The top image is answer.

sandy.bridge said:
Would you agree that the equations I presented to represent the figures are indeed correct? Perhaps the solution is wrong?

I get the same two Z equations as you do.

However, when I start to fill out a State Table for the two circuits, they behave differently. Start with the FFs cleared and X=0 at the input. The initial state has different Z values. As you clock the two circuits while holding X=0, they stabilize into two different state looping behaviors, but each of those has Z=1 at the output. When I change the input to X=1, I get a different value of Z output for the two circuits, depending on what part of the two looping state behaviors the circuits are in when the input X is changed. That doesn't seem like functionally equivalent circuits to me...

Try it yourself. Set up the two state tables and fill in some values for each clocked cycle:

For circuit #1:
Code:
STATE  X  Q1  Q1'  Q2  Q2'  --> Z
    0     0   0    1     0    1         0
    1     0   1    0     1    0         1
and so on...

For circuit #2:
Code:
STATE  X  Q1  Q1'  Q2  Q2'  --> Z
    0     0   0    1     0    1         1
    1     0   1    0     0    1         1
and so on...

I may have made some errors in my State Tables, so I'd be interested in what you find. It looks like the two circuits have different State Diagrams at least, and for me it looks like they give different outputs for at least one sequence of inputs X.
 
  • #9
I did, and I could not seem to get that they were equivalent. Hmm. Is there a strategic way to determine an equivalent circuit to the second image (green and red) using merely NAND and NOR gates? I feel as though I have been staring at this problem for LONG.
 

1. What are flip flops and how do they work?

Flip flops are digital circuits that are used to store and remember a single bit of information. They have two stable states, represented by 0 and 1, and can be triggered to change from one state to the other based on certain input signals. This makes them useful for storing data and creating sequential logic circuits.

2. What is the difference between a D flip flop and a JK flip flop?

A D flip flop has only a single input, called the data input, which controls the state of the flip flop. On the other hand, a JK flip flop has two inputs, J and K, which can be used to set or reset the flip flop's state. This allows for more flexibility in creating sequential logic circuits.

3. How are flip flops used in computer memory?

Flip flops are used in computer memory to store and retrieve data. Multiple flip flops are connected together to create a memory cell, which can store a single bit of information. By arranging these memory cells in a specific way, we can create larger memory units like bytes, kilobytes, etc.

4. What is the difference between a synchronous and asynchronous flip flop?

Synchronous flip flops use a clock signal to synchronize the state changes of the flip flop. This means that the flip flop can only change its state when the clock signal is high. On the other hand, asynchronous flip flops do not use a clock signal and can change their state at any time based on the input signals.

5. Can flip flops be combined to create more complex circuits?

Yes, flip flops can be combined to create more complex circuits like counters, shift registers, and memory units. By connecting the output of one flip flop to the input of another, we can create a chain of flip flops that can store and manipulate multiple bits of information at a time.

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