Input resistance of a circuit

Discussion in 'Homework Help' started by anhnha, Oct 30, 2013.

  1. anhnha

    Thread Starter Active Member

    Apr 19, 2012
    773
    47
    Can anyone help me find the input resistance of the circuit? I have been stuck for many hours and can't solve it.

    [​IMG]

    I find the equivalent resistance of the circuit by applying a voltage source at the input and solve for the current I.

    The input resistance:

    Rin = V/I

    However, I don't know why this method doesn't work. Rin is always a function of V and that is not right.
     
    • Rin.PNG
      Rin.PNG
      File size:
      13.8 KB
      Views:
      135
  2. The Electrician

    AAC Fanatic!

    Oct 9, 2007
    2,281
    326
    Show your work.
     
  3. WBahn

    Moderator

    Mar 31, 2012
    17,743
    4,795
    How can we possibly help you find what you did wrong when you don't show us what you did at all? We are not mind readers.

    Show us the work for the best effort that your several hours produced.
     
  4. anhnha

    Thread Starter Active Member

    Apr 19, 2012
    773
    47
    Let's call the voltage across R3 is Vx.
    Apply a voltage source V to the input and let's call the potential at the bottom is V1 => The voltage in the left side of R1 is V1 + V.

    [​IMG]

    Apply KCL:

    \frac{ V_{1} + V - V_{x} }{ R_{1} } = A*V + \frac{ V_{x} - V_{1} }{ R_{2} } +  \frac{ V_{x} }{ R_{3} }

    V_{x} ( \frac{1}{ R_{1} } +  \frac{1}{ R_{2} } +  \frac{1}{ R_{3} } ) = \frac{ V_{1} + V }{ R_{1} } - A*V +  \frac{ V_{1}}{ R_{2} }

    V_{x} =  \frac{\frac{ V_{1} + V }{ R_{1} } - A*V +  \frac{ V_{1}}{ R_{2} } }{  \frac{1}{ R_{1} } +  \frac{1}{ R_{2} } +  \frac{1}{ R_{3} } } <br />

    I =  \frac{ V_{1}+ V -  V_{x} }{ R_{1} }  =  \frac{ V_{1}+ V -  \frac{\frac{ V_{1} + V }{ R_{1} } - A*V +  \frac{ V_{1}}{ R_{2} } }{\frac{1}{ R_{1} } +  \frac{1}{ R_{2} } +  \frac{1}{ R_{3} }}  }{ R_{1}  }

    R_{in} =  \frac{V}{I }=  \frac{V*R_{1}}{  V_{1}+ V -  \frac{\frac{ V_{1} + V }{ R_{1} } - A*V +  \frac{ V_{1}}{ R_{2} } }{\frac{1}{ R_{1} } +  \frac{1}{ R_{2} } +  \frac{1}{ R_{3} }}    }


    Rin is a function of V and V1. What I am wrong here?
     
  5. Jony130

    AAC Fanatic!

    Feb 17, 2009
    3,957
    1,097
    I don't see any close loop for R3 resistor current? So how can any current flow through R3?
     
  6. Jony130

    AAC Fanatic!

    Feb 17, 2009
    3,957
    1,097
    I solve for Rin and I get this result Rin = (R1 + R2)/(1 + gm R2)
     
  7. WBahn

    Moderator

    Mar 31, 2012
    17,743
    4,795
    No, it's not. It is a function only of the circuit parameters.

    EDIT: If you read the original version, I thought the OP was someone other than anhnha. Sorry.
     
  8. anhnha

    Thread Starter Active Member

    Apr 19, 2012
    773
    47
    Jony sorry for the confusion. I am solving for a cascode amplifier. The first stage is common source and the second stage is common gate and this stage has a resistance between gate and drain.

    I want to know the voltage at the output of the first stage. If I know the input impedance, Zin of common gate stage then the voltage at drain of the common source is -gm1*Zin*Vin.

    I think this maybe should consider as there is a voltage source V1 + V connects the left side of R1 to ground and another voltage source V1 connects the bottom of R2 to ground.

    PS: the second I think it is not a common gate but a configutation in which drain and gate is connected by a resistance.
     
  9. WBahn

    Moderator

    Mar 31, 2012
    17,743
    4,795
    What you have is correct, as far as it goes. You need to apply KCL at either the ground node or at V1, in which case you will discover that the current in R3 is zero. Then it is just a matter of slogging through some algebra. But you are definitely doing it the hard way.
     
  10. WBahn

    Moderator

    Mar 31, 2012
    17,743
    4,795
    Well, you definitely need to decide if current can flow into or out of the left hand terminals other than just from one to the other and, if so, represent that in the schematic.
     
  11. anhnha

    Thread Starter Active Member

    Apr 19, 2012
    773
    47
    Here is the small signal model of the circuit. I need to know Zin.
    Then the voltage at the output of common source will be equal to -Gm1*(ro||Zin)

    [​IMG]

    PS. The first stage is common source degradation.
     
    • Zin.PNG
      Zin.PNG
      File size:
      23.1 KB
      Views:
      93
  12. Jony130

    AAC Fanatic!

    Feb 17, 2009
    3,957
    1,097
    Can you show "normal diagram" of this cascade amplifier.
    And if the second stage is common gate what is Zin for CG stage zin = 1/gm2 ??
    If so the voltage at Q1 drain is equal to:
    Vin * (1/gm2)/(1/gm1 + RS)
     
  13. anhnha

    Thread Starter Active Member

    Apr 19, 2012
    773
    47
    Jony, here is the circuit. I need it for a project. I have modified and simulated it a lot but it doesn't work. Now I decide to learn the theory carefully before make any simulation.

    [​IMG]
     
  14. Jony130

    AAC Fanatic!

    Feb 17, 2009
    3,957
    1,097
    Simulation don't work because you made a error in the diagram.
    Look where you connect M2 source. Also I don't see any capacitor between M2 gate and GND?
     
  15. anhnha

    Thread Starter Active Member

    Apr 19, 2012
    773
    47
    Thanks, yes, D and S of M2 should be swapped. I made the mistake here but in the simulation it is correct.

    yes, you are right. I used capacitor in place of R2 in the topology. However,
    last week, I go to a university and asked a friend of mine for help. He said that
    I can use resistance instead of a capacitor provided that they have the same impedance. Therefore, I draw a resistor here.
     
  16. Jony130

    AAC Fanatic!

    Feb 17, 2009
    3,957
    1,097
    I think that your small signal model should look like this
     
    • 1.PNG
      1.PNG
      File size:
      15.3 KB
      Views:
      63
  17. anhnha

    Thread Starter Active Member

    Apr 19, 2012
    773
    47
    Yes, can you tell me how to calculate Rin of the second stage?
     
  18. Jony130

    AAC Fanatic!

    Feb 17, 2009
    3,957
    1,097
    I don't know how you can find Rin for the second stage. I think that you need to analyze the whole circuit. Because R2 is connect between M2 base and M1 source. So M2 gate is not grounded, so you cannot split the circuit.
     
    anhnha likes this.
  19. anhnha

    Thread Starter Active Member

    Apr 19, 2012
    773
    47
    Jony, can you tell me steps to analyse this? I don't know where to start. It is too complicated to me.
     
  20. Jony130

    AAC Fanatic!

    Feb 17, 2009
    3,957
    1,097
    It is pain in the a... to solve this circuit manually.
    So I write four nodal equations and with help of math software solve all of them.

    For Vs node
    Vs/Rs = (Vin - Vs )*gm1 + (Vg - Vs)/R2

    VG node
    (Vg - Vs)/R2 = (Vout - Vg)/R1
    For Vout node
    (0 - Vout)/RD1 = (Vout - Vg)/R1 + ( Vg - Vd)*gm2

    For Vd node
    ( Vg - Vd)*gm2 = (Vin - Vs )*gm1

    And finally solution for Vout and Vd

    [​IMG]
     
    • 1.PNG
      1.PNG
      File size:
      16.5 KB
      Views:
      25
    • 3.PNG
      3.PNG
      File size:
      3.6 KB
      Views:
      80
    Last edited: Oct 31, 2013
    anhnha likes this.
Loading...