Calculate gain for differential amplifier

[beyondlight]

Homework Statement

Calculate gain (u_o/u_i) for a differential amplifier with symmetric output. Both transistors have the same transconductance gm. Transistor output resistance r_0 is neglected here.

circuit: http://sv.tinypic.com/view.php?pic=14cea8p&s=8#.VSf7Evl_vxM

Homework Equations

Calculate gain A = u_o/u_i

The Attempt at a Solution

We rewrite the circuit to small-signal.
Voltage sources +/- E are set to zero.
I also split the source resistance Rs, into two parts so that we can divide the differentiator between a left and right side.

v_{gs1} = u_{i}-g_{m}v_{gs1}\frac{R_{s}}{2}
v_{gs1} = \frac{u_{i}}{1+g_{m}\frac{R_{s}}{2}}v_{gs2} = 0 - g_{m}v_{gs2}\frac{R_{s}}{2}
v_{gs2}(1 + g_{m}\frac{R_{s}}{2}) = 0
v_{gs2} = 0

u_{o} = v_{o1}-v_{o2}
v_{o1} = -R_{D1}g_{m}v_{gs1}
v_{o2} = -R_{D2}g_{m}v_{gs2}

u_{o} = -R_{D1}g_{m}v_{gs1}+R_{D2}g_{m}v_{gs2} = -R_{D1}g_{m}v_{gs1}
u_{o} = -R_{D1}g_{m}(\frac{u_{i}}{1+g_{m}\frac{R_{s}}{2}} )
\frac{u_{o}}{u_{i}} = \frac{ -R_{D1}g_{m}}{1+g_{m}\frac{R_{s}}{2}} = \frac{ -(R_{D}-\frac{\Delta R}{2})g_{m}}{1+g_{m}\frac{R_{s}}{2}}The answer above is wrong, u_o/u_i = -gm*Rd is the correct one. What am I doing wrong?

 

[rude man]

Don't use small-signal equivalent circuits. All you need is g_m = dI/dV_gs for each transistor. You still can and should use small-signal parameters.
I suggest:
Ground u_in- . Apply small voltage u_in+ . Compute ΔI₁ and ΔI₂.
Then compute u_ut+ , u_ut- and finally u_ut+ - u_ut- .
Note that ΔI₁ and ΔI₂ are independent of R_D1 and R_D2.
EDIT:
Your given answer is correct if we assume differentially small ΔR_D but if not then the gain I calculated is somewhat less than g_mR_D.
Also, if 2g_mR_s >> 1 the gain also → g_mR_D.
FURTHER EDIT:
If you apply +vin/2 to the + input and -vin/2 to the - input instead of +vin to the + input and 0 to the - input, your given gain is correct at g_mR_D. This is because then the "common-mode voltage" (the average of the two input voltages) is always zero, the source voltage never changes, and i1 + i2 = 0.