Calculate gain for differential amplifier

In summary, the gain (u_o/u_i) for a differential amplifier with symmetric output can be calculated by applying small-signal voltages to the inputs, computing the change in current for each transistor, and then using the formula gmRD to calculate the gain. However, if the difference in resistance between the two transistors is significant, the calculated gain may be slightly less than gmRD. Alternatively, if the inputs are applied as +vin/2 and -vin/2, the calculated gain will be exactly gmRD.
  • #1
beyondlight
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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.

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

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

[tex]u_{o} = -R_{D1}g_{m}v_{gs1}+R_{D2}g_{m}v_{gs2} = -R_{D1}g_{m}v_{gs1} [/tex]
[tex]u_{o} = -R_{D1}g_{m}(\frac{u_{i}}{1+g_{m}\frac{R_{s}}{2}} ) [/tex]
[tex]\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}}[/tex]The answer above is wrong, u_o/u_i = -gm*Rd is the correct one. What am I doing wrong?
 
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  • #2
Don't use small-signal equivalent circuits. All you need is gm = dI/dVgs for each transistor. You still can and should use small-signal parameters.
I suggest:
Ground uin- . Apply small voltage uin+ . Compute ΔI1 and ΔI2.
Then compute uut+ , uut- and finally uut+ - uut- .
Note that ΔI1 and ΔI2 are independent of RD1 and RD2.
EDIT:
Your given answer is correct if we assume differentially small ΔRD but if not then the gain I calculated is somewhat less than gmRD.
Also, if 2gmRs >> 1 the gain also → gmRD.
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 gmRD. 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.
 
Last edited:

FAQ: Calculate gain for differential amplifier

What is a differential amplifier?

A differential amplifier is a type of electronic circuit that amplifies the difference between two input signals while rejecting any signals that are common to both inputs. It is commonly used in instrumentation, audio equipment, and other electronic devices.

How do you calculate gain for a differential amplifier?

The gain of a differential amplifier is typically calculated by taking the ratio of the output voltage to the input voltage. However, for a more accurate calculation, you can use the equation: Gain = (Rf/R1) x (1 + 2R2/R1), where Rf is the feedback resistor, R1 is the input resistor, and R2 is the resistor connecting the two inputs.

What is the purpose of a differential amplifier?

The main purpose of a differential amplifier is to amplify small signals while rejecting any common-mode signals. This is useful in applications where the desired signal is small and the noise or interference is present in both inputs.

How do you choose the right resistors for a differential amplifier?

The choice of resistors for a differential amplifier depends on the desired gain and input impedance. Typically, the input resistors should be equal (R1 = R2) for maximum common-mode rejection, while the feedback resistor (Rf) can be varied to achieve the desired gain. It is also important to consider the tolerance and temperature coefficient of the resistors to ensure accurate and stable performance.

Can a differential amplifier have a negative gain?

Yes, a differential amplifier can have a negative gain, which means that the output signal is inverted compared to the input signal. This can be achieved by configuring the input and feedback resistors in a specific way. Negative gain is often used in applications such as phase detectors and oscillators.

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