61 lines
1.5 KiB
Markdown
Executable file
61 lines
1.5 KiB
Markdown
Executable file
# Important info for lab
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Input resistance
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$R_{in}=R_d(1+A\beta)$
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Gain
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${V_o \over V_{in}}={A \over 1+A\beta}$
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output resistance
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$r_{out}={r_o\over 1+A\beta}$
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$r_d$ is the resistance between the inputs.
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$r_o$ is the resistance on the output after the source.
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The source within is represented as $A_v(v_+-v_+)$
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${R_i \over R_i + R_f}=\beta$ is the feedback factor. $\alpha={R_f\over R_i+R_f}$ is the control factor. A is the open-loop gain. (Closed-loop gain is when the output feeds back into the input)
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$R_i$ is the input resistor
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$R_f$ is the feedback resistor. It is ALWAYS back to the negative.
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# Simple feedback op-amp for examples
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## Gain
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If A is finite, and $r_d$ and $r_o$ are neglected, then the gain = ${R_i+R_f \over R_i}={1\over \beta}$
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## Input resistance
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$$
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goal\\
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r_{in}={1V\over I_{in}}\\
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.\\
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0={v_-- 0\over R_i}+{v_-- V_{in}\over r_d}+{v_-- V_o\over R_f}\\
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i_{in}={1- V_- \over r_d}\\
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0=V_-({R_iR_f+r_dR_f+r_dR_i \over R_ir_dR_f})-{1\over r_d}-{A(1-v_-) \over R_f}\\
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0=V_-({(R_iR_f+r_dR_f+r_dR_i(1-A)) \over R_ir_dR_f})-{1\over r_d}-{A\over R_f}\\
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1-i_{in}r_d=V_-\\
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0=(1-i_{in}\cancel {r_d})({(R_iR_f+r_dR_f+r_dR_i(1-A)) \over R_i\cancel {r_d}R_f})-{1\over r_d}-{A\over R_f}\\
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0=(1-i_{in})(1+{r_d\over R_i}+{r_d(1-A)\over R_f})-{1\over r_d}-{A\over R_f}\\
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0=1-i_{in}+{r_d\over R_i}+{r_d(1-A)\over R_f}-{r_di_{in}\over R_i}-{r_d(1-A)i_{in}\over R_f}-{1\over r_d}-{A\over R_f}\\
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i_{in}(1+{r_d\over R_i}+{r_d(1-A)\over R_f})={r_d\over R_i}+{r_d(1-A)\over R_f}-{1\over r_d}-{A\over R_f}
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$$
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