Quick reference for Analog Circuits.
Triode Equations
Three States
- \(v_{CE}\) from 0 to + :
- Saturation
- Active
- Cut-off (close to but still above zero)
Basic Relations
\[\begin{align} I_{CQ} &= \beta I_{BQ} \\ I_{EQ} &= I_{BQ} + I_{CQ} \\ \end{align}\]Internal Resistance
\[\begin{align} r_{be} &= r_{bb'} + (1+\beta)\frac{26\mathrm{mV}}{I_{EQ}(\mathrm{mA})} \\ &= r_{bb'} + \frac{26\mathrm{mV}}{I_{BQ}(\mathrm{mA})} \\ \end{align}\]Note
Default value for \(r_{bb'}\) is \(200\Omega\) .
Amplifier Resistance
\[\begin{align} R_i &= \begin{cases} R_{b'} \parallel [r_{be} + (1+\beta)R_e] &\text{(common-emitter)} \\ R_{b'} \parallel [r_{be} + (1+\beta)(R_e \parallel R_L)] &\text{(common-collector)} \\ R_e \parallel \frac{r_{be}}{1+\beta} &\text{(common-base)} \end{cases} \\ R_o &= \begin{cases} R_c &\text{(common-emitter)} \\ R_e \parallel \frac{r_{be} + (R_s \parallel R_{b'})}{1+\beta} &\text{(common-collector)} \\ R_c &\text{(common-base)} \end{cases} \end{align}\]Amplification
\[\begin{align} \dot{A}_V &= \frac{\dot{V}_o}{\dot{V}_i} &\text{(triode output - triode input)} \\ \dot{A}_{VS} &= \frac{\dot{V}_o}{\dot{V}_s} = \frac{R_i}{R_s+R_i}\dot{A}_V &\text{(triode output - source stimulation)} \\ \\ \dot{A}_V &= \prod_n \dot{A}_{V_{n}} \\ \end{align}\]Note
Grab onto \(I_B\) , since it contains relation between input segment and output segment.
Note
\(\dot{V}_O\) is \(R_L\)-dependant.
When cascading directly, \(R_{i_n}=R_{L_{n-1}}\) is introduced via contributing to \(\dot{V}_{O_{n-1}}\) .
MOS Equations
Three States
- \(v_{DS} \, (=v_{GS}-V_P)\) from 0 to + :
- Variant resistance
- Saturation
- Cut-off (almost 0)
Basic Relations
\[\begin{align} i_D &= K_n (v_{GS} - V_T)^2 \cdot (1 + \lambda v_{DS}) \\ g_m &= \frac{\partial{i_D}}{\partial{v_{GS}}}\vert_{v_{DS}} \\ &= 2 K_n (v_{GS} - V_T) \\ &= 2 \sqrt{K_{n/P}I_{DQ}} \\ r_{ds} &= [\lambda K_n (v_{GS} - V_T)^2]^{-1} = \frac{1}{\lambda i_D} &\text{(usually large, often ignored)} \\ \end{align}\] \[\begin{align} \text{Where} \hspace{3em} V_T &= \text{Threashold Voltage } . \\ \end{align}\]Amplifier Resistance
\[\begin{align} R_i &= \begin{cases} R_{g1} \parallel R_{g2} &\text{(common-source)} \\ R_{g1} \parallel R_{g2} &\text{(common-drain)} \\ \frac{1}{g_m} \parallel R &\text{(common-gate)} \\ \end{cases} \\ R_o &= \begin{cases} R_d \parallel r_{ds} &\text{(common-source)} \\ \frac{1}{g_m} \parallel R_{(source)} \parallel r_{ds} &\text{(common-drain)} \\ R_d \parallel r_{ds} &\text{(common-gate)} \\ \end{cases} \\ \end{align}\]Operational Amplifier
Subtraction
\[\begin{align} v_o &= - \frac{R_f}{R_n} v_{in} + (1 + \frac{R_f}{R_n}) v_{ip} \\ \end{align}\]Note
Given condition of virtual-short and virtual-open, the resistance \(R_P\) , the one before amplifier’s positive input \(v_P\) immediately, is silenced and of no influence on circuit output.
Feedback
Patterns
- Output
- \(R_L\) grounded → Shunt / Voltage sampling
- \(R_L\) not grounded → Series / Current sampling
- Input
- Same as input → Shunt / Current feedback
- Different from input → Series / Voltage feedback
Symbols
- \(\dot{F}\) → feedback index 反馈系数
- \(\dot{A}_{XF}\) → close-circuit amplification 闭环增益
- \(\dot{A}_{VF}\) → close-circuit voltage amplification 闭环电压增益
Where \(X\) could be one of \(V\), \(I\), \(R\), \(G\) accrodingly, w.r.t. feedback type.
Commonly Used Approximation
Given conditions of strong negative feedback, i.e. \(\dot{A}\) very large, have
\[\begin{align} \dot{A}_{XF} &= \frac{\dot{A}}{1 + \dot{A}\dot{F}} \approx \frac{1}{\dot{F}} \\ \end{align}\]Signal Generating Circuits
RC Bridging Sine Oscillator
\[\begin{align} f_0 &= \frac{1}{2 \pi R C} \\ \end{align}\]Require \(A_V = 1 + \frac{R_f}{R_1} > 3\), i.e. \(R_{f_{min}} = 2 R_1\), where \(R_1\) is grounding negative phase input of amplifier and \(R_f\).
Inverted-Input Hysteresis Comparer
Core concept: superposition principle
\[\begin{align} V_{T-} &= \frac{R_1 V_{OL}}{R_1 + R_3} + \frac{R_3 V_{REF}}{R_1 + R_3} \\ V_{T+} &= \frac{R_1 V_{OH}}{R_1 + R_3} + \frac{R_3 V_{REF}}{R_1 + R_3} \\ \end{align}\] \[\begin{align} \text{Where} \hspace{3em} R_1 &= \text{between reference voltage } V_{REF} \text{ and feedback resistance } R_3 \,, \\ R_2 &= \text{between input } v_i \text{ and interted input of amplifier } , \\ R_3 &= \text{between output } v_o \text{ and positive-phase input of amplifier } , \\ V_{REF} &= \text{reference voltage } . \end{align}\]Additional Notes
- Exclude \(R_L\) when calculating output resistance \(R_o\) , but do remember to include \(R_e\) / \(R_s\) (e.g. common-collector amplifier / common-source amplifier).
- Include \(R_L\) when calculating output voltage \(\dot{V}_o\) , for resistance converts \(I_C\) to voltage.
- \(r_{bb'} \approx 200 \Omega\) as default.
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When cascading,
\[\begin{align} R_i &= R_{i_1} \\ R_o &= R_{o_n} \\ \end{align}\]
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