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Key formulas
| Name | Formula | Variables / Notes |
|---|---|---|
| Diode Shockley equation | I_D = I_S (e^(V_D / n V_T) - 1) | I_D: diode current; I_S: reverse saturation current (typ. 1e-14 A); V_D: diode voltage; n: ideality factor (1 to 2); V_T: thermal voltage = kT/q = 26 mV at 300K |
| BJT collector current | I_C = beta * I_B = I_S e^(V_BE / V_T) | I_C: collector current; beta: current gain (hFE, typ. 50 to 300); I_B: base current; V_BE: base-emitter voltage (0.7 V for Si) |
| MOSFET drain current (saturation) | I_D = (1/2) u_n C_ox (W/L) (V_GS - V_th)^2 | u_n: electron mobility; C_ox: oxide capacitance per area; W/L: width-to-length ratio; V_th: threshold voltage |
| Op-amp virtual short | V+ = V- (ideal op-amp) | V+: non-inverting input voltage; V-: inverting input voltage; valid when negative feedback is applied and op-amp is not saturated |
| Inverting amplifier gain | A_v = -R_f / R_in | A_v: voltage gain; R_f: feedback resistor; R_in: input resistor; negative sign indicates phase inversion |
| Non-inverting amplifier gain | A_v = 1 + R_f / R1 | A_v: voltage gain (always >= 1); R_f: feedback resistor; R1: resistor from inverting input to ground |
Key concepts
BJT operating regions
Cut-off: both junctions reverse biased, I_C = 0. Active: V_BE = 0.7 V (forward), V_BC < 0 (reverse), I_C = beta I_B, used for amplification. Saturation: both junctions forward biased, V_CE approximately 0.2 V, used as a switch in ON state.
MOSFET regions
Cut-off: V_GS < V_th, no channel, I_D = 0. Linear (triode): V_DS < V_GS - V_th, channel present, I_D proportional to V_DS. Saturation: V_DS >= V_GS - V_th, channel pinched at drain, I_D = (1/2) k_n (V_GS - V_th)^2.
Op-amp golden rules (ideal)
Rule 1: the differential input voltage V+ - V- = 0 when negative feedback is present. Rule 2: no current flows into either input terminal. These two rules allow rapid analysis of any op-amp circuit without knowing open-loop gain.
Small-signal model (BJT)
The BJT small-signal model uses transconductance g_m = I_C / V_T and input resistance r_pi = beta / g_m. The controlled current source i_c = g_m v_be models amplification. r_o = V_A / I_C models Early effect output resistance.
Feedback types
Series-series feedback increases input and output resistance. Shunt-shunt feedback decreases both. Series-shunt (voltage amplifier) increases Rin and decreases Rout. Shunt-series (current amplifier) decreases Rin and increases Rout.
Miller theorem
An impedance Z connected between input and output of an amplifier with gain A_v can be replaced by Z/(1-A_v) at the input and Z/(1-1/A_v) at the output. This is essential for computing bandwidth of inverting amplifiers.
Tables
BJT vs MOSFET Comparison
| Parameter | BJT (NPN) | NMOS |
|---|---|---|
| Control | Current (I_B) | Voltage (V_GS) |
| Input impedance | Low (r_pi = beta/g_m) | Very high (gate insulated) |
| Transconductance | g_m = I_C / V_T | g_m = 2I_D / (V_GS-V_th) |
| ON voltage | V_BE = 0.7 V | V_GS > V_th (typ 0.5-2 V) |
| Speed | High (GHz range) | Very high (GHz range) |
| Integration | Moderate | Very high (VLSI) |
Op-Amp Configuration Summary
| Configuration | Gain | Rin | Phase |
|---|---|---|---|
| Inverting | -Rf/Rin | Rin (low) | 180 deg |
| Non-inverting | 1 + Rf/R1 | Very high | 0 deg |
| Voltage follower | 1 | Very high | 0 deg |
| Summing (inv.) | -Rf(V1/R1+V2/R2) | R1, R2 sep. | 180 deg |
| Differentiator | -RC dV/dt | C (low at DC) | 90 deg lead |
| Integrator | -(1/RC) int V dt | R | 90 deg lag |
Rectifier Circuit Comparison
| Type | V_out (peak) | Ripple freq. | Diodes used |
|---|---|---|---|
| Half-wave | V_p - 0.7 | f_in | 1 |
| Full-wave (CT) | V_p - 0.7 | 2 f_in | 2 |
| Bridge | V_p - 1.4 | 2 f_in | 4 |
Quick facts
- Thermal voltage V_T = kT/q = 25.85 mV at 300 K (approximately 26 mV). It appears in both BJT and diode equations.
- For a silicon BJT in active region, V_BE = 0.7 V and V_CE(sat) = 0.2 V are standard exam values.
- The 3 dB bandwidth of an inverting op-amp amplifier is f_3dB = GBW / |A_v|, where GBW is the gain-bandwidth product.
- A MOSFET is in saturation when V_DS >= V_GS - V_th. This is the region used for amplification.
- The Early voltage V_A models BJT output resistance: r_o = V_A / I_C. Typical V_A = 50 to 200 V.
- Op-amp slew rate limits large-signal bandwidth. For a sine wave: max frequency = SR / (2 pi V_peak).
- CMRR = 20 log10(A_differential / A_common-mode). A good op-amp has CMRR > 80 dB.
- Zener diode operates in reverse breakdown. Voltage regulation uses the flat V-I characteristic in that region.
Exam shortcuts
- DC biasing shortcut for BJT voltage divider: if beta * R_E >> R1 || R2, the base current is negligible. Then V_B = V_CC * R2/(R1+R2), V_E = V_B - 0.7, I_C = V_E/R_E. This avoids writing full KVL equations.
- Op-amp virtual short method: label the non-inverting input voltage, then set the inverting input to the same value. Apply KCL at the inverting node with zero current into the terminal. Solve for V_out. Works for any linear configuration.
- g_m calculation shortcut: for BJT, g_m = I_C / 26 mV. For I_C = 1 mA, g_m = 38.5 mA/V. For I_C = 0.5 mA, g_m = 19.2 mA/V. Memorize the 1 mA value and scale linearly.
- MOSFET region identification: check V_DS >= V_GS - V_th for saturation. If V_GS < V_th the device is off. Always confirm region before writing the drain current equation.
- Miller capacitance shortcut: for an inverting amplifier with voltage gain A_v and feedback capacitance C_f, the effective input capacitance is C_f (1 - A_v). For A_v = -10 and C_f = 1 pF, C_Miller = 11 pF. This drastically reduces bandwidth.