Full List of Formulas
Ohm's Law
\(V = I \cdot R\)
where:
- \(V\) is the voltage across the resistor
- \(I\) is the current through the resistor
- \(R\) is the resistance
Kirchhoff's Voltage Law (KVL)
\(\sum V = 0\)
where:
- \(V\) represents the voltages around a closed loop
Kirchhoff's Current Law (KCL)
\(\sum I = 0\)
where:
- \(I\) represents the currents entering and leaving a junction
Thevenin's Theorem
\(V_{th} = V_{open}\)
where:
- \(V_{th}\) is the Thevenin equivalent voltage
- \(V_{open}\) is the open-circuit voltage
Norton's Theorem
\(I_{N} = \frac{V_{th}}{R_{th}}\)
where:
- \(I_{N}\) is the Norton equivalent current
- \(V_{th}\) is the Thevenin equivalent voltage
- \(R_{th}\) is the Thevenin equivalent resistance
Impedance of a Resistor
\(Z_R = R\)
where:
- \(Z_R\) is the impedance of the resistor
- \(R\) is the resistance
Impedance of an Inductor
\(Z_L = j\omega L\)
where:
- \(Z_L\) is the impedance of the inductor
- \(j\) is the imaginary unit
- \(\omega\) is the angular frequency
- \(L\) is the inductance
Impedance of a Capacitor
\(Z_C = \frac{1}{j\omega C}\)
where:
- \(Z_C\) is the impedance of the capacitor
- \(j\) is the imaginary unit
- \(\omega\) is the angular frequency
- \(C\) is the capacitance
Total Impedance in Series
\(Z_{total} = Z_1 + Z_2 + \dots + Z_n\)
where:
- \(Z_{total}\) is the total impedance
- \(Z_1, Z_2, \dots, Z_n\) are the individual impedances in series
Total Impedance in Parallel
\(\frac{1}{Z_{total}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \dots + \frac{1}{Z_n}\)
where:
- \(Z_{total}\) is the total impedance
- \(Z_1, Z_2, \dots, Z_n\) are the individual impedances in parallel
Voltage Divider
\(V_{out} = V_{in} \cdot \frac{R_2}{R_1 + R_2}\)
where:
- \(V_{out}\) is the output voltage
- \(V_{in}\) is the input voltage
- \(R_1\) and \(R_2\) are resistances in the voltage divider
Current Divider
\(I_{out} = I_{in} \cdot \frac{R_{total}}{R_{out}}\)
where:
- \(I_{out}\) is the output current
- \(I_{in}\) is the input current
- \(R_{total}\) is the total resistance
- \(R_{out}\) is the resistance through which the current is divided
Power in DC Circuits
\(P = V \cdot I\)
where:
- \(P\) is the power
- \(V\) is the voltage
- \(I\) is the current
Power in AC Circuits
\(P = V_{rms} \cdot I_{rms} \cdot \cos{\phi}\)
where:
- \(P\) is the real power
- \(V_{rms}\) is the root mean square voltage
- \(I_{rms}\) is the root mean square current
- \(\phi\) is the phase angle between voltage and current
Reactance of an Inductor
\(X_L = \omega L\)
where:
- \(X_L\) is the inductive reactance
- \(\omega\) is the angular frequency
- \(L\) is the inductance
Reactance of a Capacitor
\(X_C = \frac{1}{\omega C}\)
where:
- \(X_C\) is the capacitive reactance
- \(\omega\) is the angular frequency
- \(C\) is the capacitance
Admittance
\(Y = \frac{1}{Z}\)
where:
- \(Y\) is the admittance
- \(Z\) is the impedance
Resonance Frequency
\(f_0 = \frac{1}{2\pi\sqrt{LC}}\)
where:
- \(f_0\) is the resonance frequency
- \(L\) is the inductance
- \(C\) is the capacitance
Quality Factor (Q Factor)
\(Q = \frac{\omega_0 L}{R}\)
where:
- \(Q\) is the quality factor
- \(\omega_0\) is the resonance angular frequency
- \(L\) is the inductance
- \(R\) is the resistance
Phasor Representation
\(V = V_m \angle \theta\)
where:
- \(V\) is the phasor voltage
- \(V_m\) is the magnitude
- \(\theta\) is the phase angle
Complex Power
\(S = V \cdot I^\*\)
where:
- \(S\) is the complex power
- \(V\) is the voltage phasor
- \(I^\*\) is the complex conjugate of the current phasor
Maximum Power Transfer Theorem
\(R_{load} = R_{th}\)
where:
- \(R_{load}\) is the load resistance
- \(R_{th}\) is the Thevenin equivalent resistance
Superposition Theorem
\(V_{total} = \sum V_i\)
where:
- \(V_{total}\) is the total voltage
- \(V_i\) are the individual voltages from each independent source
Source Transformation
\(V_{th} \leftrightarrow I_{N}\)
where:
- \(V_{th}\) is the Thevenin voltage
- \(I_{N}\) is the Norton current
Frequency Response
\(H(j\omega) = \frac{V_{out}}{V_{in}}\)
where:
- \(H(j\omega)\) is the frequency response
- \(V_{out}\) is the output voltage
- \(V_{in}\) is the input voltage
- \(\omega\) is the angular frequency
Impulse Response
\(h(t) = \mathcal{L}^{-1}{H(s)}\)
where:
- \(h(t)\) is the impulse response
- \(\mathcal{L}^{-1}\) denotes the inverse Laplace transform
- \(H(s)\) is the system transfer function
Transfer Function
\(H(s) = \frac{Y(s)}{X(s)}\)
where:
- \(H(s)\) is the transfer function
- \(Y(s)\) is the output in the Laplace domain
- \(X(s)\) is the input in the Laplace domain
Laplace Transform of a Resistor
\(Z_R(s) = R\)
where:
- \(Z_R(s)\) is the Laplace impedance of the resistor
- \(R\) is the resistance
Laplace Transform of an Inductor
\(Z_L(s) = sL\)
where:
- \(Z_L(s)\) is the Laplace impedance of the inductor
- \(s\) is the complex frequency variable
- \(L\) is the inductance
Laplace Transform of a Capacitor
\(Z_C(s) = \frac{1}{sC}\)
where:
- \(Z_C(s)\) is the Laplace impedance of the capacitor
- \(s\) is the complex frequency variable
- \(C\) is the capacitance
Bode Plot Slope
\(Slope = 20n , \text{dB/decade}\)
where:
- \(Slope\) is the rate of change of the Bode plot
- \(n\) is the number of poles or zeros
Decibel Calculation
\(G_{dB} = 20 \log_{10}\left(\frac{V_{out}}{V_{in}}\right)\)
where:
- \(G_{dB}\) is the gain in decibels
- \(V_{out}\) is the output voltage
- \(V_{in}\) is the input voltage
Voltage Gain
\(A_v = \frac{V_{out}}{V_{in}}\)
where:
- \(A_v\) is the voltage gain
- \(V_{out}\) is the output voltage
- \(V_{in}\) is the input voltage
Current Gain
\(A_i = \frac{I_{out}}{I_{in}}\)
where:
- \(A_i\) is the current gain
- \(I_{out}\) is the output current
- \(I_{in}\) is the input current
Power Gain
\(A_p = \frac{P_{out}}{P_{in}}\)
where:
- \(A_p\) is the power gain
- \(P_{out}\) is the output power
- \(P_{in}\) is the input power
Signal-to-Noise Ratio (SNR)
\(\text{SNR} = \frac{P_{signal}}{P_{noise}}\)
where:
- \(\text{SNR}\) is the signal-to-noise ratio
- \(P_{signal}\) is the power of the signal
- \(P_{noise}\) is the power of the noise
Total Harmonic Distortion (THD)
\(\text{THD} = \frac{\sqrt{V_2^2 + V_3^2 + \dots + V_n^2}}{V_1}\)
where:
- \(\text{THD}\) is the total harmonic distortion
- \(V_1\) is the fundamental voltage
- \(V_2, V_3, \dots, V_n\) are the harmonic voltages
Common-Mode Rejection Ratio (CMRR)
\(\text{CMRR} = \frac{A_d}{A_c}\)
where:
- \(\text{CMRR}\) is the common-mode rejection ratio
- \(A_d\) is the differential gain
- \(A_c\) is the common-mode gain
Gain Bandwidth Product
\(GBW = A_v \cdot f_{3dB}\)
where:
- \(GBW\) is the gain-bandwidth product
- \(A_v\) is the voltage gain
- \(f_{3dB}\) is the -3 dB bandwidth frequency
Miller Theorem
\(C_{Miller} = C(1 - A_v)\)
where:
- \(C_{Miller}\) is the Miller capacitance
- \(C\) is the original capacitance
- \(A_v\) is the voltage gain
Slew Rate
\(\text{Slew Rate} = \frac{dV}{dt}\)
where:
- \(\text{Slew Rate}\) is the rate of change of the output voltage
- \(\frac{dV}{dt}\) is the derivative of voltage with respect to time
Differential Amplifier Gain
\(A_d = \frac{V_{out}}{V_{in+} - V_{in-}}}\)
where:
- \(A_d\) is the differential gain
- \(V_{out}\) is the output voltage
- \(V_{in+}\) and \(V_{in-}\) are the input voltages
Common-Mode Gain
\(A_c = \frac{V_{out}}{\frac{V_{in+} + V_{in-}}{2}}\)
where:
- \(A_c\) is the common-mode gain
- \(V_{out}\) is the output voltage
- \(V_{in+}\) and \(V_{in-}\) are the input voltages
Input Impedance
\(Z_{in} = \frac{V_{in}}{I_{in}}\)
where:
- \(Z_{in}\) is the input impedance
- \(V_{in}\) is the input voltage
- \(I_{in}\) is the input current
Output Impedance
\(Z_{out} = \frac{V_{out}}{I_{out}}\)
where:
- \(Z_{out}\) is the output impedance
- \(V_{out}\) is the output voltage
- \(I_{out}\) is the output current
Feedback Factor
\(\beta = \frac{V_{feedback}}{V_{out}}\)
where:
- \(\beta\) is the feedback factor
- \(V_{feedback}\) is the feedback voltage
- \(V_{out}\) is the output voltage
Closed-Loop Gain with Feedback
\(A_{closed} = \frac{A}{1 + A\beta}\)
where:
- \(A_{closed}\) is the closed-loop gain
- \(A\) is the open-loop gain
- \(\beta\) is the feedback factor
Stability Criterion (Nyquist)
\(\text{Stability requires that the Nyquist plot does not encircle the -1 point}\)
where:
- Stability is determined by the Nyquist plot
- The -1 point is the critical point for encirclement
Bode Stability Criterion
\(\text{Phase margin} > 0^\circ \text{ and } \text{Gain margin} > 0 \text{ dB}\)
where:
- Phase margin is the additional phase needed for stability
- Gain margin is the additional gain before instability
Frequency Response of a First-Order Low-Pass Filter
\(H(j\omega) = \frac{1}{1 + j\omega RC}\)
where:
- \(H(j\omega)\) is the transfer function
- \(\omega\) is the angular frequency
- \(R\) is the resistance
- \(C\) is the capacitance
Frequency Response of a First-Order High-Pass Filter
\(H(j\omega) = \frac{j\omega RC}{1 + j\omega RC}\)
where:
- \(H(j\omega)\) is the transfer function
- \(\omega\) is the angular frequency
- \(R\) is the resistance
- \(C\) is the capacitance
Voltage Gain of a Common-Emitter Amplifier
\(A_v = -g_m R_C\)
where:
- \(A_v\) is the voltage gain
- \(g_m\) is the transconductance
- \(R_C\) is the collector resistor
Transconductance
\(g_m = \frac{I_C}{V_T}\)
where:
- \(g_m\) is the transconductance
- \(I_C\) is the collector current
- \(V_T\) is the thermal voltage
Power Supply Rejection Ratio (PSRR)
\(\text{PSRR} = \frac{\Delta V_{CC}}{\Delta V_{out}}\)
where:
- \(\text{PSRR}\) is the power supply rejection ratio
- \(\Delta V_{CC}\) is the change in supply voltage
- \(\Delta V_{out}\) is the change in output voltage
Differential Mode Gain
\(A_d = \frac{V_{out}}{V_{in+} - V_{in-}}\)
where:
- \(A_d\) is the differential mode gain
- \(V_{out}\) is the output voltage
- \(V_{in+}\) and \(V_{in-}\) are the input voltages
Common-Mode Rejection Ratio (CMRR)
\(\text{CMRR} = \frac{A_d}{A_c}\)
where:
- \(\text{CMRR}\) is the common-mode rejection ratio
- \(A_d\) is the differential gain
- \(A_c\) is the common-mode gain
Differential Pair Gain
\(A = \frac{g_m R_D}{2}\)
where:
- \(A\) is the differential pair gain
- \(g_m\) is the transconductance
- \(R_D\) is the drain resistor
Bandwidth of an Amplifier
\(BW = \frac{GBW}{A_v}\)
where:
- \(BW\) is the bandwidth
- \(GBW\) is the gain-bandwidth product
- \(A_v\) is the voltage gain
Slew Rate Limitation
\(\text{Slew Rate} = \frac{dV_{out}}{dt} \leq \text{Maximum Slew Rate}\)
where:
- \(\text{Slew Rate}\) is the rate of change of the output voltage
- \(\frac{dV_{out}}{dt}\) is the derivative of output voltage with respect to time
Noise Figure
\(NF = \frac{\text{SNR}*{in}}{\text{SNR}*{out}}}\)
where:
- \(NF\) is the noise figure
- \(\text{SNR}_{in}\) is the signal-to-noise ratio at the input
- \(\text{SNR}_{out}\) is the signal-to-noise ratio at the output
Total Power in a Resonant Circuit
\(P_{total} = \frac{V^2}{R}\)
where:
- \(P_{total}\) is the total power
- \(V\) is the voltage
- \(R\) is the resistance
Voltage Gain of an Operational Amplifier
\(A_v = \frac{V_{out}}{V_{in}}\)
where:
- \(A_v\) is the voltage gain
- \(V_{out}\) is the output voltage
- \(V_{in}\) is the input voltage
Bootstrap Circuit Gain
\(A_v = \frac{1 + R_2/R_1}{1}\)
where:
- \(A_v\) is the voltage gain
- \(R_1\) and \(R_2\) are resistor values in the bootstrap circuit
Feedback Network Gain
\(\beta = \frac{R_f}{R_f + R_i}\)
where:
- \(\beta\) is the feedback factor
- \(R_f\) is the feedback resistor
- \(R_i\) is the input resistor
Differential Amplifier Output
\(V_{out} = A_d (V_{in+} - V_{in-}) + A_c \left(\frac{V_{in+} + V_{in-}}{2}\right)\)
where:
- \(V_{out}\) is the output voltage
- \(A_d\) is the differential gain
- \(A_c\) is the common-mode gain
- \(V_{in+}\) and \(V_{in-}\) are the input voltages
Voltage Gain of a Differential Amplifier
\(A_v = \frac{V_{out}}{V_{in+} - V_{in-}}}\)
where:
- \(A_v\) is the voltage gain
- \(V_{out}\) is the output voltage
- \(V_{in+}\) and \(V_{in-}\) are the input voltages
Cascaded Amplifier Gain
\(A_{total} = A_1 \cdot A_2 \cdot \dots \cdot A_n\)
where:
- \(A_{total}\) is the total gain
- \(A_1, A_2, \dots, A_n\) are the individual stage gains
Power Delivered to a Load
\(P_L = \frac{V_L^2}{R_L}\)
where:
- \(P_L\) is the power delivered to the load
- \(V_L\) is the voltage across the load
- \(R_L\) is the load resistance
Transfer Function of a RLC Circuit
\(H(s) = \frac{V_{out}(s)}{V_{in}(s)} = \frac{1}{LC s^2 + RC s + 1}\)
where:
- \(H(s)\) is the transfer function
- \(V_{out}(s)\) is the output voltage in the Laplace domain
- \(V_{in}(s)\) is the input voltage in the Laplace domain
- \(L\) is the inductance
- \(C\) is the capacitance
- \(R\) is the resistance
Voltage Gain of a Non-Inverting Amplifier
\(A_v = 1 + \frac{R_f}{R_i}\)
where:
- \(A_v\) is the voltage gain
- \(R_f\) is the feedback resistor
- \(R_i\) is the input resistor
Voltage Gain of an Inverting Amplifier
\(A_v = -\frac{R_f}{R_i}\)
where:
- \(A_v\) is the voltage gain
- \(R_f\) is the feedback resistor
- \(R_i\) is the input resistor
Common-Mode Gain of an Operational Amplifier
\(A_c = \frac{V_{out}}{\frac{V_{in+} + V_{in-}}{2}}\)
where:
- \(A_c\) is the common-mode gain
- \(V_{out}\) is the output voltage
- \(V_{in+}\) and \(V_{in-}\) are the input voltages
Differential Mode Gain of an Operational Amplifier
\(A_d = \frac{V_{out}}{V_{in+} - V_{in-}}}\)
where:
- \(A_d\) is the differential mode gain
- \(V_{out}\) is the output voltage
- \(V_{in+}\) and \(V_{in-}\) are the input voltages
Phase Margin
\(\text{Phase Margin} = 180^\circ + \angle H(j\omega_{gc})\)
where:
- \(\text{Phase Margin}\) is the additional phase needed for stability
- \(\angle H(j\omega_{gc})\) is the phase angle at the gain crossover frequency
Gain Margin
\(\text{Gain Margin} = \frac{1}{|H(j\omega_{pc})|}\)
where:
- \(\text{Gain Margin}\) is the additional gain before instability
- \(|H(j\omega_{pc})|\) is the magnitude of the transfer function at the phase crossover frequency
Root Locus Equation
\(1 + A \beta = 0\)
where:
- \(A\) is the open-loop gain
- \(\beta\) is the feedback factor
Nyquist Stability Criterion
\(\text{Number of encirclements of -1} = \text{Number of right-half-plane poles}\)
where:
- Encirclements are determined by the Nyquist plot
- Right-half-plane poles affect stability
Bode Gain Equation
\(20 \log_{10}(A_v) = 20 \log_{10}\left(\frac{V_{out}}{V_{in}}\right)\)
where:
- \(A_v\) is the voltage gain
- \(V_{out}\) is the output voltage
- \(V_{in}\) is the input voltage
Differential Pair Current Equation
\(I = \frac{V_{CC} - V_{BE}}{R_E}\)
where:
- \(I\) is the current through the differential pair
- \(V_{CC}\) is the supply voltage
- \(V_{BE}\) is the base-emitter voltage
- \(R_E\) is the emitter resistor
Common-Mode Input Voltage Range
\(V_{CM} = \frac{V_{in+} + V_{in-}}{2}\)
where:
- \(V_{CM}\) is the common-mode input voltage
- \(V_{in+}\) and \(V_{in-}\) are the input voltages
Differential Input Voltage
\(V_{diff} = V_{in+} - V_{in-}\)
where:
- \(V_{diff}\) is the differential input voltage
- \(V_{in+}\) and \(V_{in-}\) are the input voltages
Voltage Swing
\(V_{swing} = V_{out(max)} - V_{out(min)}\)
where:
- \(V_{swing}\) is the voltage swing
- \(V_{out(max)}\) is the maximum output voltage
- \(V_{out(min)}\) is the minimum output voltage
Thermal Noise Voltage
\(e_n = \sqrt{4kTR\Delta f}\)
where:
- \(e_n\) is the thermal noise voltage
- \(k\) is Boltzmann's constant
- \(T\) is the absolute temperature
- \(R\) is the resistance
- \(\Delta f\) is the bandwidth
Shot Noise Current
\(i_n = \sqrt{2qI\Delta f}\)
where:
- \(i_n\) is the shot noise current
- \(q\) is the elementary charge
- \(I\) is the current
- \(\Delta f\) is the bandwidth
Flicker Noise (1/f Noise)
\(S_v(f) = \frac{K}{f}\)
where:
- \(S_v(f)\) is the flicker noise spectral density
- \(K\) is a constant
- \(f\) is the frequency
Thermal Voltage
\(V_T = \frac{kT}{q}\)
where:
- \(V_T\) is the thermal voltage
- \(k\) is Boltzmann's constant
- \(T\) is the absolute temperature
- \(q\) is the elementary charge
Bandwidth of a Differential Amplifier
\(BW = \frac{GBW}{A_d}\)
where:
- \(BW\) is the bandwidth
- \(GBW\) is the gain-bandwidth product
- \(A_d\) is the differential gain
Noise Figure of an Amplifier
\(NF = 10 \log_{10}\left(\frac{\text{SNR}*{in}}{\text{SNR}*{out}}\right)\)
where:
- \(NF\) is the noise figure
- \(\text{SNR}_{in}\) is the input signal-to-noise ratio
- \(\text{SNR}_{out}\) is the output signal-to-noise ratio
Common-Mode Rejection Ratio (CMRR) in dB
\(\text{CMRR}*{dB} = 20 \log*{10}\left(\frac{A_d}{A_c}\right)\)
where:
- \(\text{CMRR}_{dB}\) is the common-mode rejection ratio in decibels
- \(A_d\) is the differential gain
- \(A_c\) is the common-mode gain
Common-Mode Gain in Terms of CMRR
\(A_c = \frac{A_d}{\text{CMRR}}\)
where:
- \(A_c\) is the common-mode gain
- \(A_d\) is the differential gain
- \(\text{CMRR}\) is the common-mode rejection ratio
Signal Gain with Feedback
\(A_{feedback} = \frac{A}{1 + A\beta}\)
where:
- \(A_{feedback}\) is the gain with feedback
- \(A\) is the open-loop gain
- \(\beta\) is the feedback factor
Loop Gain
\(T = A \beta\)
where:
- \(T\) is the loop gain
- \(A\) is the open-loop gain
- \(\beta\) is the feedback factor
Stability Factor (K)
\(K = \frac{1 + \beta A}{1 - \beta A}\)
where:
- \(K\) is the stability factor
- \(\beta\) is the feedback factor
- \(A\) is the open-loop gain
Nyquist Criterion for Stability
\(\text{System is stable if the Nyquist plot does not encircle the -1 point}\)
where:
- Stability is determined by the Nyquist plot
- The -1 point is the critical point
Pole-Zero Cancellation
\(H(s) = \frac{(s - z_1)(s - z_2)}{(s - p_1)(s - p_2)}\)
where:
- \(H(s)\) is the transfer function
- \(z_1, z_2\) are zeros
- \(p_1, p_2\) are poles
Bode Plot Phase Equation
\(\phi(\omega) = \angle H(j\omega)\)
where:
- \(\phi(\omega)\) is the phase angle
- \(H(j\omega)\) is the transfer function
Bode Plot Magnitude Equation
\(|H(j\omega)| = 20 \log_{10}\left(\frac{V_{out}}{V_{in}}\right)\)
where:
- \(|H(j\omega)|\) is the magnitude of the transfer function
- \(V_{out}\) is the output voltage
- \(V_{in}\) is the input voltage
Euler's Formula for Phasors
\(V = V_m e^{j\theta}\)
where:
- \(V\) is the phasor voltage
- \(V_m\) is the magnitude
- \(\theta\) is the phase angle
- \(j\) is the imaginary unit
Resonant Frequency of an RLC Circuit
\(f_0 = \frac{1}{2\pi\sqrt{LC}}\)
where:
- \(f_0\) is the resonant frequency
- \(L\) is the inductance
- \(C\) is the capacitance
Damping Factor
\(\zeta = \frac{R}{2} \sqrt{\frac{C}{L}}\)
where:
- \(\zeta\) is the damping factor
- \(R\) is the resistance
- \(C\) is the capacitance
- \(L\) is the inductance
Natural Frequency
\(\omega_n = \sqrt{\frac{1}{LC}}\)
where:
- \(\omega_n\) is the natural angular frequency
- \(L\) is the inductance
- \(C\) is the capacitance
Damped Frequency
\(\omega_d = \omega_n \sqrt{1 - \zeta^2}\)
where:
- \(\omega_d\) is the damped angular frequency
- \(\omega_n\) is the natural angular frequency
- \(\zeta\) is the damping factor
Transfer Function of a Second-Order System
\(H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\)
where:
- \(H(s)\) is the transfer function
- \(\omega_n\) is the natural angular frequency
- \(\zeta\) is the damping factor
- \(s\) is the complex frequency variable
Voltage Gain of a Cascode Amplifier
\(A_v = -g_m R_C\)
where:
- \(A_v\) is the voltage gain
- \(g_m\) is the transconductance
- \(R_C\) is the collector resistor
Common-Mode Rejection Ratio (CMRR) in Terms of Differential and Common-Mode Gain
\(\text{CMRR} = \frac{A_d}{A_c}\)
where:
- \(\text{CMRR}\) is the common-mode rejection ratio
- \(A_d\) is the differential gain
- \(A_c\) is the common-mode gain
Total Harmonic Distortion (THD) in Terms of Harmonics
\(\text{THD} = \frac{\sqrt{V_2^2 + V_3^2 + \dots + V_n^2}}{V_1}\)
where:
- \(\text{THD}\) is the total harmonic distortion
- \(V_1\) is the fundamental voltage
- \(V_2, V_3, \dots, V_n\) are the harmonic voltages
Noise Figure in Decibels
\(NF_{dB} = 10 \log_{10}(NF)\)
where:
- \(NF_{dB}\) is the noise figure in decibels
- \(NF\) is the noise figure
Thermal Noise Power
\(P_n = kTB\)
where:
- \(P_n\) is the thermal noise power
- \(k\) is Boltzmann's constant
- \(T\) is the absolute temperature
- \(B\) is the bandwidth
Johnson-Nyquist Noise Voltage
\(e_n = \sqrt{4kTRB}\)
where:
- \(e_n\) is the Johnson-Nyquist noise voltage
- \(k\) is Boltzmann's constant
- \(T\) is the absolute temperature
- \(R\) is the resistance
- \(B\) is the bandwidth
Avalanche Noise Factor
\(F = \frac{SNR_{in}}{SNR_{out}}\)
where:
- \(F\) is the avalanche noise factor
- \(SNR_{in}\) is the input signal-to-noise ratio
- \(SNR_{out}\) is the output signal-to-noise ratio
Shot Noise Spectral Density
\(S_i = 2qI\)
where:
- \(S_i\) is the shot noise spectral density
- \(q\) is the elementary charge
- \(I\) is the current
Flicker Noise Spectral Density
\(S_v = \frac{K}{f}\)
where:
- \(S_v\) is the flicker noise spectral density
- \(K\) is a constant
- \(f\) is the frequency
Equivalent Noise Resistance
\(R_n = \frac{e_n^2}{4kTB}\)
where:
- \(R_n\) is the equivalent noise resistance
- \(e_n\) is the noise voltage
- \(k\) is Boltzmann's constant
- \(T\) is the absolute temperature
- \(B\) is the bandwidth
Signal-to-Noise Ratio (SNR) in Decibels
\(\text{SNR}*{dB} = 10 \log*{10}\left(\frac{P_{signal}}{P_{noise}}\right)\)
where:
- \(\text{SNR}_{dB}\) is the signal-to-noise ratio in decibels
- \(P_{signal}\) is the signal power
- \(P_{noise}\) is the noise power
Noise Figure in Terms of Noise Factors
\(NF = F \cdot G\)
where:
- \(NF\) is the noise figure
- \(F\) is the noise factor of a component
- \(G\) is the gain of the component
Cascade Noise Formula
\(NF_{total} = NF_1 + \frac{NF_2 - 1}{G_1} + \frac{NF_3 - 1}{G_1 G_2} + \dots\)
where:
- \(NF_{total}\) is the total noise figure
- \(NF_1, NF_2, \dots\) are the noise figures of individual stages
- \(G_1, G_2, \dots\) are the gains of individual stages
Noise Figure of a Two-Stage Amplifier
\(NF_{total} = NF_1 + \frac{NF_2 - 1}{G_1}\)
where:
- \(NF_{total}\) is the total noise figure
- \(NF_1\) is the noise figure of the first stage
- \(NF_2\) is the noise figure of the second stage
- \(G_1\) is the gain of the first stage
Friis Formula for Noise
\(NF_{total} = NF_1 + \frac{NF_2 - 1}{G_1} + \frac{NF_3 - 1}{G_1 G_2} + \dots\)
where:
- \(NF_{total}\) is the total noise figure
- \(NF_1, NF_2, \dots\) are the noise figures of individual stages
- \(G_1, G_2, \dots\) are the gains of individual stages
Power Gain with Noise
\(G_p = \frac{P_{out}}{P_{in}}\)
where:
- \(G_p\) is the power gain
- \(P_{out}\) is the output power
- \(P_{in}\) is the input power
Voltage Gain with Noise
\(A_v = \frac{V_{out}}{V_{in}}\)
where:
- \(A_v\) is the voltage gain
- \(V_{out}\) is the output voltage
- \(V_{in}\) is the input voltage
Current Gain with Noise
\(A_i = \frac{I_{out}}{I_{in}}\)
where:
- \(A_i\) is the current gain
- \(I_{out}\) is the output current
- \(I_{in}\) is the input current
Feedback Factor in Terms of Resistors
\(\beta = \frac{R_1}{R_1 + R_2}\)
where:
- \(\beta\) is the feedback factor
- \(R_1\) and \(R_2\) are resistors in the feedback network
Phase Shift in an RC Circuit
\(\phi = \tan^{-1}\left(-\frac{1}{\omega RC}\right)\)
where:
- \(\phi\) is the phase shift
- \(\omega\) is the angular frequency
- \(R\) is the resistance
- \(C\) is the capacitance
Phase Shift in an RL Circuit
\(\phi = \tan^{-1}(\omega L / R)\)
where:
- \(\phi\) is the phase shift
- \(\omega\) is the angular frequency
- \(L\) is the inductance
- \(R\) is the resistance
Total Phase Shift in a Series RLC Circuit
\(\phi = \tan^{-1}\left(\frac{\omega L - 1/\omega C}{R}\right)\)
where:
- \(\phi\) is the total phase shift
- \(\omega\) is the angular frequency
- \(L\) is the inductance
- \(C\) is the capacitance
- \(R\) is the resistance
Voltage Gain of a Differential Amplifier with Feedback
\(A_v = \frac{A_d}{1 + A_d \beta}\)
where:
- \(A_v\) is the voltage gain with feedback
- \(A_d\) is the differential gain
- \(\beta\) is the feedback factor
Common-Mode Gain with Feedback
\(A_{c,feedback} = \frac{A_c}{1 + A_d \beta}\)
where:
- \(A_{c,feedback}\) is the common-mode gain with feedback
- \(A_c\) is the common-mode gain
- \(A_d\) is the differential gain
- \(\beta\) is the feedback factor
Loop Gain for Stability
\(T = A \beta\)
where:
- \(T\) is the loop gain
- \(A\) is the open-loop gain
- \(\beta\) is the feedback factor
Gain-Bandwidth Product (GBW)
\(GBW = A_v \cdot f_{3dB}\)
where:
- \(GBW\) is the gain-bandwidth product
- \(A_v\) is the voltage gain
- \(f_{3dB}\) is the -3 dB bandwidth frequency
Slew Rate Limitation in Amplifiers
\(\text{Slew Rate} = \frac{dV_{out}}{dt} \leq \text{Max Slew Rate}\)
where:
- \(\text{Slew Rate}\) is the rate of change of the output voltage
- \(\frac{dV_{out}}{dt}\) is the derivative of output voltage with respect to time
Clipping in Amplifiers
\(V_{out} = V_{saturation}\)
where:
- \(V_{out}\) is the output voltage
- \(V_{saturation}\) is the saturation voltage of the amplifier
Distortion Factor
\(\text{Distortion} = \frac{V_{out,distorted} - V_{out,ideal}}{V_{out,ideal}}\)
where:
- \(\text{Distortion}\) is the distortion factor
- \(V_{out,distorted}\) is the distorted output voltage
- \(V_{out,ideal}\) is the ideal output voltage
Load Line Analysis
\(V = V_{CC} - I R\)
where:
- \(V\) is the voltage across the load
- \(V_{CC}\) is the supply voltage
- \(I\) is the current through the load
- \(R\) is the resistance
Q Factor of a Resonant Circuit
\(Q = \frac{\omega_0 L}{R}\)
where:
- \(Q\) is the quality factor
- \(\omega_0\) is the resonant angular frequency
- \(L\) is the inductance
- \(R\) is the resistance
Bandwidth of a Resonant Circuit
\(BW = \frac{\omega_0}{Q}\)
where:
- \(BW\) is the bandwidth
- \(\omega_0\) is the resonant angular frequency
- \(Q\) is the quality factor
Transimpedance Gain
\(Z_t = \frac{V_{out}}{I_{in}}\)
where:
- \(Z_t\) is the transimpedance gain
- \(V_{out}\) is the output voltage
- \(I_{in}\) is the input current
Transadmittance
\(Y_t = \frac{I_{out}}{V_{in}}\)
where:
- \(Y_t\) is the transadmittance
- \(I_{out}\) is the output current
- \(V_{in}\) is the input voltage
Voltage Transfer Characteristic
\(V_{out} = f(V_{in})\)
where:
- \(V_{out}\) is the output voltage
- \(V_{in}\) is the input voltage
- \(f\) is the transfer function
Current Transfer Characteristic
\(I_{out} = f(I_{in})\)
where:
- \(I_{out}\) is the output current
- \(I_{in}\) is the input current
- \(f\) is the transfer function
Hysteresis in Amplifiers
\(V_{out} = \begin{cases} V_{high} & \text{if } V_{in} > V_{th} \ V_{low} & \text{if } V_{in} < V_{tl} \end{cases}\)
where:
- \(V_{out}\) is the output voltage
- \(V_{high}\) and \(V_{low}\) are the high and low output levels
- \(V_{th}\) and \(V_{tl}\) are the upper and lower threshold voltages
Schmitt Trigger Transfer Function
\(V_{out} = \begin{cases} V_{high} & \text{if } V_{in} > V_{th} \ V_{low} & \text{if } V_{in} < V_{tl} \end{cases}\)
where:
- \(V_{out}\) is the output voltage
- \(V_{high}\) and \(V_{low}\) are the high and low output levels
- \(V_{th}\) and \(V_{tl}\) are the upper and lower threshold voltages
Transfer Function of an Op-Amp Inverter
\(H(s) = -\frac{R_f}{R_{in}}\)
where:
- \(H(s)\) is the transfer function
- \(R_f\) is the feedback resistor
- \(R_{in}\) is the input resistor
Transfer Function of an Op-Amp Non-Inverter
\(H(s) = 1 + \frac{R_f}{R_{in}}\)
where:
- \(H(s)\) is the transfer function
- \(R_f\) is the feedback resistor
- \(R_{in}\) is the input resistor
Common-Mode Gain Reduction with Feedback
\(A_{c,feedback} = \frac{A_c}{1 + A_d \beta}\)
where:
- \(A_{c,feedback}\) is the common-mode gain with feedback
- \(A_c\) is the original common-mode gain
- \(A_d\) is the differential gain
- \(\beta\) is the feedback factor
Differential Gain Reduction with Feedback
\(A_{d,feedback} = \frac{A_d}{1 + A_d \beta}\)
where:
- \(A_{d,feedback}\) is the differential gain with feedback
- \(A_d\) is the original differential gain
- \(\beta\) is the feedback factor
Phase Shift in a Feedback Network
\(\phi = \angle(1 + A\beta)\)
where:
- \(\phi\) is the phase shift
- \(A\) is the open-loop gain
- \(\beta\) is the feedback factor
Stability in Feedback Systems
\(\text{System is stable if } 1 + A\beta \neq 0 \text{ for all frequencies}\)
where:
- \(A\) is the open-loop gain
- \(\beta\) is the feedback factor
Loop Gain and Stability
\(T = A\beta\)
where:
- \(T\) is the loop gain
- \(A\) is the open-loop gain
- \(\beta\) is the feedback factor
Feedback Factor in a Voltage Divider
\(\beta = \frac{R_2}{R_1 + R_2}\)
where:
- \(\beta\) is the feedback factor
- \(R_1\) and \(R_2\) are resistors in the voltage divider
Feedback Factor in a Current Divider
\(\beta = \frac{R_1}{R_1 + R_2}\)
where:
- \(\beta\) is the feedback factor
- \(R_1\) and \(R_2\) are resistors in the current divider
Closed-Loop Bandwidth
\(BW_{closed} = \frac{GBW}{A_{closed}}\)
where:
- \(BW_{closed}\) is the closed-loop bandwidth
- \(GBW\) is the gain-bandwidth product
- \(A_{closed}\) is the closed-loop gain
Unity Gain Bandwidth
\(f_{unity} = GBW\)
where:
- \(f_{unity}\) is the unity gain bandwidth
- \(GBW\) is the gain-bandwidth product
Phase Margin Definition
\(\text{Phase Margin} = 180^\circ + \angle H(j\omega_{gc})\)
where:
- \(\text{Phase Margin}\) is the phase margin
- \(\angle H(j\omega_{gc})\) is the phase angle at the gain crossover frequency
Gain Margin Definition
\(\text{Gain Margin} = \frac{1}{|H(j\omega_{pc})|}\)
where:
- \(\text{Gain Margin}\) is the gain margin
- \(|H(j\omega_{pc})|\) is the magnitude of the transfer function at the phase crossover frequency
Nyquist Plot Encirclements
\(\text{Number of encirclements} = P - N\)
where:
- \(P\) is the number of poles in the right half-plane
- \(N\) is the number of encirclements of the -1 point
Root Locus on Real Axis
\(\text{Segments on the real axis are to the left of an odd number of poles and zeros}\)
where:
- Real axis segments are determined by the number of poles and zeros to their right
Transfer Function of an Inverting Amplifier
\(H(s) = -\frac{R_f}{R_{in}}\)
where:
- \(H(s)\) is the transfer function
- \(R_f\) is the feedback resistor
- \(R_{in}\) is the input resistor
Transfer Function of a Non-Inverting Amplifier
\(H(s) = 1 + \frac{R_f}{R_{in}}\)
where:
- \(H(s)\) is the transfer function
- \(R_f\) is the feedback resistor
- \(R_{in}\) is the input resistor
Voltage Gain of a Common-Collector Amplifier
\(A_v \approx 1\)
where:
- \(A_v\) is the voltage gain
- Common-collector amplifiers have a voltage gain close to 1
Voltage Gain of a Common-Base Amplifier
\(A_v \approx \frac{R_C}{r_e}\)
where:
- \(A_v\) is the voltage gain
- \(R_C\) is the collector resistor
- \(r_e\) is the emitter resistance
Power Gain of a Common-Emitter Amplifier
\(A_p = A_v \cdot A_i\)
where:
- \(A_p\) is the power gain
- \(A_v\) is the voltage gain
- \(A_i\) is the current gain
Power Gain of a Common-Collector Amplifier
\(A_p \approx \beta\)
where:
- \(A_p\) is the power gain
- \(\beta\) is the current gain
Miller Effect in Feedback Amplifiers
\(C_M = C(1 - A_v)\)
where:
- \(C_M\) is the Miller capacitance
- \(C\) is the original capacitance
- \(A_v\) is the voltage gain
Voltage Gain with Miller Capacitance
\(A_v = \frac{1}{1 + j\omega C_M R}\)
where:
- \(A_v\) is the voltage gain
- \(\omega\) is the angular frequency
- \(C_M\) is the Miller capacitance
- \(R\) is the resistance
Differential Amplifier Common-Mode Gain Reduction
\(A_{c,feedback} = \frac{A_c}{1 + A_d \beta}\)
where:
- \(A_{c,feedback}\) is the common-mode gain with feedback
- \(A_c\) is the common-mode gain
- \(A_d\) is the differential gain
- \(\beta\) is the feedback factor
Voltage Gain of a Differential Amplifier with Feedback
\(A_{v} = \frac{A_d}{1 + A_d \beta}\)
where:
- \(A_{v}\) is the voltage gain with feedback
- \(A_d\) is the differential gain
- \(\beta\) is the feedback factor
Output Impedance with Feedback
\(Z_{out,feedback} = \frac{Z_{out}}{1 + A\beta}\)
where:
- \(Z_{out,feedback}\) is the output impedance with feedback
- \(Z_{out}\) is the original output impedance
- \(A\) is the open-loop gain
- \(\beta\) is the feedback factor
Input Impedance with Feedback
\(Z_{in,feedback} = Z_{in} \cdot (1 + A\beta)\)
where:
- \(Z_{in,feedback}\) is the input impedance with feedback
- \(Z_{in}\) is the original input impedance
- \(A\) is the open-loop gain
- \(\beta\) is the feedback factor
Total Harmonic Distortion (THD) in dB
\(\text{THD}*{dB} = 20 \log*{10}\left(\frac{V_{total,harmonics}}{V_1}\right)\)
where:
- \(\text{THD}_{dB}\) is the total harmonic distortion in decibels
- \(V_{total,harmonics}\) is the total voltage of harmonics
- \(V_1\) is the fundamental voltage
Signal Integrity in High-Frequency Circuits
\(|H(j\omega)| = \frac{V_{out}}{V_{in}}\)
where:
- \(|H(j\omega)|\) is the magnitude of the transfer function
- \(V_{out}\) is the output voltage
- \(V_{in}\) is the input voltage
Reflection Coefficient
\(\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}\)
where:
- \(\Gamma\) is the reflection coefficient
- \(Z_L\) is the load impedance
- \(Z_0\) is the characteristic impedance
Standing Wave Ratio (SWR)
\(SWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}\)
where:
- \(SWR\) is the standing wave ratio
- \(\Gamma\) is the reflection coefficient
Impedance Matching Condition
\(Z_L = Z_0^\*\)
where:
- \(Z_L\) is the load impedance
- \(Z_0^\*\) is the complex conjugate of the characteristic impedance
Power Delivered to a Matched Load
\(P_L = \frac{|V_{in}|^2}{4 Z_0}\)
where:
- \(P_L\) is the power delivered to the load
- \(V_{in}\) is the input voltage
- \(Z_0\) is the characteristic impedance
Transmission Line Voltage Equation
\(V(z) = V^+ e^{-j\beta z} + V^- e^{j\beta z}\)
where:
- \(V(z)\) is the voltage at position \(z\)
- \(V^+\) is the forward traveling voltage wave
- \(V^-\) is the backward traveling voltage wave
- \(\beta\) is the phase constant
- \(z\) is the position along the transmission line
Transmission Line Current Equation
\(I(z) = \frac{V^+}{Z_0} e^{-j\beta z} - \frac{V^-}{Z_0} e^{j\beta z}\)
where:
- \(I(z)\) is the current at position \(z\)
- \(V^+\) is the forward traveling voltage wave
- \(V^-\) is the backward traveling voltage wave
- \(Z_0\) is the characteristic impedance
- \(\beta\) is the phase constant
- \(z\) is the position along the transmission line
Characteristic Impedance of a Transmission Line
\(Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}\)
where:
- \(Z_0\) is the characteristic impedance
- \(R\) is the resistance per unit length
- \(L\) is the inductance per unit length
- \(G\) is the conductance per unit length
- \(C\) is the capacitance per unit length
- \(\omega\) is the angular frequency
Propagation Constant
\(\gamma = \alpha + j\beta = \sqrt{(R + j\omega L)(G + j\omega C)}\)
where:
- \(\gamma\) is the propagation constant
- \(\alpha\) is the attenuation constant
- \(\beta\) is the phase constant
- \(R\), \(L\), \(G\), \(C\) are transmission line parameters
- \(\omega\) is the angular frequency
Attenuation Constant
\(\alpha = \text{Re}(\gamma)\)
where:
- \(\alpha\) is the attenuation constant
- \(\gamma\) is the propagation constant
Phase Constant
\(\beta = \text{Im}(\gamma)\)
where:
- \(\beta\) is the phase constant
- \(\gamma\) is the propagation constant
Voltage Reflection Coefficient
\(\Gamma_V = \frac{V^-}{V^+}\)
where:
- \(\Gamma_V\) is the voltage reflection coefficient
- \(V^+\) is the forward traveling voltage wave
- \(V^-\) is the backward traveling voltage wave
Current Reflection Coefficient
\(\Gamma_I = \frac{I^-}{I^+}\)
where:
- \(\Gamma_I\) is the current reflection coefficient
- \(I^+\) is the forward traveling current wave
- \(I^-\) is the backward traveling current wave
Power Reflection Coefficient
\(\Gamma_P = |\Gamma_V|^2\)
where:
- \(\Gamma_P\) is the power reflection coefficient
- \(\Gamma_V\) is the voltage reflection coefficient
Minimum SWR
\(\text{SWR}_{min} = 1\)
where:
- \(\text{SWR}_{min}\) is the minimum standing wave ratio
Maximum SWR
\(\text{SWR}_{max} = \infty\)
where:
- \(\text{SWR}_{max}\) is the maximum standing wave ratio
Voltage Standing Wave Ratio (VSWR)
\(\text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}\)
where:
- \(\text{VSWR}\) is the voltage standing wave ratio
- \(\Gamma\) is the reflection coefficient
Power Transfer in Transmission Lines
\(P = \frac{|V^+|^2}{2 Z_0} (1 - |\Gamma|^2)\)
where:
- \(P\) is the power transferred
- \(V^+\) is the forward traveling voltage wave
- \(Z_0\) is the characteristic impedance
- \(\Gamma\) is the reflection coefficient
Propagation Delay
\(\tau = \frac{l}{v_p}\)
where:
- \(\tau\) is the propagation delay
- \(l\) is the length of the transmission line
- \(v_p\) is the phase velocity
Phase Velocity
\(v_p = \frac{\omega}{\beta}\)
where:
- \(v_p\) is the phase velocity
- \(\omega\) is the angular frequency
- \(\beta\) is the phase constant
Group Velocity
\(v_g = \frac{d\omega}{d\beta}\)
where:
- \(v_g\) is the group velocity
- \(\omega\) is the angular frequency
- \(\beta\) is the phase constant
Velocity Factor
\(VF = \frac{v_p}{c}\)
where:
- \(VF\) is the velocity factor
- \(v_p\) is the phase velocity
- \(c\) is the speed of light
Electrical Length of a Transmission Line
\(\theta = \beta l\)
where:
- \(\theta\) is the electrical length
- \(\beta\) is the phase constant
- \(l\) is the physical length of the transmission line
Reflection Loss
\(\text{Reflection Loss} = -20 \log_{10}(|\Gamma|)\)
where:
- \(\text{Reflection Loss}\) is the loss due to reflections
- \(\Gamma\) is the reflection coefficient
Transmission Line Attenuation
\(A = e^{-\alpha l}\)
where:
- \(A\) is the attenuation
- \(\alpha\) is the attenuation constant
- \(l\) is the length of the transmission line
Total Transmission Line Loss
\(P_{out} = P_{in} e^{-2\alpha l}\)
where:
- \(P_{out}\) is the output power
- \(P_{in}\) is the input power
- \(\alpha\) is the attenuation constant
- \(l\) is the length of the transmission line
Reflection Coefficient Magnitude
\(|\Gamma| = \sqrt{(\text{Re}(\Gamma))^2 + (\text{Im}(\Gamma))^2}\)
where:
- \(|\Gamma|\) is the magnitude of the reflection coefficient
- \(\text{Re}(\Gamma)\) and \(\text{Im}(\Gamma)\) are the real and imaginary parts of \(\Gamma\)
Transmission Coefficient
\(T = 1 + \Gamma\)
where:
- \(T\) is the transmission coefficient
- \(\Gamma\) is the reflection coefficient
Impedance Matching for Maximum Power Transfer
\(Z_L = Z_0^\*\)
where:
- \(Z_L\) is the load impedance
- \(Z_0^\*\) is the complex conjugate of the characteristic impedance
Power Delivered to a Matched Load
\(P_L = \frac{|V_{in}|^2}{8 R_0}\)
where:
- \(P_L\) is the power delivered to the load
- \(V_{in}\) is the input voltage
- \(R_0\) is the characteristic impedance
Transmission Line Voltage Standing Wave Ratio (VSWR)
\(\text{VSWR} = \frac{V_{max}}{V_{min}}\)
where:
- \(\text{VSWR}\) is the voltage standing wave ratio
- \(V_{max}\) is the maximum voltage on the line
- \(V_{min}\) is the minimum voltage on the line
Transmission Line Current Standing Wave Ratio (ISWR)
\(\text{ISWR} = \frac{I_{max}}{I_{min}}\)
where:
- \(\text{ISWR}\) is the current standing wave ratio
- \(I_{max}\) is the maximum current on the line
- \(I_{min}\) is the minimum current on the line
Characteristic Impedance of a Coaxial Cable
\(Z_0 = \frac{1}{2\pi} \sqrt{\frac{\mu}{\epsilon}} \ln{\frac{b}{a}}\)
where:
- \(Z_0\) is the characteristic impedance
- \(\mu\) is the permeability of the dielectric
- \(\epsilon\) is the permittivity of the dielectric
- \(a\) is the radius of the inner conductor
- \(b\) is the inner radius of the outer conductor
Return Loss
\(\text{Return Loss} = -20 \log_{10}(|\Gamma|)\)
where:
- \(\text{Return Loss}\) is the loss due to reflections
- \(\Gamma\) is the reflection coefficient
Transmission Line Differential Equations
\(\frac{dV}{dz} = -Z_0 I\)
\(\frac{dI}{dz} = -Y_0 V\)
where:
- \(V\) is the voltage
- \(I\) is the current
- \(Z_0\) is the characteristic impedance
- \(Y_0\) is the admittance
- \(z\) is the position along the transmission line
Voltage and Current Relations on a Lossless Transmission Line
\(V(z) = V^+ e^{-j\beta z} + V^- e^{j\beta z}\)
\(I(z) = \frac{V^+}{Z_0} e^{-j\beta z} - \frac{V^-}{Z_0} e^{j\beta z}\)
where:
- \(V(z)\) is the voltage at position \(z\)
- \(I(z)\) is the current at position \(z\)
- \(V^+\) and \(V^-\) are the forward and backward traveling voltage waves
- \(Z_0\) is the characteristic impedance
- \(\beta\) is the phase constant
- \(z\) is the position along the transmission line
Wave Impedance
\(Z_w = \frac{V}{I}\)
where:
- \(Z_w\) is the wave impedance
- \(V\) is the voltage
- \(I\) is the current
Power on a Transmission Line
\(P(z) = V(z) \cdot I(z)\)
where:
- \(P(z)\) is the power at position \(z\)
- \(V(z)\) is the voltage at position \(z\)
- \(I(z)\) is the current at position \(z\)
Maximum Power Transfer in AC Circuits
\(R_L = \sqrt{R_{th}^2 + (X_{th})^2}\)
where:
- \(R_L\) is the load resistance
- \(R_{th}\) is the Thevenin resistance
- \(X_{th}\) is the Thevenin reactance
Reactive Power in AC Circuits
\(Q = V \cdot I \cdot \sin{\phi}\)
where:
- \(Q\) is the reactive power
- \(V\) is the voltage
- \(I\) is the current
- \(\phi\) is the phase angle between voltage and current
Apparent Power in AC Circuits
\(S = V \cdot I\)
where:
- \(S\) is the apparent power
- \(V\) is the voltage
- \(I\) is the current
Power Triangle Relationship
\(S^2 = P^2 + Q^2\)
where:
- \(S\) is the apparent power
- \(P\) is the real power
- \(Q\) is the reactive power
Power Factor
\(\text{Power Factor} = \cos{\phi}\)
where:
- \(\text{Power Factor}\) is the power factor
- \(\phi\) is the phase angle between voltage and current
Lagging and Leading Power Factor
\(\text{Lagging: } \phi > 0^\circ\)
\(\text{Leading: } \phi < 0^\circ\)
where:
- \(\phi\) is the phase angle between voltage and current
Complex Power in Phasor Form
\(S = V \cdot I^\*\)
where:
- \(S\) is the complex power
- \(V\) is the voltage phasor
- \(I^\*\) is the complex conjugate of the current phasor
Maximum Power Transfer Condition in AC Circuits
\(Z_L = Z_{th}^\*\)
where:
- \(Z_L\) is the load impedance
- \(Z_{th}^\*\) is the complex conjugate of the Thevenin impedance
Quality Factor in Resonant Circuits
\(Q = \frac{\omega_0 L}{R}\)
where:
- \(Q\) is the quality factor
- \(\omega_0\) is the resonant angular frequency
- \(L\) is the inductance
- \(R\) is the resistance
Bandwidth of a Resonant Circuit
\(BW = \frac{\omega_0}{Q}\)
where:
- \(BW\) is the bandwidth
- \(\omega_0\) is the resonant angular frequency
- \(Q\) is the quality factor
Frequency Response of a Low-Pass Filter
\(H(j\omega) = \frac{1}{1 + j\omega RC}\)
where:
- \(H(j\omega)\) is the transfer function
- \(\omega\) is the angular frequency
- \(R\) is the resistance
- \(C\) is the capacitance
Frequency Response of a High-Pass Filter
\(H(j\omega) = \frac{j\omega RC}{1 + j\omega RC}\)
where:
- \(H(j\omega)\) is the transfer function
- \(\omega\) is the angular frequency
- \(R\) is the resistance
- \(C\) is the capacitance
Gain-Bandwidth Product of an Op-Amp
\(GBW = A_v \cdot f_{3dB}\)
where:
- \(GBW\) is the gain-bandwidth product
- \(A_v\) is the voltage gain
- \(f_{3dB}\) is the -3 dB bandwidth frequency
Transfer Function of a First-Order System
\(H(s) = \frac{1}{1 + sRC}\)
where:
- \(H(s)\) is the transfer function
- \(s\) is the complex frequency variable
- \(R\) is the resistance
- \(C\) is the capacitance
Transfer Function of a Second-Order System
\(H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\)
where:
- \(H(s)\) is the transfer function
- \(\omega_n\) is the natural angular frequency
- \(\zeta\) is the damping factor
- \(s\) is the complex frequency variable
Frequency Response of a Band-Pass Filter
\(H(j\omega) = \frac{j\omega RC}{1 + j\omega RC + (j\omega)^2 LC}\)
where:
- \(H(j\omega)\) is the transfer function
- \(\omega\) is the angular frequency
- \(R\) is the resistance
- \(C\) is the capacitance
- \(L\) is the inductance
Frequency Response of a Band-Stop Filter
\(H(j\omega) = \frac{1 + (j\omega)^2 LC}{1 + j\omega RC + (j\omega)^2 LC}\)
where:
- \(H(j\omega)\) is the transfer function
- \(\omega\) is the angular frequency
- \(R\) is the resistance
- \(C\) is the capacitance
- \(L\) is the inductance
Transfer Function of an Operational Amplifier Integrator
\(H(s) = -\frac{1}{sRC}\)
where:
- \(H(s)\) is the transfer function
- \(s\) is the complex frequency variable
- \(R\) is the resistance
- \(C\) is the capacitance
Transfer Function of an Operational Amplifier Differentiator
\(H(s) = -sRC\)
where:
- \(H(s)\) is the transfer function
- \(s\) is