Active Filters Fundamentals
Active vs Passive Filters
Active Filters:
-
Use active components (Op-Amps, transistors)
-
Provide gain (amplification)
-
No inductors required
-
Input/output impedance can be controlled
-
Require power supply
Advantages over Passive:
-
High input impedance, low output impedance
-
Gain adjustment possible
-
No loading effects
-
Better isolation between stages
Filter Classification
By Frequency Response:
-
Low Pass Filter (LPF): \(H(j\omega) = \dfrac{A}{1 + j\omega/\omega_c}\)
-
High Pass Filter (HPF): \(H(j\omega) = \dfrac{A \cdot j\omega/\omega_c}{1 + j\omega/\omega_c}\)
-
Band Pass Filter (BPF): Passes specific frequency band
-
Band Stop Filter (BSF): Rejects specific frequency band
By Order:
-
First Order: \(20~\mathrm{dB/decade}\) roll-off
-
Second Order: \(40~\mathrm{dB/decade}\) roll-off
-
Higher Order: \(n \times 20~\mathrm{dB/decade}\) roll-off
Frequency Response Parameters
Key Parameters:
-
Passband: Frequency range where \(|H(j\omega)| \geq -3dB\)
-
Stopband: Frequency range where signal is significantly attenuated
-
Transition band: Region between passband and stopband
-
Passband ripple: Maximum variation in passband gain
-
Stopband attenuation: Minimum attenuation in stopband
Bode Plot Characteristics:
-
Magnitude plot: \(20\log|H(j\omega)|\) vs \(\log\omega\)
-
Phase plot: \(\angle H(j\omega)\) vs \(\log\omega\)
-
Corner frequency: Where response is 3dB down
-
Asymptotic approximations for analysis
Single-Stage Active Filters
First Order Low Pass Filter
Non-inverting Configuration:
-
Gain: \(A_v = 1 + \dfrac{R_2}{R_1}\)
-
Cut-off frequency: \(f_c = \dfrac{1}{2\pi RC}\)
-
Transfer function: \(H(j\omega) = \dfrac{A_v}{1 + j\omega RC}\)
-
Roll-off: 20 dB/decade beyond \(f_c\)
Inverting Configuration:
-
Gain: \(A_v = -\dfrac{R_2}{R_1}\)
-
Same cut-off frequency formula
-
\(180^{\circ}\) phase shift
First Order High Pass Filter
Non-inverting Configuration:
-
Gain: \(A_v = 1 + \dfrac{R_2}{R_1}\)
-
Cut-off frequency: \(f_c = \dfrac{1}{2\pi RC}\)
-
Transfer function: \(H(j\omega) = \dfrac{A_v \cdot j\omega RC}{1 + j\omega RC}\)
-
Roll-off: 20 dB/decade below \(f_c\)
Key Points:
-
Capacitor and resistor positions are swapped vs LPF
-
Phase leads by \(90^{\circ}\) at very high frequencies
-
Gain approaches \(A_v\) at high frequencies
Band Pass Filter
Cascade of HPF and LPF:
-
\(f_{c1} < f_{c2}\) (HPF cut-off < LPF cut-off)
-
Bandwidth: \(BW = f_{c2} - f_{c1}\)
-
Center frequency: \(f_0 = \sqrt{f_{c1} \cdot f_{c2}}\)
-
Quality factor: \(Q = \dfrac{f_0}{BW}\)
Single Op-Amp BPF:
-
Multiple feedback topology
-
Gain: \(A_v = -\dfrac{R_2}{2R_1}\)
-
\(f_0 = \dfrac{1}{2\pi RC}\)
-
\(Q = \dfrac{1}{3 - A_v}\)
All-Pass Filter
Characteristics:
-
Constant magnitude response: \(|H(j\omega)| = A\) (constant)
-
Variable phase response: \(\angle H(j\omega) = f(\omega)\)
-
Used for phase correction and delay equalization
First Order All-Pass:
-
Transfer function: \(H(s) = \dfrac{s - \omega_0}{s + \omega_0}\)
-
Phase shift: \(\phi(\omega) = -2\tan^{-1}(\omega/\omega_0)\)
-
Phase shift at \(\omega_0\): \(-90^{\circ}\)
-
Group delay: \(\tau_g = \dfrac{2\omega_0}{\omega_0^2 + \omega^2}\)
Second Order All-Pass:
-
Higher phase shift range: \(0^{\circ}\) to \(-180^{\circ}\)
-
Used in multi-stage phase shifters
Integrator and Differentiator
Op-Amp Integrator:
-
Transfer function: \(H(s) = -\dfrac{1}{sRC}\)
-
Magnitude: \(|H(j\omega)| = \dfrac{1}{\omega RC}\)
-
Phase: \(\angle H(j\omega) = -90^{\circ}\) (constant)
-
Slope: \(-20\) dB/decade
-
DC gain: Infinite (requires reset mechanism)
Op-Amp Differentiator:
-
Transfer function: \(H(s) = -sRC\)
-
Magnitude: \(|H(j\omega)| = \omega RC\)
-
Phase: \(\angle H(j\omega) = +90^{\circ}\) (constant)
-
Slope: \(+20\) dB/decade
-
Prone to noise amplification at high frequencies
Practical Considerations:
-
Integrator needs feedback resistor for DC stability
-
Differentiator needs series resistor for stability
Sallen Key Topology
Sallen Key - Introduction
Characteristics:
-
Voltage Controlled Voltage Source (VCVS) topology
-
Non-inverting configuration
-
Unity gain buffer or non-inverting amplifier
-
Two-pole (second-order) response
-
Popular for realizing standard filter responses
General Form:
-
Uses 2 resistors, 2 capacitors, 1 op-amp
-
Feedback from output to input through components
-
Gain: \(K = 1 + \dfrac{R_4}{R_3}\) (if gain \(> 1\))
Sallen Key Low Pass Filter
Equal Component Design:
-
\(R_1 = R_2 = R\), \(C_1 = C_2 = C\)
-
Cut-off frequency: \(f_c = \dfrac{1}{2\pi RC}\)
-
Damping factor: \(\zeta = \dfrac{3-K}{2}\)
-
For critical damping: \(K = 1\) (unity gain)
Transfer Function:
Design Constraints:
-
For stability: \(K < 3\)
-
For Butterworth response: \(K = 1.586\)
Sallen Key High Pass Filter
Equal Component Design:
-
\(R_1 = R_2 = R\), \(C_1 = C_2 = C\)
-
Cut-off frequency: \(f_c = \dfrac{1}{2\pi RC}\)
-
Same damping factor: \(\zeta = \dfrac{3-K}{2}\)
Transfer Function:
Key Difference:
-
Resistors and capacitors are interchanged
-
High frequency gain = K
-
Roll-off below cut-off: 40 dB/decade
Sallen Key Band Pass Filter
Configuration:
-
More complex than LPF/HPF
-
Uses different component arrangement
-
Can be realized using multiple feedback
Parameters:
-
Center frequency: \(f_0 = \dfrac{1}{2\pi\sqrt{R_1R_2C_1C_2}}\)
-
Quality factor: \(Q = \dfrac{1}{2\zeta}\)
-
Gain at \(f_0\): \(A_0 = \dfrac{K}{2\zeta}\)
Design Considerations:
-
Higher Q requires more precision
-
Stability issues with high Q
-
Component tolerance effects
Butterworth Filters
Butterworth Response
Characteristics:
-
Maximally flat response in passband
-
No ripple in passband
-
Monotonic roll-off in stopband
-
Also called "maximally flat" filter
Transfer Function (nth order):
Properties:
-
At \(\omega_c\): \(|H(j\omega_c)| = \dfrac{1}{\sqrt{2}} = -3dB\)
-
Roll-off: \(n \times 20\) dB/decade
-
Phase response: \(n \times 90^{\circ}\) at \(\omega \gg \omega_c\)
Butterworth Polynomial
Normalized Polynomials:
-
\(B_1(s) = s + 1\)
-
\(B_2(s) = s^2 + 1.414s + 1\)
-
\(B_3(s) = (s + 1)(s^2 + s + 1)\)
-
\(B_4(s) = (s^2 + 0.765s + 1)(s^2 + 1.848s + 1)\)
Pole Locations:
-
Poles lie on unit circle in s-plane
-
Equally spaced by \(\pi/n\) radians
-
Left half-plane poles for stable filter
Damping Factors:
-
2nd order: \(\zeta = 0.707\)
-
4th order: \(\zeta_1 = 0.383\), \(\zeta_2 = 0.924\)
Butterworth Filter Design
Order Determination:
Where:
-
\(A_p\): Passband attenuation (dB)
-
\(A_s\): Stopband attenuation (dB)
-
\(\omega_p\): Passband edge frequency
-
\(\omega_s\): Stopband edge frequency
Frequency Scaling:
-
Replace \(s\) with \(s/\omega_c\)
-
Scale all component values accordingly
Higher Order Butterworth Realization
Cascade Approach:
-
Decompose into 1st and 2nd order sections
-
Each section implemented using Sallen Key
-
Odd orders: One 1st order + multiple 2nd order
-
Even orders: Only 2nd order sections
Design Steps:
-
Determine required order
-
Factor the polynomial
-
Design each section with appropriate Q
-
Cascade the sections
Practical Considerations:
-
Component tolerance effects
-
Op-amp bandwidth limitations
-
Noise considerations
Chebyshev Filters
Type I Chebyshev:
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Ripple in passband, monotonic in stopband
-
Steeper roll-off than Butterworth
-
Passband ripple: \(\epsilon\) parameter
-
\(|H(j\omega)|^2 = \dfrac{1}{1 + \epsilon^2 T_n^2(\omega/\omega_c)}\)
-
\(T_n(x)\): Chebyshev polynomial of first kind
Type II Chebyshev (Inverse):
-
Monotonic in passband, ripple in stopband
-
Finite transmission zeros
-
Better phase response than Type I
Comparison with Butterworth:
-
Chebyshev: Faster roll-off, phase distortion
-
Butterworth: Slower roll-off, better phase response
-
Trade-off between magnitude and phase performance
Elliptic (Cauer) Filters
Characteristics:
-
Ripple in both passband and stopband
-
Steepest roll-off for given order
-
Minimum order for given specifications
-
Complex design procedure
Comparison Summary:
Filter Type | Passband | Stopband | Roll-off |
---|---|---|---|
Butterworth | Flat | Monotonic | Moderate |
Chebyshev I | Ripple | Monotonic | Steep |
Chebyshev II | Monotonic | Ripple | Steep |
Elliptic | Ripple | Ripple | Steepest |
Selection Criteria:
-
Butterworth: General purpose, good phase response
-
Chebyshev: When steep roll-off is critical
-
Elliptic: Minimum order requirement
Bessel Filters
Characteristics:
-
Maximally flat group delay in passband
-
Linear phase response
-
Constant time delay for all frequencies
-
Slower roll-off than Butterworth
Transfer Function:
-
Based on Bessel polynomials
-
Denominator: \(B_n(s) = \sum_{k=0}^{n} \dfrac{(2n-k)!}{2^{n-k}k!(n-k)!}s^k\)
-
Poles lie on circle in s-plane (different from Butterworth)
Applications:
-
Pulse shaping circuits
-
Data communication systems
-
Any application requiring minimal phase distortion
Group Delay:
-
Definition: \(\tau_g = -\dfrac{d\phi}{d\omega}\)
-
Bessel filters have constant group delay
-
Important for preserving waveform shape
Filter Transformations
Frequency Transformations:
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Low-pass to High-pass: \(s \rightarrow \dfrac{\omega_c^2}{s}\)
-
Low-pass to Band-pass: \(s \rightarrow \dfrac{s^2 + \omega_0^2}{s \cdot BW}\)
-
Low-pass to Band-stop: \(s \rightarrow \dfrac{s \cdot BW}{s^2 + \omega_0^2}\)
Impedance Scaling:
-
Multiply all resistors by \(K_R\)
-
Divide all capacitors by \(K_R\)
-
Inductors remain unchanged
Frequency Scaling:
-
Divide all reactive components by \(K_f\)
-
Resistors remain unchanged
-
New frequency = \(K_f \times\) old frequency
Multiple Feedback Topology
Characteristics:
-
Inverting configuration
-
Single op-amp implementation
-
Good for band-pass filters
-
Multiple feedback paths
MFB Band-Pass Filter:
-
Gain: \(A_v = -\dfrac{R_2}{2R_1}\)
-
Center frequency: \(f_0 = \dfrac{1}{2\pi\sqrt{R_1R_2C_1C_2}}\)
-
Quality factor: \(Q = \dfrac{\pi f_0 R_2 C_2}{1}\)
-
Bandwidth: \(BW = \dfrac{1}{\pi R_2 C_2}\)
Advantages:
-
Single op-amp design
-
Independent gain and Q adjustment
-
Lower component count
State Variable Filters
Configuration:
-
Uses 3 op-amps
-
Simultaneous LP, HP, BP outputs
-
Universal filter topology
-
Also called KHN (Kerwin-Huelsman-Newcomb) filter
Transfer Functions:
-
Low-pass: \(H_{LP}(s) = \dfrac{\omega_0^2}{s^2 + \dfrac{\omega_0}{Q}s + \omega_0^2}\)
-
High-pass: \(H_{HP}(s) = \dfrac{s^2}{s^2 + \dfrac{\omega_0}{Q}s + \omega_0^2}\)
-
Band-pass: \(H_{BP}(s) = \dfrac{\dfrac{\omega_0}{Q}s}{s^2 + \dfrac{\omega_0}{Q}s + \omega_0^2}\)
Advantages:
-
Independent control of \(\omega_0\) and Q
-
Multiple outputs available
-
Good stability
-
Easy to tune
Switched Capacitor Filters
Principle:
-
Capacitor switched between two nodes
-
Equivalent resistance: \(R_{eq} = \dfrac{1}{f_s C}\)
-
\(f_s\): Switching frequency
-
CMOS technology compatible
Advantages:
-
No external resistors required
-
High integration density
-
Programmable frequency response
-
Good matching in IC process
Disadvantages:
-
Requires clock signal
-
Limited frequency range
-
Aliasing considerations
-
Clock feedthrough
Applications:
-
Audio processing
-
Telecommunications
-
Anti-aliasing filters
Gyrator-Based Filters
Gyrator Concept:
-
Converts capacitor to inductor behavior
-
Impedance: \(Z_L = \dfrac{s R_1 R_2 C}{1}\)
-
Simulated inductance: \(L = R_1 R_2 C\)
-
No actual inductor required
Implementation:
-
Two op-amps and few passive components
-
Realizes LC ladder filters
-
Good for low-frequency applications
Applications:
-
Low-frequency filters
-
Precision filters
-
Replacement for bulky inductors
Limitations:
-
Requires matched components
-
Sensitive to op-amp characteristics
-
Limited frequency range
Sensitivity Analysis
Definition:
-
Sensitivity: \(S_x^y = \dfrac{x}{y} \dfrac{\partial y}{\partial x}\)
-
Measures how output parameter y changes with component x
-
Important for practical filter design
Component Sensitivity:
-
Frequency sensitivity: \(S_R^{\omega_0} = S_C^{\omega_0} = -\dfrac{1}{2}\)
-
Gain sensitivity varies with topology
-
Q sensitivity is critical for high-Q filters
Design Guidelines:
-
Low sensitivity to component variations
-
Use precision components for critical parameters
-
Consider temperature effects
-
Active filters generally have lower sensitivity than passive
Practical Impact:
-
Component tolerance effects
-
Aging and temperature drift
-
Manufacturing variations
Filter Specifications and Testing
Key Specifications:
-
Passband gain and ripple
-
Stopband attenuation
-
Transition bandwidth
-
Group delay variation
-
Input/output impedances
Testing Methods:
-
Frequency response measurement
-
Step response analysis
-
Impulse response testing
-
Distortion measurements
-
Noise figure measurements
Common Problems:
-
Oscillation due to instability
-
Frequency response deviation
-
Excessive noise
-
Distortion at high levels
-
Temperature drift
Design Verification:
-
Simulation vs measurement
-
Component tolerance analysis
-
Temperature testing
-
Long-term stability
Voltage Controlled Oscillators (VCOs)
VCO Fundamentals
Definition:
-
Oscillator whose frequency is controlled by input voltage
-
Linear relationship: \(f_{out} = f_0 + K_v \cdot V_{in}\)
-
\(K_v\): VCO sensitivity (Hz/V)
-
\(f_0\): Free-running frequency
Applications:
-
Frequency modulation (FM)
-
Phase-locked loops (PLLs)
-
Function generators
-
Frequency synthesizers
Key Parameters:
-
Frequency range
-
Linearity
-
Temperature stability
-
Phase noise
VCO Topologies
RC Oscillators:
-
Wien bridge oscillator
-
Phase shift oscillator
-
Voltage-controlled through varactor diodes
LC Oscillators:
-
Hartley oscillator
-
Colpitts oscillator
-
Higher frequency operation
Integrated VCOs:
-
555 timer as VCO
-
Dedicated VCO ICs
-
Voltage-to-frequency converters
Wien Bridge VCO
Basic Operation:
-
Frequency: \(f = \dfrac{1}{2\pi RC}\)
-
Voltage control through FET resistance
-
Good frequency stability
Control Methods:
-
Varactor diode in parallel with C
-
FET as voltage-controlled resistor
-
Dual-gate MOSFET control
Design Considerations:
-
Gain must be exactly 3 for oscillation
-
Amplitude stabilization required
-
Temperature compensation
VCO Applications in PLLs
Phase-Locked Loop:
-
VCO is controlled element
-
Phase detector compares input and VCO
-
Loop filter smooths control voltage
-
Negative feedback system
PLL Equation:
Loop Parameters:
-
\(K_d\): Phase detector gain (V/rad)
-
\(K_v\): VCO gain (rad/s/V)
-
\(F(s)\): Loop filter transfer function
Timers
555 Timer - Introduction
Internal Structure:
-
Two comparators
-
SR flip-flop
-
Discharge transistor
-
Voltage divider (three \(5~\mathrm{k \Omega}\) resistors)
Reference Voltages:
-
Upper threshold: \(V_{CC}/3 \times 2 = 2V_{CC}/3\)
-
Lower threshold: \(V_{CC}/3\)
Pin Configuration:
-
Pin 1: Ground, Pin 2: Trigger, Pin 3: Output
-
Pin 4: Reset, Pin 5: Control, Pin 6: Threshold
-
Pin 7: Discharge, Pin 8: \(V_{CC}\)
555 Timer - Monostable Mode
Operation:
-
Triggered by negative edge on trigger pin
-
Output goes HIGH for predetermined time
-
Time period: \(T = 1.1 \times R \times C\)
-
Returns to LOW state after timeout
Timing Sequence:
-
Trigger pulse (LOW) starts timing
-
Output goes HIGH, discharge transistor OFF
-
Capacitor charges through R
-
When \(V_C = 2V_{CC}/3\), output goes LOW
-
Discharge transistor ON, capacitor discharges
Applications:
-
Pulse width modulation
-
Time delays
-
Missing pulse detection
555 Timer - Astable Mode
Operation:
-
Continuous oscillation
-
No external trigger required
-
Square wave output
Timing Equations:
-
HIGH time: \(t_H = 0.693(R_A + R_B)C\)
-
LOW time: \(t_L = 0.693 R_B C\)
-
Period: \(T = t_H + t_L = 0.693(R_A + 2R_B)C\)
-
Frequency: \(f = \dfrac{1.44}{(R_A + 2R_B)C}\)
-
Duty cycle: \(D = \dfrac{R_A + R_B}{R_A + 2R_B}\)
Design Notes:
-
Minimum duty cycle \(\approx 50\%\)
-
For 50% duty cycle: \(R_A \ll R_B\)
555 Timer - VCO Configuration
Voltage Controlled Oscillator:
-
Control voltage applied to pin 5
-
Changes threshold voltages
-
Frequency varies with control voltage
Modified Equations:
-
Upper threshold: \(V_{control}\)
-
Lower threshold: \(V_{control}/2\)
-
Frequency: \(f = \dfrac{1.44}{(R_A + 2R_B)C} \times \dfrac{V_{CC}}{V_{control}}\)
Characteristics:
-
Linear frequency vs voltage relationship
-
Frequency range: few Hz to several MHz
-
Simple implementation
Other Timer ICs
556 Dual Timer:
-
Two independent 555 timers
-
Separate power and ground pins
-
Can be cascaded or used independently
LM555 vs NE555:
-
CMOS version: Lower power consumption
-
Better temperature stability
-
Higher input impedance
Precision Timers:
-
ICM7555: CMOS version of 555
-
LTC1799: Resistor-set oscillator
-
XR-2206: Function generator IC
Additional GATE Calculation Examples
Filter Order Calculation:
-
Given: \(A_p = 1dB\) at \(f_p = 1kHz\), \(A_s = 40dB\) at \(f_s = 10kHz\)
-
Butterworth: \(n = \dfrac{\log\sqrt{10^4 - 1} - \log\sqrt{10^{0.1} - 1}}{2\log(10)} = 2.9\)
-
Choose \(n = 3\) (next higher integer)
Component Scaling:
-
Normalized design: \(R = 1\Omega\), \(C = 1F\), \(f_c = 1Hz\)
-
For \(f_c = 1kHz\): \(K_f = 1000\)
-
For \(R = 10k\Omega\): \(K_R = 10^4\)
-
Final: \(R = 10k\Omega\), \(C = \dfrac{1}{K_f \times K_R} = 100nF\)
Q Calculation:
-
BPF with \(f_0 = 1kHz\), \(f_1 = 900Hz\), \(f_2 = 1100Hz\)
-
\(BW = f_2 - f_1 = 200Hz\)
-
\(Q = \dfrac{f_0}{BW} = \dfrac{1000}{200} = 5\)
Advanced Timer Applications
555 Timer as Schmitt Trigger:
-
Upper threshold: \(\dfrac{2V_{CC}}{3}\)
-
Lower threshold: \(\dfrac{V_{CC}}{3}\)
-
Hysteresis: \(\dfrac{V_{CC}}{3}\)
-
Applications: Noise immunity, waveform shaping
Precision Timing:
-
Crystal oscillators for accurate timing
-
Temperature compensation techniques
-
Voltage regulation for stability
Timer Cascading:
-
Frequency division
-
Sequential timing
-
Complex waveform generation
Power Considerations:
-
CMOS vs bipolar power consumption
-
Supply voltage effects on timing
-
Load current limitations
GATE Specific Points
Important Formulas for GATE
Filter Cutoff Frequencies:
-
Single RC: \(f_c = \dfrac{1}{2\pi RC}\)
-
Sallen Key: \(f_c = \dfrac{1}{2\pi RC}\) (equal components)
-
Butterworth: Same, but with specific Q values
Quality Factor:
-
\(Q = \dfrac{f_0}{BW}\)
-
For Butterworth 2nd order: \(Q = 0.707\)
-
Higher Q \(\to\) sharper response
555 Timer:
-
Monostable: \(T = 1.1RC\)
-
Astable: \(f = \dfrac{1.44}{(R_A + 2R_B)C}\)
Common GATE Questions
Filter Analysis:
-
Given circuit, find cutoff frequency
-
Calculate gain at specific frequency
-
Determine filter order and type
-
Find component values for given specs
Timer Calculations:
-
Time period calculations
-
Duty cycle problems
-
Frequency calculations
-
Component selection
Op-Amp Considerations:
-
Gain-bandwidth product effects
-
Slew rate limitations
-
Input offset voltage effects
Design Guidelines
Active Filter Design:
-
Use standard component values
-
Consider op-amp bandwidth limitations
-
Account for component tolerances
-
Ensure stability (\(K < 3\) for Sallen Key)
Practical Considerations:
-
Power supply requirements
-
Temperature effects
-
Noise considerations
-
Loading effects
GATE Exam Tips:
-
Memorize key formulas
-
Practice frequency response plots
-
Understand pole-zero concepts
-
Know standard filter responses
GATE Problem-Solving Strategy
Filter Analysis Steps:
-
Identify filter type and order
-
Write transfer function
-
Find poles and zeros
-
Calculate key parameters (\(f_c\), Q, gain)
-
Verify stability conditions
Common Mistakes to Avoid:
-
Confusing magnitude and phase responses
-
Incorrect pole-zero identification
-
Wrong stability analysis
-
Unit conversion errors
-
Ignoring op-amp limitations
Quick Check Methods:
-
Dimensional analysis
-
Limiting case verification
-
Symmetry properties
-
Physical realizability
Summary
Key Takeaways:
-
Active filters provide gain and good isolation
-
Sallen Key topology is popular for 2nd order filters
-
Butterworth filters have maximally flat response
-
VCOs are essential for frequency synthesis
-
555 timer is versatile for timing applications
For GATE Success:
-
Practice numerical problems
-
Understand circuit analysis methods
-
Know applications of each topology
-
Remember key formulas and relationships