20.309:Recitation 092107
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Review of ideal circuit elements
Linear passive
Independent sources
Dependent sources
Element  Symbol  Equation  Comments 

Op Amp  V_{o} = A(V_{ + } − V_{ − }) 

Modeling real components
 The ideal elements comprise a vocabulary of mathematical relationships that are useful for modeling real systems.
 In many situations, a single ideal element models the behavior of a real component well.
 At other times, such as high frequency, high load, or high gain, it may be necessary to create a more sophisticated model
Example: modeling a battery
Consider models for common AAA and D cell batteries.
AAA and D cell
Simple model: ideal voltage source
 The voltage output of both the AAA and the D cell is 1.6 V
 A simple model for both batteries is a voltage source with a value of 1.6V
 This model works well with small loads
 This model does not capture any difference between the two sizes
Simple Battery Model
Measuring output resistance
 Output resistance can be estimated from open circuit voltage and short circuit current.
AAA  D  

Measured open circuit voltage (V_{oc}),  1.6 V  1.6 V 
Measured short circuit current (I_{sc})  5.4 A  11.8 A 
Computed output resistance  0.30 Ω  0.14 Ω 
Better model: output resistance
 Output resistance is simply add in series with the voltage source.
Battery Model including Output Resistance
Comparing the two models under heavy loading (0.2 Ω)
AAA  D  

Battery Voltage Under 0.2 Ω Load Predicted by Simple Model  1.6 V  1.6 V 
Battery Voltage Under 0.2 Ω Load Predicted by Output Resistance Model  0.9 V  1.2 V 
Measured Battery Voltage Under 0.2 Ω Load  0.9 V  1.1 V 
 The improved model predicts the output voltage and current flow under a load of 0.2 Ohms
 This model is much more accurate under heavy load conditions.
Op amp model
 Just as with the battery, it is possible to make a more accurate op amp model by adding elements
 Offset voltage, input bias current, input offset current can be modeled by adding simple souces to the ideal op amp
 Unfortunately, these parameters are temperature dependent
 Noninfinite gain and frequency dependence can be modeled by cascading a transfer function at the output of the op amp
 Regardless of the connection, the product of gain and bandwidth is constant. This parameter is usually specified in the datasheet.
Sallen Key circuit
Key concepts
 Even though the circuit is complicated, solving it requires only two techniques: the golden rules and Kirchoff' Current Law.
 Because of the buffer amplifer at its output, the Sallen Key circuit has low output impedence.
 The circuit implements a second order low pass filter. (There are also band and high pass versions.)
 The cutoff frequency, ω_{c}, occurs where the magnitude of the transfer funcion falls to .
 Above the cutoff frequency, the second order transfer function falls off twice as fast as the first order — 40 dB per decade versus 20.
Approach to solving the Sallen Key circuit
 Apply the Golden Rules
 Apply KCL at the V_{x} node
 Apply KCL at the V_{ − } node
 Solve one equation for V_{x} and substitute into the other
 Rewrite the result in the form of a transfer function V_{o} / V_{i}
ALthough the details are different, the same approach will work to solve the op amp question on Homework #1
The gruesome details
Apply the Golden Rules
In the Sallen Key circuit, a wire connects V_{ − } to V_{o}. Therefore, V_{ − } = V_{ + } = V_{o}. This will be a useful substitution when applying KCL.
Apply KCL at the V_{x} node
KCL
 (1)
Multiply by R_{1}R_{2}
 R_{2}(V_{i} − V_{x}) + R_{1}(V_{o} − V_{x}) + R_{1}R_{2}C_{1}s(V_{o} − V_{x}) = 0 (2)
Gather terms
 R_{2}V_{i} − (R_{2} + R_{1} + R_{1}R_{2}C_{1}s)V_{x} + (R_{1} + R_{1}R_{2}C_{1}s)V_{o} = 0 (3)
Apply KCL at the V_{ − } node
KCL
 (4)
Multiply by R_{2}
 V_{x} = V_{o}(1 + R_{2}C_{2}s) (5)
Substitute for V_{x} (equation 5 into equation 3)
Sustitute
 R_{2}V_{i} − (R_{2} + R_{1} + R_{1}R_{2}C_{1}s)(1 + R_{2}C_{2}s)V_{o} + (R_{1} + R_{1}R_{2}C_{1}s)V_{o} = 0 (6)
Multiply and collect terms
 (7)
 V_{i} = {R_{1}R_{2}C_{1}C_{2}s^{2} + C_{2}(R_{1} + R_{2})s + 1}V_{o} (8)
Rewrite as a transfer function
 (9)
Comparison of first and second order transfer functions
Compare the Sallen Key transfer function with the first order low pass filter we considered in class:
Because its transfer function includes an s^{2} term, the Sallen Key filter is called a second order filter. The magnitude of both functions is plotted below. Although they have similar cutoff frequencies, the Sallen Key function falls off twice as fast as the passive RC filter (40 dB/decade versus 20 dB/decade).
Here is the Matlab code that generated the plot:
%Clear the plot axes clf; %Component values for Sallen Key circuit (gives Fc = 1) R1=1; R2=1; C1=1; C2=1; %Generate a vector for the frequency axis with exponential spacing w = 2:.01:2; w = 10 .^ w; s = Fc * w * j; % Sallen Key transfer function Vsk = 1 ./ (R1 * R2 * C1 * C2 * s .^2 + (R1 + R2) * C2 * s + 1); %Plot the magnitude of the transfer function on a loglog scale vs. %frequency loglog( abs(s), abs(Vsk), 'b', 'LineWidth', 3); %Maintain the current plot hold on %Adjust axes for a little headroom around the plot axis([ min(abs(s)), max(abs(s)), 1.2 * min(abs(Vsk)), 1.2 * max(abs(Vsk))]) %Component values for 1st order circuit R = 1; C = 1; % Single order low pass transfer function Vlp = 1 ./(R * C * s + 1); %Plot the magnitude of the transfer function on athe same axes loglog(abs(s), abs(Vlp), 'r', 'LineWidth', 3); %Add a chart title, legend, and axis labels title('First and Second Order Filter Responses', 'FontSize', 18) legend('Second Order', 'First Order', 'Location', 'SouthWest'); ylabel('H(s)', 'FontSize', 14); xlabel('Frequency', 'FontSize', 14);