1 1st Order Circuits Any circuit with a single energy storage element, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 1. Any voltage or current in such a circuit is the solution to a 1st order differential equation.

2 A First Order RC Circuit One capacitor and one resistor The source and resistor may be equivalent to a circuit with many resistors and sources. R + - Cv s (t) + - v c (t) +- v r (t)

3 The Differential Equation(s) KVL around the loop: v r (t) + v c (t) = v s (t) R + - Cv s (t) + - v c (t) +- v r (t)

4 Differential Equation(s)

5 A First Order RL Circuit One inductor and one resistor The source and resistor may be equivalent to a circuit with many resistors and sources. v(t) i s (t) RL + -

6 The Differential Equation(s) KCL at the top node: v(t) i s (t) RL + -

7 Why? (Superposition)

8 Solving First Order Circuits 1.Draw the circuit for t=0 - and find v(0 - ), or i(0 - ) 2.Use the continuity of the capacitor voltage, or inductor current, draw the circuit for t=0 + to find v(0 + ), or i(0 + ) 3.Find v( ), or i( ) at steady state 4.Find the time constant  –For an RC circuit,  = RC –For an RL circuit,  = L/R 5.The solution is:

9 The Time Constant For an RC circuit,  = RC For an RL circuit,  = L/R -1/  is the initial slope of an exponential with an initial value of 1 Also,  is the amount of time necessary for an exponential to decay to 36.7% of its initial value

10 Implications of the Time Constant Should the time constant be large or small: –Computer RAM –The low-pass filter for the envelope detector –The sample-and-hold circuit –The electrical motor

11 2nd Order Circuits Any circuit with a single capacitor, a single inductor, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 2. Any voltage or current in such a circuit is the solution to a 2nd order differential equation.

12 A 2nd Order RLC Circuit The source and resistor may be equivalent to a circuit with many resistors and sources. R + - Cv s (t) i (t) L

13 Applications Modeled by a 2nd Order RLC Circuit Filters –A bandpass filter such as the IF amp for the AM radio. –A lowpass filter with a sharper cutoff than can be obtained with an RC circuit.

14 The Differential Equation KVL around the loop: v r (t) + v c (t) + v l (t) = v s (t) R + - Cv s (t) + - v c (t) + - v r (t) L +- v l (t) i (t)

15 Differential Equation (cont’d)

16 The Differential Equation Most circuits with one capacitor and inductor are not as easy to analyze as the previous circuit. However, every voltage and current in such a circuit is the solution to a differential equation of the following form:

17 Example response: Over Damped

18 Example Response: Under Damped