Second-Order Circuits Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C.

Contents Introduction Finding Initial and Final Values The Source-Free Series RLC Circuit The Source-Free Parallel RLC Circuit Step Response of a Series RLC Circuit Step Response of a Parallel RLC Circuit General Second-Order Circuits Duality Applications

Introduction A second-order circuit is characterized by a second-order differential equation It consists of resistors and the equivalent of two energy storage elements

Finding Initial and Final Values v and i are defined according to the passive sign convention Continuity properties Capacitor voltage Inductor current v i + _

Example

Example (Cont’d)

Example (Cont’d)

The Source-Free Series RLC Circuit

Cont’d Natural frequencies Damping factor Resonant frequency Characteristic equation

Summary Three cases discussed Overdamped case :  > 0 Critically damped case :  = 0 Underdamped case :  < 0

Overdamped Case ( > 0) i(t)

Critically damped Case ( = 0)

Critically damped Case (Cont’d)

Underdamped Case ( < 0)

Underdamped Case (Cont’d) i(t)

Finding The Constants A1,2

Conclusions The concept of damping Oscillatory response is possible The gradual loss of the initial stored energy Due to the resistance R Oscillatory response is possible The energy is transferred between L and C Ringing denotes the damped oscillation in the underdamped case With the same initial conditions, the overdamped case has the longest settling time. The critically damped case has the fastest decay.

Example Find i(t). t < 0 t > 0

Example (Cont’d) t < 0 t > 0

The Source-Free Parallel RLC Circuit

Summary Overdamped case :  > 0 Critically damped case :  = 0 Underdamped case :  < 0

Finding The Constants A1,2

Comparisons Series RLC Circuit Parallel RLC Circuit

Example 1 Find v(t) for t > 0. v(0) = 5 V, i(0) = 0 Consider three cases: R = 1.923  R = 5  R =6.25 

Example 1 (Cont’d)

Example 1 (Cont’d)

Example 2 Find v(t). Get x(0). Get x(), dx(0)/dt, s1,2, A1,2.

Example 2 (Cont’d) t > 0 t < 0

Step Response of A Series RLC Circuit

Characteristic Equation

Summary (Overdamped) (Critically damped) (Underdamped)

Example Find v(t), i(t) for t > 0. Consider three cases: R = 5  Get x(0). Get x(), dx(0)/dt, s1,2, A1,2. t < 0 t > 0

Case 1: R = 5 

Case 2: R = 4 

Case 3: R = 1 

Example (Cont’d)

Step Response of A Parallel RLC Circuit

Characteristic Equation

Summary (Overdamped) (Critically damped) (Underdamped)

General Second-Order Circuits Steps required to determine the step response Determine x(0), dx(0)/dt, and x() Find the transient response xt(t) Apply KCL and KVL to obtain the differential equation Determine the characteristic roots (s1,2) Obtain xt(t) with two unknown constants (A1,2) Obtain the steady-state response xss(t) = x() Use x(t) = xt(t) + xss(t) to determine A1,2 from the two initial conditions x(0) and dx(0)/dt

Example Find v, i for t > 0. Get x(0). Get x(), dx(0)/dt, s1,2, A1,2. t < 0 t > 0

Example (Cont’d) t < 0 t > 0

Example (Cont’d) t > 0

Duality Duality means the same characterizing equations with dual quantities interchanged. Table for dual pairs Resistance R Conductance G Inductance L Capacitance C Voltage v Current i Voltage source Current source Node Mesh Series path Parallel path Open circuit Short circuit KVL KCL Thevenin Norton

Example 1 Series RLC Circuit Parallel RLC Circuit

Example 2

Application: Smoothing Circuits Output from a D/A vs v0