1. ECE53A Introduction to Analog and Digital Circuits Lecture Notes Second-Order Analog Circuit and System

2. ECE 53A Fall 2007 Order of an analog circuit or system depends on the number of energy storing elements L’s and C’s. Simplest second-order circuits are ones containing two C’s plus resistors and their dual networks containing two L’s; e.g. Simplest RLC-circuit is the series RLC-circuit shown in Figure 2(a) and its dual, the parallel (or shunt) RLC network. R1R1 C1C1 R2R2 C2C2 RLRL R1R1 R2R2 RLRL L1L1 L2L2 + - + - + - RCL V(t) R C L + - i(t) Figure 1(a) Figure 1(b) Figure 2(a) Single loop RLC-circuit Figure 2(b) Single node-pair RLC-circuit

3. ECE 53A Fall 2007 Series-RLC Circuit Figure 1 Single-loop series RLC circuit R C L i(t) i L (t) + - V c (t) + - First of all, even before we write the network equations to solve for the single unknown which is the loop current i(t), we need to satisfy two initial conditions (I.C) Vc(0) and iL(0). KVL (loop) equation: Differentiate Eq.(1) using (2), [Note i L (t)=i(t)] [Note i c (t)=i(t)] or

4. Eq.(3) is in the form of a generic second-order (n=2) ORDINARY differential equation with constant coefficients, Where can be a current or voltage and is the input or forcing function. The circuit we are considering is linear and lumped containing two(2) energy storage circuit elements (In our case, the circuit contains a inductor and a capacitor). For the series RLC circuit, to solve for, we need two(2) I.C, vc(0) and iL(0). Let, substituting this into the homogeneous equation We obtain the quadratic equation This is known as the characteristic equation. For the second-order circuits, the generic characteristic equation can be written as ECE 53A Fall 2007

5. Where Comparing Eq.(6) and (7), we have If (No damping) and the characteristic eq. becomes Let be, then for, This justifies the naming of as the un-damped natural or the un-damped resonant frequency of the circuit. The roots of the generic quadratic characteristic equation are ECE 53A Fall 2007

6. ECE 53A If can be shown that does not affect the behavior of the response of the circuit by using frequency, thus, the behavior of the circuit response depends solely on the damping factor. Without lose of generality, let us assume. Thus, Eq.(10) reduces to Case 1: If,(roots are real and equal. Double order) Critical damping Case 2: If, (both roots are real and unequal) Un-damped Case 3: If, (roots are complex and complex conjugal) Over- damped Case 1Case 2Case 3 Fall 2007

7. ECE 53A Standard method is used to solve for the complete solutions of the ordinary second-order differential equation as where Fall 2007

8. KCL (Nodal) Equation: Since, we have ECE 53A Parallel-RLC Circuit Figure 2. Single node-pair parallel RLC circuit or Note this equation is identical to (3) if In fact the parallel circuit in Fig. 2 is the dual network for the series circuit in Fig. 1. Fall 2007 RCL i L (t) + - V c (t)

9. Write KCL (Nodal) Equation by inspection, we have ECE 53A Parallel-RLC Circuit Figure 3. Second-order passive RC-circuits Under-damped values of R’s and C’s Fall 2007 + - R1R1 C1C1 R2R2 C2C2 + -

10. ECE 53A Fall 2007 R1R1 RLRL L1L1 C1C1 C2C2 + - +- N=4

11. By inspection, nodal (or KVL) equations are given by ECE 53A Second-Order RC-Network (or its dual: RL-Network) Node Eq.(3) is in the form of the nodal admittance matrix and can be obtained from the circuit in Figure 1. Fall 2007 Figure 1. Second-order RC-network R1R1 R2R2 R3R3 C1C1 C2C2 + - + - Node

12. To treat Equation (1) and (2) (or(3)) simultaneously, we first use (1) to determine in terms of, and then substitute the result into (2) to obtain a second-order differential equation in. We have Note that Eq.(5) is of the generic form ECE 53A Fall 2007

13. Also, note that Eq.(1) and (2) are already in the form of state-variable formation. Thus, rewriting (1) and (2) as ECE 53A Therefore Fall 2007 Since

14. For the special case of RC-network, the nodal equations are indeed the state equations using state variables. The roots of the characteristic equation are constrained to be only the negative -axis as shown below ECE 53A Where Fall 2007

15. Can use Loop (KVL) analysis (3 loops) or use Nodal (KCL) analysis (2 node-pairs) Better get use state-variable analysis ECE 53A High-Order (n=4) RLC-Network Let Set up equations is straight forward – use a mixture of KCL equations (tree branch) and KVL equations (Cuts). Solve by computing state and output equations. to obtain 4 coupled 1st-rder differential equations. Fall 2007 RgRg RLRL L1L1 C1C1 C2C2 + - +- L2L2 (1)(2) + - N=4 (2L’s and 2C’s)

16. Tree: Connect all nodes – No closed path Tree branches: Loops: Use a selected mixture of KCL and KVL equations ECE 53A To obtain 4 coupled 1st-order differential equations which we can put into a compact matrix from I.C. Question: what kind of network is this? Look at behavior of element (L’s and C’s) at Fall 2007 First obtain directed graph:  4 KCL equations  3 KVL equations  No integration only 1st-order derivatives. RgRg L1L1 L2L2 RLRL C1C1 C2C2 V g1

17. Let the capacitor C be charged to a voltage V 0 and at t=0 let the switch be closed. The value of the resistance R will determine whether the system is (a) over-damped (b) critically damped, (c) under-damped ECE 53A Fall 2007 Figure E-1. RC-network used to illustrate the finding the particular solution from the general solution Example E-1 R L i(t)C V0V0 VcVc K + - iLiL Figure E2-2. Network response for the three cases: (a) over-damped (b) critically damped and © under-damped

18. In Eq.(1-3), K 1 and K 2 are arbitrary constant of integration. Eq.(3) can be rewritten equivalently as ECE 53A General solutions in terms of Fall 2007

19. ECE 53A Cascaded inverters with parasitic wiring inductance and gate capacitance Fall 2007 Two Cascaded Inverters RLRL RLRL RLRL RLRL V in1 V out2 V in2 V out1 V in1 V out2 V in2 V out1 L1L1 C gs

20. ECE 53A Fall 2007 Circuit model of the cascaded inverters when the input at V in1 is low RLRL RLRL V in1 V out2 V in2 V out1 L1L1 C gs2 R ON Series-RLC circuit: R on L 1 C gs2

21. ECE 53A Fall 2007 DUALITY and DUAL NETWORKS VsVs Dual L C R Dual Quantities: Loop current and node-pair voltage Loop and node pair Short circuit and open circuit Example 1: KCL and KVL Example 2: Dual Network: + - i(t) isis LRC V(t) L C R VsVs + - isis L RC 1 3 3 1 2 2

22. ECE 53A State-variable method I. First-order circuits and systems For LTI systems, a,b,c,d are constants. II. Second-order circuits and systems For continuous-time LTI systems, A,B,C and D are constant matrices. For LTI analog lumped circuits containing R, L, C and transformers. I.C needed to solve the state equations: