Net work analysis Dr. Sumrit Hungsasutra Text : Basic Circuit Theory, Charles A. Desoer & Kuh, McGrawHill

Linear Time-invariant Circuits Definition and properties Node and mesh analysis Input-output representation Responses to an arbitrary input Computation of convolution integrals

Definition and properties Linear circuit contains linear elements. Linear time invariant circuit contains linear time invariant elements and independent sources. Circuits with nonlinear elements are nonlinear circuits. Circuits with time varying elements are time varying circuits.

Definition and properties Voltage sources and current sources play important roles in circuit analysis. Dependent sources are non-linear and time varying. All sources are inputs to the circuit. The input is a waveform of either independent voltage or current source. The wave form can be of a constant, step and a function of time.

Definition and properties The output response is a branch voltage or branch current at the desired point or charge in a capacitor or flux in an inductor. Differential equations can be written on all lumped circuit from which branch currents or branch voltages are solved. The unique solution (circuit response) requires the input information and its initial solution.

Definition and properties Common initial conditions are capacitor’s voltage and inductor’s current. State of a circuit at time t 0 to any set of initial conditions together with the inputs uniquely determine all the network variables of the circuit at time t>t 0. If all initial conditions are zero, it is called zero state.

Definition and properties In linear circuit with zero state and no inputs all network variables remain equal to zero forever after. When inputs are applied to the circuit, initial states (can also be zero state) are required to solve for all circuit variables. Zero-state response is the solution of the circuit with inputs and zero state.

Definition and properties Zero-input response is the response with no inputs. Complete responses are the responses of the circuit to both inputs and initial states (zero states). For linear time-invariant or time-varying circuits:  Complete response is the sum of zero-input response and zero-state response.  Zero-state response is a linear function of input.  Zero-input response is a linear function of initial states

Node and mesh analyses Simple topology circuits can be analyzed more easy (simple loop simple node circuits) using KCL and KVL. More complex circuits requires a more advanced techniques. Simple circuit Fig 1

Node and mesh analysis Pick a reference node as a datum (ground) Apply KCL at each node Redrawn circuit of Fig1 Fig 2

Node and mesh analysis KCL at node 1 KCL at node 2 Initial condition

Node and mesh analysis Adding the two equations Diff KCL from node 2 or

Node and mesh analysis substitute Differential equation for voltage at node 2

Mesh and mesh analysis Redrawn Fig 1 using Thevenin equivalent Fig 3

Mesh and mesh analysis Mesh 1 KVL Mesh 2 KVL Initial condition

Mesh and mesh analysis Adding the two equations Or Diff both sides

Mesh and mesh analysis Substitute for i 1 Initial conditions Substitute

From this simple example we can see the general fact that “Given any single-input single-output linear time-invariant circuit, it is always possible to write a single differential equation relating the output to the input.” input output

Input-output Representation General equation for a single-input single-output Constants depend on circuit element values and network topology. The initial conditions are Is the output from the circuit Is the input to the circuit

Input-output Representation Zero-input response The general equation becomes homogeneous and the nth-degree characteristic Polynomial is The zeroes of this polynomial are the natural frequency of the network variable The solution of the homogeneous equation becomes If s 1 is a repeated root

Input-output Representation Zero-state response The general zero-state response of the circuit is Where is any particular solution of the circuit due to the input w

Input-output Representation Example 1 Fig 4 Figure 4 shows a simple RC circuit with zero initial condition. The input is KVL: or

Input-output Representation Initial condition: From KVL Tina simulation

Input-output Representation Impulse Response From the general equation, With initial conditions If the RHS are impulse and its derivatives and the response is

The impulse function and derivatives and

Input-output Representation Example 2 Suppose that the differential equation relating the output y and the input w of a circuit is Find the impulse response of the circuit. The characteristic polynomial is and the roots are The response

Input-output Representation Solve for constants from

Input-output Representation Substitute and we have and

Response to an arbitrary input An arbitrary input signal can be divided into many impulse functions.

Response to an arbitrary input As the output response is the sum of all impulse responses Conclusion 1 Determine the impulse response 2 Calculate the integral 3 This type of integral is called convolution integral

Response to an arbitrary input The complete response Where is the zero-input response Note that the complete response is a linear function of input only if the Zero-input response is identically zero.

Computation of convolution integrals From For the unit impulse at

Computation of convolution integrals Let Then Thus Convolution integral is symmetric role

Computation of convolution integrals Example 3 Let the input be a step function and the impulse response be a triangular waveform

Computation of convolution integrals

Example 4 Find the zero-state response for the input and impulse response shown

Computation of convolution integrals For

Computation of convolution integrals Example 5 Find the zero-state response for the input and impulse response shown

Computation of convolution integrals For

Computation of convolution integrals