Lecture 4: Electrical Circuits• Electric circuits: resistors, capacitors, inductors, and OP-

Amp

• Basic laws governing electric circuits� Ohm’s Law

� Kirchhoff’s current law (KCL, node law): The sum of currents entering a node () is equal to the sum of currents leaving the same node () .

=

� Kirchhoff’s voltage law (KVL, loop law): The sum of voltage around any closed loop in an electrical circuit is zero.

1

Electrical Circuits• Find Voltage and current across

2

Electric Circuit

Electrical Circuits• LRC circuit:

� Find mathematical equations� Find transfer function:

4

LRC circuit

• (• (

Electrical Circuits• Applying Kirchhoff’s voltage law

Taking the Laplace transform,

Transfer function

5

LRC circuit

Loop 1

Loop 2

Loop 1 Loop 2

• (• (

Electrical Circuits• Write the dynamic equations and find the transfer

function

6

• (• (

Electric Circuit

Electric Circuits• Find the transfer function

Electric Circuits• Applying KVL

• Taking Laplace transform

𝑖3

+=-

Complex Impedances• Transfer function

• Complex impedances� Resistance: � Capacitance: � Inductance:

Operational Amplifiers

• Op Amp is used to amplify signals.• Output• K: differential gain ()• No current flows into either input

of the op-amp (Input impedance is infinite)• Output impedance is zero

Operational Amplifiers• Obtain the transfer function

Operational Amplifiers• Obtain the transfer function • Since = =0

Hence • Transfer function

=

Inverting amplifier

Operational Amplifiers• Obtain the transfer function

Operational Amplifiers• Obtain the transfer function

𝑒′

Since

=

𝑒′

Noninverting amplifier

Operational Amplifiers• Obtain the transfer function

Operational Amplifiers• Obtain the transfer function

First order lag circuit

Filters

Filters

Low pass filter High pass filter

Impedance of capacitor decreases as frequency increases.

1𝑅𝐶𝑠+1

𝑅𝐶𝑆𝑅𝐶𝑠+1

Operational Amplifiers• Obtain the transfer function • What kind of filter is this?

Operational Amplifiers• Obtain the transfer function

Low pass filter

Operational Amplifiers• Obtain the transfer function

Low pass filter

Operational Amplifiers• Obtain the transfer function • What kind of filter is this?

Operational Amplifiers• Obtain the transfer function

High pass filter