1. 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 ( 𝐼 𝑖𝑛 ) is equal to the sum of currents leaving the same node ( 𝐼 𝑜𝑢𝑡 ) .
𝐼 𝑖𝑛 = 𝐼 𝑜𝑢𝑡
Kirchhoff’s voltage law (KVL, loop law): The sum of voltage around any closed loop in an electrical circuit is zero.
∆ 𝑉 𝑐𝑙𝑜𝑠𝑒𝑑_𝑙𝑜𝑜𝑝 =0

2. Electrical Circuits
Find Voltage 𝑉3 and current 𝐼3 across 𝑅3

3. Electric Circuit

4. Electrical Circuits LRC circuit: Find mathematical equations
Find transfer function:
𝑣=𝑅𝑖
𝑖=𝐶 𝑑𝑣 𝑑𝑡 (𝑣= 1 𝐶 𝑖 dt)
𝑣=𝐿 𝑑𝑖 𝑑𝑡 (𝑖= 1 𝐿 𝑣 dt)
LRC circuit

5. Electrical Circuits Applying Kirchhoff’s voltage law LRC circuit 𝑣=𝑅𝑖
Taking the Laplace transform,
Transfer function
𝑣=𝑅𝑖
𝑖=𝐶 𝑑𝑣 𝑑𝑡 (𝑣= 1 𝐶 𝑖 dt)
𝑣=𝐿 𝑑𝑖 𝑑𝑡 (𝑖= 1 𝐿 𝑣 dt)
Loop 1 
Loop 2 
Loop 2
Loop 1
LRC circuit

6. Electrical Circuits
Write the dynamic equations and find the transfer function
𝑣=𝑅𝑖
𝑖=𝐶 𝑑𝑣 𝑑𝑡 (𝑣= 1 𝐶 𝑖 dt)
𝑣=𝐿 𝑑𝑖 𝑑𝑡 (𝑖= 1 𝐿 𝑣 dt)

7. Electric Circuit

8. Electric Circuits
Find the transfer function

9. Electric Circuits Applying KVL Taking Laplace transform
𝑖 3 = 𝑖 1 - 𝑖 2
𝑖 1 = 𝑖 2 + 𝑖 3
𝑖 3

10. Complex Impedances Transfer function Complex impedances Resistance: 𝑅
Capacitance: 1/𝐶𝑠
Inductance: 𝐿𝑠

11. 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

12. Operational Amplifiers
Obtain the transfer function eo ei

13. Operational Amplifiers
Obtain the transfer function eo ei
Since 𝑒 0 =𝐾 0− 𝑒 ′
𝑒 ′ = eo 𝐾 ≅ eo  =0
Hence
Transfer function
eo ei =
Inverting amplifier

14. Operational Amplifiers
Obtain the transfer function eo ei

15. Operational Amplifiers
Obtain the transfer function eo ei
𝑒 ′
𝑒 ′
Noninverting amplifier
Since 𝐾=
eo ei =

16. Operational Amplifiers
Obtain the transfer function eo ei

17. Operational Amplifiers
Obtain the transfer function eo ei
First order lag circuit

18. Filters

19. Filters 1 𝑅𝐶𝑠+1 𝑅𝐶𝑆 𝑅𝐶𝑠+1 Low pass filter High pass filter
Impedance of capacitor decreases as frequency increases.

20. Operational Amplifiers
Obtain the transfer function eo ei
What kind of filter is this?

21. Operational Amplifiers
Obtain the transfer function eo ei
Low pass filter

22. Operational Amplifiers
Obtain the transfer function eo ei
Low pass filter

23. Operational Amplifiers
Obtain the transfer function eo ei
What kind of filter is this?

24. Operational Amplifiers
Obtain the transfer function eo ei
High pass filter