CP2 Circuit Theory Revision Lecture

1. CP2 Circuit Theory Revision Lecture Basics, Kirchoff’s laws, Theveninand Norton’s theorem, Capacitors, Inductors • June 2004 q7 • September 2010 q7 AC theory, complex notation, LCR circuits • September 2005 q8 • June 2002 q7 Op-amps Sam Henry (Tony Weidberg) www.physics.ox.ac.uk/users/weidberg/teaching

2. +IR1 + – –V0 +IR2 + + – – +IR1 + – Kirchoff’s laws I1 I2 I1+I2–I3–I4=0 I3 I4 V0–IR1–IR2–IR3=0 R1 I R2 V0 R3

3. Thevenin and Norton theorem Req Veq  Req In Practice, to find Veq, Req… RL (open circuit) IL0 Veq=VL RL0 (short circuit) VL0 Req = resistance between terminals when all voltages sources shorted

4. Capacitors Q = CV +Q C -Q Capacitors resist change in voltage • Capacitors in series • Capacitors in parallel • Stored energy stored in form of electric field C1 C2 CN

5. Inductors • Inductors in series • Inductors in parallel • Stored energy stored in form of magnetic field Inductors resist a change of current L “back emf” Self-inductance L1 L2

6. RC and RL circuits R R + + V0 V0 C L

7. CP2 June 2004

12. Check Energy conservation: Total energy dissipated by resistor = Energy delivered by source

13. CP2 September 2010

18. AC circuit theory • Voltage represented by complex exponential • Impedance relates current and voltage V=ZIin complex notation: Resistance  R Inductance  jL Capacitance 1/jCand combinations thereof • Impedance has magnitude and phase represented by real component of easily shown from Q=VC

19. Current is given by • So |Z| gives the ratio of magnitudes of V and I, and  give the phase difference by which current lags voltage

20. CP2 September 2005

21. The complex impedance is given by the voltage in a circuit divided by the current when both are expressed in complex form. The magnitude of the complex impedance is the ratio of the magnitudes of the voltage and current; the phase is the phase lag between the voltage and current.

26. Physics 3 June 2002

27. An oscillating voltage of frequency ω, V(t)=V0cos(ωt) is represented in complex notation by V0ejωt. The current in the circuit is then given by I=V/Z, where Z is the complex impedance Z=|Z|ejφ, where |Z| is the ratio of the magnitude of the voltage to the magnitude of the current, and φ is the phase by which the current leads the voltage. The real component of Z is the resistance, and the imaginary component is the reactance where C the capacitance and L is the inductance of the circuit. Complex impedances allow circuits when inductance and capacitance to be analysed using linear circuit theory in the same way as resistance

30. The sum of these peak voltages is not equal to V0 as VR, VC and VL all have different phases. When the complex voltages are summed VR+VC+VL=V0

31. LCR circuit + L R V0 C I(t) overdamped solution critically damped solution underdamped (oscillatory) solution

32. Inverting Amplifier Circuit R2 i R1 v– v+ VIN VIN VOUT – i VOUT v+ v– + R1 R2 Non-Inverting Amplifier Circuit + – Gain is very large (A) Inputs draw no current (ZIN=) Feedback  v+=v– Op-amps