Kirchhoffs Rules Kirchhoffs Voltage Rule sign convention Use only - PDF document

Kirchhoff’s Rules

Kirchhoff’s Voltage Rule – sign convention Use only currents as unknown variables.  V can always be written in 1. terms of currents or derivatives or integrals of currents. 2. Assign current direction to every path in a circuit. Apply Kirchhoff’s current rule as much as you can to reduced the number of unknown variables. 3. Across every component in the circuit determine which end has a higher potential (mark it with a + sign) and which end has a lower potential (mark it with a - sign). This will depend on the current direction you assume in step 2. 4. Pick up a loop and travel around it either clockwise or anticlockwise. If you travel from – to + across a component, then  V across the component is positive. If you travel from + to – across a component, then  V across the component is negative. You can reverse this convention as long as you do it consistently for the whole loop. 5. If you get a negative current, that means the current direction you assume in step 2 is wrong and you should reverse that direction.

Kirchhoff’s Voltage Rule – sign convention Example: C + - I Q +       + IR 0   R C  -   V V - R bat  V C As you travel around a loop, if you find yourself moving from + to - , make  V across that component negative (C and R in this example).

Class 25: RC Circuits

RC Circuits – Charging At t=0, capacitance is uncharged and C Q=0 (initial condition). At t=0, switched is closed, it the capacitor has no charge, it behaves like a conductor and I=  /R.  R After the capacitor is completely charged, Q=C  ,  V C =  and  V R =0. I=0 and the capacitors behave like an insulator.

RC Circuits – Charging q q d q        IR R C C d t C    CR dq (C - q) dt dq 1   Integration constant - dt  q - C CR t      n( q - C ) - K'  CR R t -     K' CR q - C Ke (K e ) t -     q C K e CR          At t 0, q 0 0 C K K - C t -     q C ( 1 e CR )   t t dq C - -    I e CR e CR d t CR R t -     V IR e CR R  V R +  V C =  t q -     CR V (1 - e ) C C

RC time constant  =RC is known as the RC time constant. It indicates the response time (how fast you can charge up the capacitor) of the RC circuit.  t -  CR I e R   I  e 2.72 R -  1 e 0 . 37   - 1 I e R  ~ 0 . 37 R t t=RC t -    CR q C ( 1 e ) q   C  2 1.414    -1 q ( 1 e ) C  1 ~ 0 . 63 C  0 . 707 2 t Nothing to do with RC circuits t=RC

RC Circuits – Discharging At t=0, capacitance is charged with a charge Q (initial condition). At t=0, switched is closed, the capacitor starts to discharge. C R After the capacitor is completely discharged, Q=0,  V C = 0,  V R =0 and I=0.

RC Circuits – Discharging q q d q d q       0 IR R 0 (I - ) C C d t d t   CR dq - q dt dq 1   Integration constant - dt q CR t    C  n q - K' R CR t -    K' q Ke CR (K e ) t -   q K e CR     At t 0, q Q Q K t -   CR q Qe t dq Q -    CR I e d t RC t Q -     CR V IR e R C  V R +  V C = 0 t q Q -    V e CR C C C

In Summary For both charge and discharge, Q, I,  V C , and  V R must be one of the following two cases: y y 0 t - y  RC y 0 e t y y  t -  RC y y (1 - e )  y can be Q, I,  V C , or  V R t