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S3   `̗ ??"    }E 2002 University of North Carolina at Charlotte, ALL RIGHTS RESERVEDFF&dF  "p 0t"   0G3#" `r B T3??"`f  C 0Abottom logo"HB  s *޽h ? 3f3fff̙3ff̙y___PPT10Y+D=' = @B + Capsules  P  B(     64f3#" `p  @ "  <xG0*"pp  @   < "4    W#Click to edit Master subtitle style$ $vF     "  z"  B 6G3#" ` R r   Z3??#" `L  3   ` ??"   }E 2002 University of North Carolina at Charlotte, ALL RIGHTS RESERVEDFF&    `) ??"NnR  T Click to edit Master title style! !f  C 0Abottom logo"\  C &Acrown2"RB  s *޽h ? 3f3fff̙3ff̙y___PPT10Y+D=' = @B + 0,(  , , 0 P    X*  , 003     Z*  , 6L9 _P   X*  , 6= _   Z* H , 0޽h ? 3380___PPT10.jipl @#(  x  c $8) NnR ) x  c $ ) 0p   )   TT ) ?? 0   g)University of North Carolina at Charlotte"*(2*f3H  0޽h ? 3f3fff̙3ff̙y___PPT10Y+D=' = @B +Hz  ((%+((  x  c $\o)@  )  0 # 0e0eA#  ?@ 5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?  H $D  0 )  To) ??@P,$ 0 HKirchoff s voltage law or KVL states the sum of voltages around a closed loop taken in the same direction equals zero volts.4} 2$X4  T(v) ??P ,$ 0 | V = 0 V 2%    Tz) ?? ` ,$ 0 :or 2(  T$) ?? ` ,$ 0 p V = 0 V 2   T) ?? 0 ,$ 0 RV14 2    T) ??0 ,$ 0 RV24 2    T؋) ?? ,$ 0 RV34 2    Tď) ??  ,$ 0 RV44 2   T`) ?? 0  ,$ 0 =+  2   T@) ?? ` ,$ 0 =-  2  T) ??P0 7,$  0 =+  2  T) ??0 ,$ 0 > -  2  T4) ??PP 7,$ 0 =+  2  T) ??P @ ,$ 0 > -  2  TP) ?? P  ,$ 0 =+  2  T<) ?? P @ ,$ 0 : - 2A  T8) ??P ,$" 0 QNote, these voltages can be voltage sources or IR voltage drops across resistors.R 2RB @ # lDp??P,$D# 0   T) ??  ,$ 0 RKVL tells us&  2w  T) ?? @,$ 0 V1+V2-V3-V4 = 0 2      T) ?? p ,$ 0 :or 2w  T) ?? ,$ 0 V4+V3-V2-V1 = 0 2      T@) ??00P,$ 0 G Same equation(2B   fDp??,$D  0B  # lDp??,$D!  0B @  fDp??` ,$D 0B  # lDp??,$D   02   fGH--I#3f??"`P,$@ 0 #  fA* ? ??`p`8 $@  0 $ T8) ??@ ,$ 0 `((Voltages summed in clockwise direction)) 2)  & T) ?? @ < 2 * ( T) ??@],$  0 r0(Voltages summed in counter clockwise direction)&1 20  * T) ??  @-,$ 0 O- +0 2 + T) ??p,$  0 I+ -* 2H  0޽h ? 3f3fff̙3ff̙iQaQ___PPT10AQ..zw+-DYK' = @B DK' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D+' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D8' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =%(DT' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'checkerboard(down)*<3<*$D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D8' =%( DG' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-6B'checkerboard(down)*<3<*#D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =%( DT' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B'checkerboard(down)*<3<*(D' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D|' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dl' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*++0+c ++0+c ++0+c ++0+c ++0+c ++0+ c ++0+ c ++0+ c ++0+ c ++0+ c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+$c ++0+(c ++0+*c ++0++c +*s  $$%,N$(  b   A$  ,$D  0  B=c ?@ L8Kirchoff s Voltage Law (KVL)T  T>c ??@P,$ 0 TNow, lets examine the same loop with a different polarity assigned to its voltage V4,U 2S   TDc ?? ` ,$ 0 :or 2   THc ?? 0 ,$ 0 RV14 2    TMc ??0 ,$ 0 RV24 2    TRc ?? ,$ 0 RV34 2    T(Vc ??  ,$ 0 RV44 2   TZc ?? 0  ,$ 0 =+  2   TOc ?? ` ,$ 0 =-  2  Tac ??P0 7,$ 0 =+  2  T0ec ??0 ,$ 0 > -  2  Tic ??PP 7,$ 0 =+  2  Tlc ??P @ ,$ 0 > -  2  Tpjc ?? P  ,$ 0 =+  2  Ttc ?? P @ ,$ 0 : - 2  Twc ??  ,$ 0 `(Again, KVL tells us&  2w  T0|c ?? @,$ 0 V1+V2-V3+V4 = 0 2      Tc ?? p ,$ 0 :or 2x  T+B#style.visibility<*%(DU-' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-6B'blinds(horizontal)*<3<*"D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-6B'blinds(horizontal)*<3<*#D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =%(DT' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'checkerboard(down)*<3<*$D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-6B'blinds(horizontal)*<3<*'D8' =%( DG' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B'checkerboard(down)*<3<*(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =%( DT' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-6B'checkerboard(down)*<3<**D' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D|' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dl' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*++0+c ++0+c ++0+c ++0+ c ++0+ c ++0+ c ++0+ c ++0+ c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+c ++0+"c ++0+$c ++0+%c ++0+&c ++0+'c ++0+*c ++0++c +b  !!E!(  b   A7P0 ,$@ 0  Z u ??,$  0 XV1: 2   Zu ??p ,$ 0 XV2: 2 Q   Zu ??  @,$  0 )Step 1. Assume a current direction for I1N* 2&   Zpu ?? ,$ 0 TR16 2 1  TP!u ??  & ,$ 0 y7Step 3. Write a KVL equation in the clockwise direction&8 27  Z%u ??$ ,$ 0 TR26 2   B)u ?p j<Kirchoff s Voltage Law Example H   Z.u ??`@C,$ 0 8Given: consider the single-loop circuit pictured below 69 21H   Zp3u ??@,$ 0 8Find: The current flowing through this circuit using KVL69 24   Z<9u ??@s` ,$ 0 YVin: 2    Z5u ??0 S P,$ 0 Q Solution:,  2   J0e0e B0CDE4F@ 5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E? 8xhX 8(P0H @     ,$@   0  ZTCu ??  ,$  0 TI16 2   Z0Hu ??,$ 0 C+& 2  ZMu ??t,$ 0 C-& 2  ZpQu ??@ ,$ 0 C+& 2  ZUu ??0 @P ,$ 0 C-& 2.  TPZu ??  ,$  0 v4Step 2. Define the voltage drop across each resistor&5 24W  Tl^u ??0 ` P ,$ 0 Vin - V1 - V2 = 0r 2     Zfu ?? 4  > 2   Tiu ??` ` ; ,$ 0 Vin = I1R1 + I1R2 2     B   `D?? ` ,$@  0B   fJD?? ` ,$D  0  Ttu ?? ` ,$ 0 Vin = I1(R1 + R2) 2       T u ?? ,$ 0 I1 = Vin /(R1 + R2) 2    S  T,u ??P ` pp ,$ 0  Vin = V1 + V2r 2   H  0޽h ? 3f3fff̙3ff̙@@___PPT10q@.P`-+iD=;' = @B D:' = @BA?%,( < +O%,( < +D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(Dl ' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D(' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D`' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*++0+u ++0+u ++0+ u ++0+ u ++0+u ++0+u ++0+u ++0+ u ++0+ u ++0+ u ++0+ u ++0+u ++0+u ++0+u ++0+u ++0+u ++0+u ++0+u ++0+u ++0+u ++0+u ++0+u ++0+u +"f  ##[#(  `   Zu ??P p,$  0 *Step 1. Assume a current direction for I1.\+ 2& 4  T|u ??P V ,$ 0 |:Step 3. Write the KVL equation in the clockwise direction.&; 2:   Zu ??,$  0 XV1: 2 b   A80 ,$@ 0   Z\u ??@ ,$ 0 XV2: 2 %  Tu ??@ F ,$  0 m5Step 2. Define the voltage drop across each resistor.6 26  Zu ??$ ,$ 0 TR26 2   Zhu ?? ,$ 0 TR16 2   Nu ?@  u n  Tdu ??`@f,$ 0 FGiven: Now, consider the same single-loop circuit with Vin reversed TG 23 H  Zw ??@@`,$ 0 8Find: The current flowing through this circuit using KVL69 24   Zdw ??@s` ,$ 0 UVin6 2    Z@w ??` # ,$ 0 Q Solution:,  2  Zw ??0 P ,$  0 JI1, 2   Zw ??,$ 0 C+& 2  Zw ??t,$ 0 C-& 2  Zxw ??@ ,$ 0 C+& 2  Zw ??0 @P ,$ 0 C-& 2f  T$w ??` ` ] ,$ 0 V2 + V1 + Vin = 0  2     Z /w ??P d p > 2   Z1w ??` `  ,$ 0 Vin = -I1R1 - I1R2 2     B   `D?? ` ,$@  0B   fJD?? ` ,$D  0  T =w ?? ` ,$ 0 Vin = -I1(R1 + R2) 2       Z8Gw ?? ,$ 0 I1 = -Vin /(R1 + R2) 2      TXRw ?? p` X,$ 0 Notice the negative sign appearing in the equation for I1. This means current flows in the direction opposite from our assumption. : 28 K]  TXw ?? `  ,$ 0 Vin = (-V1) + (-V2)v 2      0e0e BC@DE|F4A@ 5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?`08@P0 h@h`P@(P  @@ ( |@@          "`@P ,$@   0H  0޽h ? 3f3fff̙3ff̙AA___PPT10A.Ul+I Dk<' = @B D&<' = @BA?%,( < +O%,( < +D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dl ' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(Dq' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D(' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D`' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+@+0+ w ++0+ w ++0+w ++0+ w ++0+ w ++0+w ++0+w ++0+w ++0+w ++0+w ++0+w ++0+ w ++0+ w ++0+w ++0+w ++0+w ++0+w ++0+w ++0+w ++0+w ++0+w ++0+w ++0+w ++0+w +`  ''$.n'(  b   A7 `p ,$@ 0 %  fA, ? ?? 8 $@ 0b   A7 0 p ,$@ 0r  S Xw@  w   Tw ??p@j ZContinued& *  2 f3  Tw ??@|,$ 0 An important fact to observe when using KVL is that you can sum in either clockwise or counter clockwise directions and the answer will be the same. $ 2  Z4w ??`p,$ 0 XV1: 2   ZTw ??pp ,$ 0 XV2: 2   ZXw ?? ,$  0 TR16 2    Z\w ??p$ ,$  0 TR26 2    Z`w ??s0 ,$  0 YVin: 2     J0e0e B0CDE4F@ 5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E? 8xhX 8(P0H @     ,$@  0   Zw ?? ,$  0 TI16 2    Zw ??,$ 0 C+& 2  Zhw ??t,$ 0 C-& 2  Zw ??@ ,$ 0 C+& 2  Z$w ?? @ ,$ 0 C-& 2  Zw ??`?,$ 0 XV1: 2   Zw ??p ,$ 0 XV2: 2   Z@w ??0d ,$ 0 TR16 2   ZDw ??@P` ,$ 0 TR26 2   ZHw ??p0 ,$ 0 YVin: 2    TLw ??0 P ,$ 0 TI26 2   Zx ??@ ,$ 0 C+& 2  Z@x ?? ,$ 0 C-& 2  Zhx ??,$ 0 C+& 2  Zhx ?? T ,$ 0 C-& 2Z & T x ??@ 00 ` ,$ 0 Vin - V1 - V2 = 0t 2   Y ' T<x ??@  ` ,$ 0 V1 + V2 - Vin = 0t 2    ( T4#x ?? @|,$" 0 It is important to use the same summing direction for each circuit. If you use a consistent sign convention your answers will be consistent regardless of the summing direction. 2" ) TP(x ??` p,$ 0 jNotice I1 & I2@ 2  2 * T-x ?? 0 ,$ 0 zDSumming in these directions gives& # 2# , T@2x ??@ ` `` ,$ 0 Gidentical results(2B -@  `D??  ,$@  0B .  `D?? 0 ,$D!  0H  0޽h ? 3f3fff̙3ff̙*"___PPT10+[lD}' = @B D}' = @BA?%,( < +O%,( < +D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(Dmv' =%(Dh' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*%D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*%D' =-g6B fade*<3<*%DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(DN' =%(Dv' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-u6Bdiamond(in)*<3<*,D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-u6Bwipe(right)*<3<*-D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-s6Bwipe(left)*<3<*.DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(++0+x ++0+x ++0+x ++0+x ++0+ x ++0+ x ++0+ x ++0+ x ++0+x ++0+x ++0+x ++0+x ++0+x ++0+x ++0+x ++0+x ++0+x ++0+x ++0+x ++0+x ++0+x ++0+&x ++0+'x ++0+(x ++0+)x ++0+*x ++0+,x +]  (  x  c $(x@  x r  Tx ??@,$ 0 Kirchoff s voltage law can also be observed in an alternate form.B 2B  THx ??@P0,$ 0 B Such that,  2 B  Tx ??  ,$ 0  Vleft side of loop from bottom to top =  Vright side of loop from bottom to top W(2 !   (+*b   A$ ,$@ 0   Tx ?? ` ,$  0 RV14 2   TTx ??` ,$ 0 RV24 2   Tx ?? ` P ,$  0 =+  2   Tx ?? P w ,$ 0 =-  2   Tx ??` ,$ 0 =+  2   Tx ?? ` P w ,$ 0 > -  2    T@x ??P  ,$  0 RV34 2   T4x ??  ,$ 0 RV44 2  T x ?? @,$  0 =+  2  Ttx ?? pw ,$  0 > -  2  T`x ?? @ ,$ 0 =+  2  T x ??` pG ,$ 0 : - 2B   `Dp?? ,$D  0   *0e0e BCDEF@ p8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?0H@  X ,$D  0   00e0e BCDEF@ p8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?0H@  P ,$D  0  Zx ?? p ,$ 0 WVleft6 2   T,x ??`  ,$ 0 YVright 6 2 q  THx ??0 t,$ 0  V1+V2 = V4+V3 2      Tx ?? p@,$ 0 ;Then, 2  T$x ??pj ZContinued& *  2 f3H  0޽h ? 3f3fff̙3ff̙f=^=___PPT10>=.z\R8+@ DB8' = @B D7' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D` ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*++0+| ++0+| ++0+| ++0+| ++0+| ++0+ | ++0+ | ++0+ | ++0+ | ++0+ | ++0+ | ++0+| ++0+| ++0+| ++0+| ++0+| ++0+| ++0+| ++0+| ++0+| ++0+| ++0+| +{  4),)$%4((  4: 4 T %| ?? ,$ 0 LFirst, assume a current direction and the voltage drop across each resistor.M 2M, %4 T,*| ?? ,$ 0 t>In this example, however, the voltage drop is already assumed.? 2? 40 # 0e0eA9   ?@ 5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?0@ H  $D 0 |x 4 c $(|@  | ' 4 Z$/| ??@ ,$ 0 i!Given: The circuit pictured below," 2p 4 TL4| ??@,$ 0 fFind: The current through this circuit using KVL in its alternate form 6g 2a 4 Z:| ??@@q` ,$ 0 TVs6 2  4 Z?| ??0d0 ,$ 0 TR16 2  4 ZdE| ??@O ` ,$  0 TV46 2   4 ZJ| ??  ,$  0 TR26 2    4 TO| ??  ,$ 0 TV36 2 B  4  fDp??Ppp,$D  0  4 ZT| ??@;`,$ 0 [Vleft: 2 "  4 ZY| ??@`,$ 0 dVrightB 2  B 4  fDp??P @,$@ 0B 4  fDp??P @,$@ 0 4 T8`| ?? ,$ 0 [ Solution:6  2< 4 Ze| ?? @ 5 ,$ 0 ~Vleft = VrightT 2   4 Zm| ??p @  ,$ 0 Vs = V3+V2-V4+V1 2      4 Zw| ?? @ @,$" 0 Vs+V4-V3 = IR2+IR1 2       4 ZT| ??@@ ;`,$# 0 Vs+V4-V3 = I(R2+R1) 2       4 ZH| ??0p `P,$$ 0 I = (Vs+V4-V3 )/ (R2+R1) 2     B 4 ZD??@,$D  0B 4  `D??,$D  0 4 T(| ??,$  0 5I 2 4 T| ??   ,$ 0 V And begin writing your equations! 2! 4 T| ??  ,$  0 HV2* 2  4 T$| ??`  ,$  0 9+ 2 33 4 T | ??  - ,$  0 9- 233 4 TL| ??0,$ 0 HV1* 2  4 T̴| ??P,$ 0 9+ 2 33  4 T|| ?? ,$ 0 9- 233B "4  `D8c??` P ,$D!  0B #4  `D8c?? `P ,$@   0  $4 T| ??  ,$ 0 UThen develop a line of symmetry 2 H 4 0޽h ? 3f3fff̙3ff̙ZRRR___PPT102R+=2YD.L' = @B DK' = @BA?%,( < +O%,( < +DE' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*4D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*4D' =-g6B fade*<3<*4D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-s6Bwipe(down)*<3<*4D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-s6Bwipe(left)*<3<*4D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-s6Bwipe(left)*<3<*4D0' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%4%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$4%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =-s6Bwipe(down)*<3<* 4D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#4%(D' =-o6Bwipe(up)*<3<*#4D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"4%(D' =-o6Bwipe(up)*<3<*"4D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(++0+4| ++0+4| ++0+%4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+ 4| ++0+ 4| ++0+ 4| ++0+ 4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+4| ++0+ 4| ++0+$4| +H  0$($P )#(    T} ?? @,$& 0 rNotice with voltage division, it s the resistor you want to find the voltage across divided by the sum of resistors in series, regardless of the number of resistors there are in series.T 2',= $  fA= 4? ??^` 8 4$D 0   fAL "? ??  8 "$D% 0x  c $}@  } o  T} ??N@T,$ 0 Voltage division provides a relationship for dividing voltage among series resistors in direct proportion to their resistances. 2  Z"} ?? ",$" 0 ZVoltage Division states,& 2  T'} ??p`,$ 0 `(For example, consider the circuit below:) 2)  Z+} ??n?,$ 0 TV16 2    T(1} ?? ` ,$  0 XV2: 2    Z5} ??0d ,$ 0 XR1: 2    Zp;} ??  ,$ 0 XR2: 2    ZA} ??p5,$  0 E+( 2   ZPE} ??P,$  0 E-( 2  T>} ?? U  ,$  0 E+( 2  T9} ?? 4 ,$  0 E-( 2   ZQ} ??~ @ ,$ 0 KVin, 2   ZV} ?? . ,$$ 0 Dor& 22   ` ?? P ,$D'  02   ` ??N n ,$D(  0   fAK !? ?? p 8 !$D# 0t  T\} ??~ ,$ 0 >Using KVL, we can derive a voltage relationship for V1 and V2.b? 25  U  Zd} ??^ q~ ,$ 0  Vin=V1+V2r  2     Zm} ??n  ,$ 0 Vin=IinR1+IinR2 2     |  Zx} ??  ,$ 0 Vin=Iin(R1+R2) 2      Z} ?? .,$ 0 Iin = Vin/(R1+R2) (1) 2    B    `D??~ n ,$@  0B !  `D??~ `n ,$D  0 " Z\} ??^ ~,$ 0 Iin=V1/R1 = V2/R2 (2) 2     \ # TT} ??N p ,$ 0 lSubstituting equ.2 into equ.1, gives rise to a fundamental voltage relationship known as voltage division. m 2mB ( ZD??Pp,$@  0 ) T} ??> p^,$ 0 KIin,(2 H  0޽h ? 3f3fff̙3ff̙^^___PPT10^+dD$X' = @B DW' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*$D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*$D' =-g6B fade*<3<*$D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D(' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-s6Bwipe(left)*<3<*(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(Do' =%(D&' =%(Dc' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* Dc' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bwipe(up)*<3<*!D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*#%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*"%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*!%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(DG' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'checkerboard(down)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DG' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'checkerboard(down)*<3<*D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*Dv' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+X+0+} ++0+} ++0+} ++0+} ++0+} ++0+ } ++0+ } ++0+ } ++0+ } ++0+ } ++0+} ++0+} ++0+} ++0+} ++0+} ++0+} ++0+} ++0+} ++0+} ++0+} ++0+} ++0+} ++0+} ++0+} ++0+"} ++0+"} ++0+#} ++0+#} ++0+)} +g xGg$'h ujT+-ԝRPRoBǙݲlvg7;dμ{%Ǘ{Y[n0hLl,J+47 ?_R|$L.3p>2$޿Nq%\WU[ 4V)kڸNuq!n&)n%n[6-n#;.+'{tp <g!x(x$ģh<gl<qx<'Ix2ix:gYx6yx>^\/KR /+p^Wx kq^|<>O'S𩸰W0wY_%;Zv7}w5>6yE11oh~ܚ?c:~)<sP~2U# 43?p̦ͯq,W8|݅W|5ZW;x=_slcu%&>9y x[v? 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