GTU BE03009031 Electrical Circuit Analysis Summer 2026 Solved Paper & Solutions

An ideal source is a source whose output does not change with the load.

• An ideal voltage source gives a fixed terminal voltage $V$ for any current. Its internal resistance is zero.
• An ideal current source gives a fixed current $I$ for any voltage. Its internal resistance is infinite.

Statement: In a linear bilateral network with more than one independent source, the current in any branch (or voltage across any element) is the algebraic sum of the values produced by each source acting alone, with all the other independent sources removed.

How to apply it:

1. Keep one source and replace the others: a voltage source by a short circuit, a current source by an open circuit. Internal resistances stay.
2. Find the required current or voltage due to this source.
3. Repeat for every source and add the results with their signs.

Example: a 10 V source is in series with a 5 $\Omega$ resistor, and a 2 A source is connected across this branch, both pushing current the same way. With 10 V alone the current is $10/5 = 2$ A. With the 2 A source alone it is 2 A in the same direction through the 5 $\Omega$ resistor. Total current $= 2 + 2 = 4$ A.

It holds only for linear circuits, so it cannot be used for power, as power depends on the square of the current.

Check: short $a$ and $b$. The short circuit current is $\dfrac{30}{15} + \dfrac{10}{15} = 2.667$ A.

The time constant $\tau$ is the time in which the transient current or voltage of a first order circuit falls to 36.8% ($1/e$) of its initial value, or reaches 63.2% of its final value.

• For a series RL circuit, $\tau = \dfrac{L}{R}$ seconds.
• For a series RC circuit, $\tau = RC$ seconds.

The response is $i(t) = I_{final} + (I_{initial} - I_{final})e^{-t/\tau}$. After about $5\tau$ the transient is over (99.3% complete). A small $\tau$ means a fast response.

Statement: A load takes the maximum power from a DC source when the load resistance equals the Thevenin (source) resistance.

Derivation: Any linear DC network seen from the load terminals can be replaced by its Thevenin equivalent. Let the source be $V_{th}$ with resistance $R_{th}$, and the load be $R_L$, which we can vary. The current is

The power in the load is

For the maximum, put $\dfrac{dP}{dR_L} = 0$:

The second derivative is negative at this point, so it is a maximum, and not a minimum. Putting $R_L = R_{th}$ in the power equation gives the largest power the load can get:

The efficiency at this point is only 50%, as half of the power is lost in $R_{th}$. So the condition is used where the power is small and the efficiency is not important, such as in communication circuits.

Given: all meshes are taken clockwise. The dependent source is $\tfrac{1}{9}v_x$, where $v_x$ is the voltage across the 3 $\Omega$ resistor.

Check: $v_x = 3(17 - 11) = 18$ V, and $\dfrac{v_x}{9} = 2$ A, which equals $i_3 - i_1 = 2$ A.

Let $V$ and $I$ be the rms values and $\phi$ the phase angle between them.

• Apparent power $S = VI$ in VA.
• Average (real) power $P = VI\cos\phi$ in W.
• Reactive power $Q = VI\sin\phi$ in var.

They form a right angled power triangle, with $P$ along the base, $Q$ upright and $S$ as the hypotenuse:

The power factor is $\cos\phi = P/S$. $P$ is the power turned into work or heat, and $Q$ is the power that swings between the source and the inductor or capacitor.

At $t = 0^+$ an inductor and a capacitor can be replaced by simple sources, because the inductor current and the capacitor voltage cannot change suddenly.

Reason: the current in an inductor cannot jump, so an uncharged inductor carries zero current at once (open circuit). The voltage across a capacitor cannot jump, so an uncharged capacitor has zero voltage at once (short circuit). When the parts are charged, the initial values are kept: the inductor still carries $I_0$ and the capacitor still has $V_0$ across it. After $t = 0^+$ these sources are not valid, and the real $L$ and $C$ are used again.

Check: at $t = 0$ it gives $0.6$ A, and as $t \to \infty$ it gives $0.5$ A.

The h (hybrid) parameters of a two-port network are called hybrid because they mix a voltage and a current as the independent variables. They relate $V_1$ and $I_2$ to the independent variables $I_1$ and $V_2$:

• $h_{11} = V_1/I_1$ with $V_2 = 0$: short circuit input impedance ($\Omega$).
• $h_{12} = V_1/V_2$ with $I_1 = 0$: open circuit reverse voltage gain.
• $h_{21} = I_2/I_1$ with $V_2 = 0$: short circuit forward current gain.
• $h_{22} = I_2/V_2$ with $I_1 = 0$: open circuit output admittance (S).

Let a DC voltage $V$ be applied at $t = 0$ to a series RL circuit, with zero initial current. This is called the step response, as the voltage is a step $Vu(t)$, which is zero before $t = 0$ and $V$ after it. We write the circuit equation, change it to the $s$-domain, find $I(s)$ and then go back to the time domain.

The current rises from zero to $V/R$ with the time constant $L/R$. At $t = 0$ it is zero, as the inductor opposes a sudden change, and after $5L/R$ the inductor behaves like a short circuit and the current is the DC value $V/R$.

Statement: If $f(t)$ and $f'(t)$ are Laplace transformable and $F(s) = \mathcal{L}\{f(t)\}$, then the value of $f(t)$ at $t = 0^+$ is

It lets us find the initial value from $F(s)$ with no inverse transform, which is useful for finding the starting current or voltage in a circuit. It is valid when the limit exists, and when $F(s)$ is a proper fraction (the degree of the numerator is less than that of the denominator).

Proof:

so

Example 2: for a series RL circuit with $I(s) = \dfrac{V}{s(Ls + R)}$, $i(0^+) = \lim\limits_{s \to \infty}\dfrac{V}{Ls + R} = 0$. This agrees with the fact that the inductor current cannot jump.

Example: for $F(s) = \dfrac{5}{s + 2}$, $f(0^+) = \lim\limits_{s \to \infty}\dfrac{5s}{s + 2} = 5$. The inverse is $5e^{-2t}$, which is $5$ at $t = 0$. So the theorem agrees with the direct method.

Where it is used in circuits: to find $i(0^+)$ or $v(0^+)$ just after switching, and to find the value of $di/dt$ at $t = 0^+$ by applying the theorem to $sF(s) - f(0^+)$. The matching theorem for the final value is $\lim\limits_{t \to \infty} f(t) = \lim\limits_{s \to 0} sF(s)$.

The energy stored in an inductor is $\tfrac{1}{2}Li^2$ and in a capacitor $\tfrac{1}{2}Cv^2$. A sudden change of $i$ or $v$ would change the stored energy in zero time, which needs infinite power, as $p = dW/dt$.

• Voltage across an inductor is $v_L = L\dfrac{di}{dt}$. A sudden jump in $i$ gives $\dfrac{di}{dt} \to \infty$, so an infinite voltage.
• Current in a capacitor is $i_C = C\dfrac{dv}{dt}$. A sudden jump in $v$ gives an infinite current.

Real sources cannot give infinite power, so $i_L$ and $v_C$ are continuous.

Statement: A network made of passive, linear and bilateral elements (R, L, C) is reciprocal, and for it $y_{12} = y_{21}$. Reciprocity means that if a voltage at one port gives a certain short circuit current at the other port, the same voltage at the other port gives the same current at the first.

Derivation: The y parameters are $I_1 = y_{11}V_1 + y_{12}V_2$ and $I_2 = y_{21}V_1 + y_{22}V_2$.

Case A: apply voltage $V_a$ at port 1 and short port 2 ($V_2 = 0$). The short circuit current at port 2 is

Case B: apply voltage $V_b$ at port 2 and short port 1 ($V_1 = 0$). The short circuit current at port 1 is

By the reciprocity theorem, the ratio of the response current to the applied voltage is the same when the source and the response are exchanged:

So only three of the four y parameters are independent for a reciprocal network. A network with a dependent source (like a transistor model) is not reciprocal.

ABCD parameters (transmission or chain parameters) relate the sending end quantities $V_1$ and $I_1$ to the receiving end quantities $V_2$ and $-I_2$. The current $I_2$ is taken into port 2, so the receiving end current, going out of the port, is $-I_2$. They are called chain parameters as they are the natural choice when networks are joined one after another.

or in matrix form $\begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix}\begin{bmatrix} V_2 \\ -I_2 \end{bmatrix}$.

Meaning of each parameter:

Properties:

• Reciprocal network: $AD - BC = 1$, so only three of the four parameters are independent.
• Symmetrical network: $A = D$, so the network looks the same from both ends.
• Lossless network: $A$ and $D$ are real, and $B$ and $C$ are purely imaginary.

The parameters are found by the open circuit and short circuit tests described in the table: open port 2 to get $A$ and $C$, short port 2 to get $B$ and $D$.

Cascade connection: for two networks in cascade the overall matrix is the product of the individual matrices, in the order of the connection. This is the main use of ABCD parameters, as in transmission lines.

Why they are useful: they give the sending end voltage and current directly from the load values, so they suit transmission lines and long chains of networks such as filter sections, where the other parameter sets would need a lot more algebra. Any other parameter set (Z, Y, h) can be converted to ABCD.

Example: a single series impedance $Z$ has $A = 1$, $B = Z$, $C = 0$, $D = 1$. A single shunt admittance $Y$ has $A = 1$, $B = 0$, $C = Y$, $D = 1$.